diff --git "a/2tFLT4oBgHgl3EQfrC9d/content/tmp_files/load_file.txt" "b/2tFLT4oBgHgl3EQfrC9d/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/2tFLT4oBgHgl3EQfrC9d/content/tmp_files/load_file.txt" @@ -0,0 +1,789 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf,len=788 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='12142v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='DG] 28 Jan 2023 THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS HUI ZHANG AND ZAILI YAN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' We consider the moment map m : PVn → iu(n) for the action of GL(n) on Vn = ⊗2(Cn)∗ ⊗ Cn, and study the critical points of the functional Fn = ∥m∥2 : PVn → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Firstly, we prove that [µ] ∈ PVn is a critical point if and only if Mµ = cµI + Dµ for some cµ ∈ R and Dµ ∈ Der(µ), where m([µ]) = Mµ ∥µ∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then we show that any algebra µ admits a Nikolayevsky derivation φµ which is unique up to automorphism, and if moreover, [µ] is a critical point of Fn, then φµ = − 1 cµ Dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Secondly, we characterize the maxima and minima of the functional Fn : An → R, where An denotes the projectivization of the algebraic varieties of all n- dimensional associative algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Furthermore, for an arbitrary critical point [µ] of Fn : An → R, we also obtain a description of the algebraic structure of [µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Finally, we classify the critical points of Fn : An → R for n = 2, 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Introduction Lauret has studied the moment map for the variety of Lie algebras and obtained many remarkable results in [7], which turned out to be very important in proving that every Einstein solvmanifold is standard ([9]) and in the characterization of solitons ([1, 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Apart from the Lie algebras, the study of the moment map in other classes of algebras was also initiated by Lauret, see [11] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Motivated by this, the authors have recently extended the study of the moment map to the variety of 3-Lie algebras (see [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In this paper, we study the moment map for the variety of associative algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let GL(n) be the complex reductive Lie group acting naturally on the complex vector space Vn = ⊗2(Cn)∗ ⊗ Cn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=', the space of all n-dimensional complex algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The usual Hermitian inner product on Cn naturally induces an U(n)-invariant Hermitian inner product on Vn, which is denoted by ⟨·, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since gl(n) = u(n) + iu(n), we may define a function as follows m : PVn → iu(n), (m([µ]), A) = (dρµ)eA ∥µ∥2 , 0 � µ ∈ Vn, A ∈ iu(n), where (·, ·) is an Ad(U(n))-invariant real inner product on iu(n), and ρµ : GL(n) → R is defined by ρµ(g) = ⟨g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The function m is the moment map from symplectic geometry, corresponding to the Hamiltonian action U(n) of Vn on the symplectic manifold PVn (see [4, 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In this paper, we study the critical points of the functional Fn = ∥m∥2 : PVn → R, with an emphasis on the critical points that lie in the projectivization of the algebraic variety of all n-dimensional associative algebras An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 14L30, 17B30, 53D20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moment map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Variety of associative algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This work is supported by NSFC (Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 11701300, 11626134) and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Wong Magna Fund in Ningbo University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1 2 HUI ZHANG AND ZAILI YAN The paper is organized as follows: In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, we recall some basic concepts and results of complex associative algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 3, we first give the explicit expression of the moment map m : PVn → iu(n) in terms of Mµ, that is, m([µ]) = Mµ ∥µ∥2 for any [µ] ∈ PVn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then we show that [µ] ∈ PVn is a critical point of Fn if and only if Mµ = cµI + Dµ for some cµ ∈ R and Dµ ∈ Der(µ) (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4, we first show that any algebra µ ∈ Vn admits a Nikolayevsky derivation φµ which is unique up to automorphism, the eigenvalues of φµ are necessarily rational, and moreover, φµ = − 1 cµ Dµ if [µ] is a critical point of Fn (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then we study the extremal points of Fn : An → R, proving that the minimum value is attained at semisimple associative algebras (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6), and the maximum value at the direct sum of a two-dimensional commutative associative algebra with the trivial algebra (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In the context of Lie algebras ([7]), Lauret proved that any µ for which there exists [λ] ∈ GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] such that all eigenvalues of Mλ are negative, must be semisimple, and we prove that this result also holds for associative algebras (Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Besides, the structure for an arbitrary critical point [µ] of Fn : An → R is discussed (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='10 and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 5, we classify the critical points of Fn : An → R for n = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It shows that every two- dimensional associative algebra is isomorphic to a critical point of F2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' and there exists only one three- dimensional associative algebra which is not isomorphic to any critical point of F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Finally, based on the discussion in previous sections, we collect some natural and interesting questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Preliminaries In this section, we recall some basic definitions and results of associative algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The ambient field is always assumed to be the complex number field C unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' A vector space A over C with a bilinear operation A × A → A, denoted by (x, y) �→ xy, is called an associative algebra, if x(yz) = (xy)z for all x, y, z ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' A derivation of an associative algebra A is a linear transformation D : A → A satisfying D(xy) = (Dx)y + x(Dy), for x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It is easy to see that the set of all derivations of A is a Lie algebra, which is denoted by Der(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' A vector subspace I of A is called an ideal if AI, IA ⊂ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 3 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let A be an associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The center of A is defined by C(A) = {x ∈ A : xy = yx, ∀y ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The annihilator of A is defined by ann(A) = {x ∈ A : xy = yx = 0, ∀y ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Clearly, C(A) is a subalgebra of A, and ann(A) is an ideal of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let I be an ideal of an associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then I is called nilpotent, if Ik = 0 for some integer k ≥ 1, where Ik = I· · ·I· · ·I ���������� k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' If I, J are any two nilpotent ideals of an associative algebra A, then I + J is also a nilpotent ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So the maximum nilpotent ideal of A is unique, which is called the radical and denoted by N(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that N(A) coincides with the Jacobson radical of A since A is an associative algebra over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, N(A) = {x ∈ A : xy, yx are nilpotent elements for any y ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let A be an associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' If A has no ideals except itself and 0, we call A simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Denote by Mn(C) the set of all n × n complex square matrices, which is clearly an associative algebra with respect to the usual matrix addition and multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In fact, Mn(C) is a simple associative algebra for any n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, it follows from Wedderburn-Artin theorem that any finite-dimensional simple associative algebra over C is isomorphic to Mn(C) for some integer n ≥ 1 ([15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' An associative algebra A is called semisimple if its radical N(A) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The following theorem is well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6 ([15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' An associative algebra over C is semisimple if and only if it is a direct sum of simple ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' That is, a semisimple associative algebra is isomorphic to Mn1(C) × Mn2(C) × · · · × Mns(C) for some positive integers n1, n2, · · · , ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The moment map for complex algebras Let Cn be the n-dimensional complex vector space and Vn = ⊗2(Cn)∗ ⊗ Cn be the space of all complex n-dimensional algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The natural action of GL(n) = GL(Cn) on Vn is given by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ(X, Y) = gµ(g−1X, g−1Y), g ∈ GL(n), X, Y ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1) Clearly, GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ is precisely the isomorphism class of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that lim t→∞ gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ = 0, gt = tI ⊂ GL(n), t > 0, we see that 0 lies in the boundary of the orbit GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ for each µ ∈ Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By differentiating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1), we obtain the natural action gl(n) on Vn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ(X, Y) = Aµ(X, Y) − µ(AX, Y) − µ(X, AY), A ∈ gl(n), µ ∈ Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2) 4 HUI ZHANG AND ZAILI YAN It follows that A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ = 0 if and only if A ∈ Der(µ), where Der(µ) denotes the derivation algebra of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that the usual Hermitian inner product on Cn gives an U(n)-invariant Hermitian inner product on Vn as follows ⟨µ, λ⟩ = � i, j,k ⟨µ(Xi, X j), Xk⟩⟨λ(Xi, X j), Xk⟩, µ, λ ∈ Vn, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3) where {X1, X2, · · · , Xn} is an arbitrary orthonormal basis of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let u(n) denote the Lie algebra of U(n), then it is easy to see that gl(n) = u(n) + iu(n) decomposes into skew-Hermitian and Hermitian transfor- mations of Vn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, there is an Ad(U(n))-invariant Hermitian inner product on gl(n) given by (A, B) = tr AB∗, A, B ∈ gl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4) The moment map from symplectic geometry, corresponding to the Hamiltonian action of U(n) on the symplectic manifold PVn, is defined as follows m : PVn → iu(n), (m([µ]), A) = (dρµ)eA ∥µ∥2 , 0 � µ ∈ Vn, A ∈ iu(n), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='5) where ρµ : GL(n) → R is given by ρµ(g) = ⟨g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Clearly, (dρµ)eA = ⟨A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, µ⟩ + ⟨µ, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩ = 2⟨A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, µ⟩ for any A ∈ iu(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The square norm of the moment map is denoted by Fn : PVn → R, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6) where Fn([µ]) = ∥m([µ])∥2 = (m([µ]), m([µ])) for any [µ] ∈ PVn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In order to express the moment map m explicitly, we define Mµ ∈ iu(n) as follows Mµ = 2 � i Lµ Xi(Lµ Xi)∗ − 2 � i (Lµ Xi)∗Lµ Xi − 2 � i (Rµ Xi)∗Rµ Xi, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='7) where the left and right multiplication Lµ X, Rµ X : Cn → Cn by X of the algebra µ, are given by Lµ X(Y) = µ(X, Y) and Rµ X(Y) = µ(Y, X) for all Y ∈ Cn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It is not hard to prove that ⟨MµX, Y⟩ =2 � i, j ⟨µ(Xi, X j), X⟩⟨µ(Xi, X j), Y⟩ − 2 � i, j ⟨µ(Xi, X), X j⟩⟨µ(Xi, Y), X j⟩ − 2 � i, j ⟨µ(X, Xi), X j⟩⟨µ(Y, Xi), X j⟩ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='8) for X, Y ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that if the algebra µ is commutative or anticommutative, then the second and third term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='8) are the same, and in this case, Mµ coincides with [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any µ ∈ Vn, we have (Mµ, A) = 2⟨µ, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩, ∀A ∈ gl(n) = u(n) + iu(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In particular, m([µ]) = Mµ ∥µ∥2 for any 0 � µ ∈ Vn THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any A ∈ gl(n), we have (A, Mµ) = tr AM∗ µ = tr AMµ = tr MµA, and tr MµA = 2 tr � i Lµ Xi(Lµ Xi)∗A �������������������������������������� I − 2 tr � i ((Lµ Xi)∗Lµ Xi + (Rµ Xi)∗Rµ Xi)A �������������������������������������������������������������������������� II =: I − II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that I =2 � i tr Lµ Xi(Lµ Xi)∗A =2 � i tr(Lµ Xi)∗ALµ Xi =2 � i, j ⟨(Lµ Xi)∗ALµ Xi(X j), X j⟩ =2 � i, j ⟨Aµ(Xi, X j), µ(Xi, X j)⟩, and II =2 tr � i ((Lµ Xi)∗Lµ Xi + (Rµ Xi)∗Rµ Xi)A =2 � i, j ⟨((Lµ Xi)∗Lµ Xi + (Rµ Xi)∗Rµ Xi)AX j, X j⟩ =2 � i, j ⟨µ(Xi, AX j), µ(Xi, X j)⟩ − 2 � i, j ⟨µ(AX j, Xi), µ(X j, Xi)⟩ =2 � i, j ⟨µ(AXi, X j) + µ(Xi, AX j), µ(Xi, X j)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2), it follows that (A, Mµ) = tr MµA = 2⟨A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, µ⟩, so (Mµ, A) = 2⟨µ, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩ for any A ∈ gl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This proves the first statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For A ∈ iu(n), we have ⟨A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, µ⟩ = ⟨µ, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='5), we conclude that m([µ]) = Mµ ∥µ∥2 for any 0 � µ ∈ Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This completes proof Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1 □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any µ ∈ Vn, then (i) tr MµD = 0 for any D ∈ Der(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (ii) tr Mµ[A, A∗] ≥ 0 for any A ∈ Der(µ), and equality holds if and only if A∗ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For (i), it follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1 that tr MµD = 2⟨D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, µ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For (ii), it follows from that tr Mµ[A, A∗] = 2⟨[A, A∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, µ⟩ = 2⟨A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩ ≥ 0, ∀A ∈ Der(µ), and the fact A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ = 0 if and only if A∗ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The moment map m : PVn → iu(n), the functional square norm of the moment map Fn = ∥m∥2 : PVn → R and the gradient of Fn are, respectively, given by Fn([µ]) = tr M2 µ ∥µ∥4 , grad(Fn)[µ] = 8π∗(Mµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ ∥µ∥4 , [µ] ∈ PVn, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='9) 6 HUI ZHANG AND ZAILI YAN where π∗ denotes the derivative of π : Vn\\{0} → PVn, the canonical projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, the following statements are equivalent: (i) [µ] ∈ PVn is a critical point of Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (ii) [µ] ∈ PVn is a critical point of Fn|GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (iii) Mµ = cµI + Dµ for some cµ ∈ R and Dµ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1, we have Fn([µ]) = tr M2 µ ∥µ∥4 for any [µ] ∈ PVn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' To prove the second one, we only need to compute the gradient of Fn : Vn \\ {0} → R, Fn(µ) = tr M2 µ ∥µ∥4 , and then to project it via π∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' If µ, λ ∈ Vn with µ � 0, then Re⟨grad(Fn)µ, λ⟩ = d dt �����t=0 Fn(µ + tλ) = d dt �����t=0 1 ∥µ + tλ∥4 (Mµ+tλ, Mµ+tλ) = − 4 Re⟨Fn(µ) ∥µ∥2 µ, λ⟩ + 2 ∥µ∥4 ( d dt �����t=0 Mµ+tλ, Mµ) We claim that ( d dt ���t=0 Mµ+tλ, A) = 4 Re⟨A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, λ⟩ for any A ∈ iu(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Indeed, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1, ( d dt ���t=0 Mµ+tλ, A) = d dt ���t=0 (Mµ+tλ, A) = 2 d dt ���t=0 ⟨µ + tλ, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (µ + tλ)⟩ = 2⟨λ, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩ + 2⟨µ, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='λ⟩ = 4 Re⟨A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ, λ⟩ for any A ∈ iu(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows that grad(Fn)µ = −4 Fn(µ) ∥µ∥2 µ + 8(Mµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ ∥µ∥4 , and consequentely grad(Fn)[µ] = 8π∗(Mµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ ∥µ∥4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Thus the first part of the theorem is proved, and the following is to prove the equivalence among the statements (i), (ii) and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (i) ⇔ (ii) : The equivalence follows from that grad(Fn) is tangent to the GL(n)-orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Indeed grad(Fn)[µ] = 8π∗(Mµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ ∥µ∥4 = 8 ∥µ∥4 π∗( d dt �����t=0 etMµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ) = 8 ∥µ∥4 d dt �����t=0 etMµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] ∈ T[µ](GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (iii) ⇒ (i) : By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2), we know that I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ = −µ, and (Mµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ = (cµI + Dµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ = −cµµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows that grad(Fn)[µ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (i) ⇒ (iii) : Since grad(Fn)[µ] = 0, then (Mµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ ∈ ker π∗µ = Cµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So Mµ = cI + D for some c ∈ C and D ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Clearly [D, D∗] = [Mµ − cI, Mµ − ¯cI] = 0, we conclude by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2 that D∗ is also a derivation of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In particular, (c − ¯c)I = (D∗ − D) ∈ Der(µ), thus c = ¯c ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let [µ] be a critical point of Fn and [λ] be a critical point of Fm, then [µ ⊕ cλ] is a critical point of Fn+m for a suitable c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Indeed, assume that Mµ = cµI + Dµ for some cµ ∈ R, Dµ ∈ Der(µ), and Mλ = cλI +Dλ for some cλ ∈ R, Dλ ∈ Der(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Noting that Mtλ = |t|2Mλ for any t ∈ C, we can choose t0 such that cµ = |t0|2cλ, then it follows that [µ ⊕ t0λ] is a critical point of Fn+m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In the frame of algebras, a remarkable result due to Ness can be stated as follows Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='5 ([12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' If [µ] is a critical point of the functional Fn : PVn → R, then THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 7 (i) Fn|GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] attains its minimum value at [µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (ii) [λ] ∈ GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] is a critical point of Fn if and only if [λ] ∈ U(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In fact, the above theorem implies that up to U(n)-orbit, GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] contains at most one critical point for each [µ] ∈ PVn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let [µ] ∈ PVn be a critical point of Fn with Mµ = cµI+Dµ for some cµ ∈ R and Dµ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then we have (i) cµ = tr M2 µ tr Mµ = − 1 2 tr M2 µ ∥µ∥2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (ii) If tr Dµ � 0, then cµ = − tr D2 µ tr Dµ and tr Dµ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since Mµ = cµI + Dµ, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2 we have tr Mµ = (Mµ, I) = 2⟨µ, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='µ⟩ = −2∥µ∥2 < 0, tr M2 µ = tr Mµ(cµI + Dµ) = cµ tr Mµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So cµ = tr M2 µ tr Mµ = − 1 2 tr M2 µ ∥µ∥2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' If tr Dµ � 0, then 0 = tr MµDµ = cµ tr Dµ + tr D2 µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So cµ = − tr D2 µ tr Dµ and tr Dµ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In fact, tr Dµ = 0 if and only if Dµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Indeed, it follows from that 0 = tr MµDµ = cµ tr Dµ + tr D2 µ and Dµ is Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The critical points of the variety of associative algebras The space An of all n-dimensional associative algebras is an algebraic set since it is given by polyno- mial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Denote by An the projective algebraic variety obtained by projectivization of An .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that An is GL(n)-invariant, then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3, the critical points of Fn : An → R are precisely the critical points of Fn : PVn → R that lie in An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The Nikolayevsky derivation and the rationality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' A derivation of φ of an algebra (µ, Cn) is called a Nikolayevsky derivation, if it is semisimple with all eigenvalues real, and tr φψ = tr ψ for any ψ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This notion is motivated by [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let (µ, Cn) be an arbitrary algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then (1) (µ, Cn) admits a Nikolayevsky derivation φµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (2) The Nikolayevsky derivation φµ is determined up to automorphism of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (3) All eigenvalues of φµ are rational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 8 HUI ZHANG AND ZAILI YAN If moreover, [µ] is a critical point of Fn : PVn → R with Mµ = cµI+Dµ for some cµ ∈ R and Dµ ∈ Der(µ), then − 1 cµ Dµ is the Nikolayevsky derivation of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (1) The complex Lie algebra Der(µ) is algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let Der(µ) = s ⊕ t ⊕ n be its Levi-Mal’cev decomposition, where s is semisimple, t ⊕ n is the radical of Der(µ), n is the set of all nilpotent elements in t ⊕ n (and is the nilradical of t ⊕ n), t is an abelian subalgebra consisting of semisimple elements, and [s, t] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Define the bilinear form B on Der(µ) by B(ψ1, ψ2) := tr ψ1ψ2, ∀ψ1, ψ2 ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then, in general, B is degenerate, and Ker B = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since s is semisimple, then B(s, t) = B([s, s], t) = B(s, [s, t]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Clearly, B is nondegenerate on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since t is reductive, we have t = a + ia, where a consists of semisimple elements with all eigenvalues real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows that there exists a unique element φ ∈ a such that B(φ, ψ) = tr ψ for any ψ ∈ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Thus tr φψ = tr ψ for any ψ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (2) The subalgebra s ⊕ t is a maximal fully reducible subalgebra of Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since the maximal fully reducible subalgebras of Der(µ) are conjugate by inner automorphism of Der(µ) (which corresponds to an automorphism of µ), and then the center t of s ⊕ t, is defined uniquely, up to automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So the Nikolayevsky derivation is determined up to automorphism of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (3) The case φµ = 0 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In the following, we assume that φµ is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that φµ is simisimple with all eigenvalues real, we have the following decomposition Cn = l1 ⊕ l2 ⊕ · · · ⊕ lr, where li = {X ∈ Cn|φµX = ciX} are eigenspaces of φµ corresponding to eigenvalues c1 < c2 < · · · < cr ∈ R, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Set di = dim li ∈ N, 1 ≤ i ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since φµ is a derivation, we have the following relations µ(li, lj) ⊂ lk if ci + cj = ck, for all 1 ≤ i, j, k ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Conversely, if we define a linear transformation ψ : Cn → Cn by ψ|li = aiIdli, where a1, a2, · · · , ar ∈ R satisfy ai + aj = ak for all 1 ≤ i, j, k ≤ r such that ci + cj = ck, then ψ is a derivation of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Clearly, all such derivations form a real vector space, which can be identified with W := {(w1, w2, · · · , wr) ∈ Rr|wi + w j = wk if ci + cj = ck}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' We endow Rr with the usual inner product, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' ⟨x, y⟩ = � i xiyi, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1) for any x = (x1, x2, · · · , xr), y = (y1, y2, · · · , yr) ∈ Rr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any derivation ψ ∈ W, we have tr(φµ − I)ψ = tr φµψ − tr ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 9 Then we see that (d1(c1 − 1), d2(c2 − 1), · · · , dr(cr − 1)) ⊥ W relative to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Put F := W⊥, then by definition we have F = span1≤i, j,k≤r{ei + ej − ek : ci + cj = ck}, where ei belongs to Rr having 1 in the i-th position and 0 elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let {ei1 +ej1 −ek1, · · · , eis +ejs −eks} be a basis of F, then (d1(c1 − 1), d2(c2 − 1), · · · , dr(cr − 1)) = s � p=1 bp(eip + ejp − ekp), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2) for some b1, b2, · · · , bs ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Put E = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed ei1 + ej1 − ek1 ei2 + ej2 − ek2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' eis + ejs − eks \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ∈ Zs×r, then EET ∈ GL(s, Z), and (EET)−1 ∈ GL(s, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2) and the definition of E, we have \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed d1(c1 − 1) d2(c2 − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' dr(cr − 1) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 r×1 = ET \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed b1 b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' bs \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s×1 , E \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed c1 c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' cr \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 r×1 = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 0 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s×1 , E \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 r×1 = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s×1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By the left multiplication of E on (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2), we have \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 0 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s×1 − \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s×1 = ED−1ET \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed b1 b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' bs \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s×1 , where D = diag(d1, d2, · · · , dr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It is easy to see that (ED−1ET) ∈ GL(s, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Consequently D \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed c1 − 1 c2 − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' cr − 1 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 r×1 = −ET(ED−1ET)−1 \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s×1 , and \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed c1 c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' cr \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 r×1 = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 r×1 − D−1ET(ED−1ET)−1 \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s×1 ∈ Qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So all eigenvalues of φµ are rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For the last statement, by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2 we know that 0 = tr Mµψ = cµ tr ψ+tr Dµψ for any ψ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since Dµ is Hermitian, we conclude that − 1 cµ Dµ is the Nikolayevsky derivation of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1, it is easy to obtain the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 10 HUI ZHANG AND ZAILI YAN Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let [µ] ∈ PVn be a critical point of Fn : PVn → R with Mµ = cµI + Dµ for some cµ ∈ R and Dµ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then there exists a constant c > 0 such that the eigenvalues of cDµ are integers prime to each other, say k1 < k2 < · · · < kr ∈ Z with multiplicities d1, d2, · · · , dr ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The data set (k1 < k2 < · · · < kr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' d1, d2, · · · , dr) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2 is called the type of the critical point [µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let [µ] ∈ PVn be a critical point of Fn with type α = (k1 < k2 < · · · < kr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' d1, d2, · · · , dr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then we have (i) If α = (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' n), then Fn([µ]) = 4 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (ii) If α � (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' n), then Fn([µ]) = 4 � n − (k1d1+k2d2+···+krdr)2 k2 1d1+k2 2d2+···+k2r dr �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' We suppose that Mµ = cµI + Dµ, ∥µ∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since tr Mµ = −2⟨µ, µ⟩ = −2, then tr M2 µ = tr Mµ(cµI + Dµ) = cµ tr Mµ = −2cµ, and Fn([µ]) = tr Mµ2 ∥µ∥4 = tr Mµ2 = −2cµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For (i), we have Dµ = 0, so Mµ = cµI and cµn = tr Mµ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Thus cµ = − 2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Fn([µ]) = −2cµ = 4 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For (ii), we have Dµ � 0, and cµ = − tr D2 µ tr Dµ by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6 and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that Fn([µ]) = tr Mµ2 = tr(cµI + Dµ)2 = c2 µn + cµ tr Dµ = 1 4Fn([µ])2n − 1 2Fn([µ]) tr Dµ, so we have 1 Fn([µ]) = 1 4n − 1 2Fn([µ]) tr(Dµ) = 1 4n + 1 4cµ tr Dµ = 1 4 \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8edn − (tr Dµ)2 tr D2µ \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows that Fn([µ]) = 4 � n − (k1d1+k2d2+···+krdr)2 k2 1d1+k2 2d2+···+k2r dr �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The minima of Fn : An → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Assume [µ] ∈ PVn, then [µ] is a critical point of Fn : PVn → R with type (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' n) if and only if Fn([µ]) = 4 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, 4 n is the minimum value of Fn : PVn → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any 0 � µ ∈ Vn, we use x1, x2, · · · , xn ∈ R denote the eigenvalues of Mµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that tr Mµ = −2∥µ∥2, then we have Fn([µ]) = tr Mµ2 ∥µ∥4 = 4 tr Mµ2 (tr Mµ)2 = 4 (x2 1 + x2 2 + · · · + x2 n) (x1 + x2 + · · · + xn)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It is easy to see that Fn([µ]) ≥ 4 n with equality holds if and only if x1 = x2 = · · · = xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So [µ] is a critical point of Fn : PVn → R with type (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' n) if only if Mµ is a constant multiple of I, if and only Fn attains its minimum value 4 n at [µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 11 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The functional Fn : An → R attains its minimum value at a point [λ] ∈ GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] if and only if µ is a semisimple associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In such a case, Fn([λ]) = 4 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Consider the simple associative algebra Mm(C) for an integer m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' We endow Mm(C) with the following Hermitian inner product ⟨A, B⟩ := tr AB∗, A, B ∈ Mm(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3) Then {Eij : 1 ≤ i, j ≤ m} is an orthonormal basis, where Eij denote the matrices having 1 in the (i, j)- position and 0 elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Set ν := (Mm(C), ⟨·, ·⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Clearly (Lν A)∗ = LA∗, (Rν A)∗ = RA∗ for any A ∈ Mm(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Thus by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='7), we have Mν = 2 � ij Lν Eij(Lν Eij)∗ − 2 � ij (Lν Eij)∗Lν Eij − 2 � ij (Rν Eij)∗Rν Eij = 2 � ij Lν EijLν Eji − 2 � ij Lν EjiLν Eij − 2 � ij Rν EjiRν Eij = 2 � ij Lν EijEji − 2 � ij Lν EjiEij − 2 � ij Rν EijEji = 2m � i Lν Eii − 2m � i Lν Eii − 2m � i Rν Eii = 2mLν I − 2mLν I − 2mRν I = 2mIm2 − 2mIm2 − 2mIm2 = −2mIm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So [ν] is a critical point of type (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since µ is a complex semisimple associative algebra, by Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6, µ is isomorphic to Mn1(C) × Mn2(C) × · · · × Mns(C) for some positive integers n1, n2, · · · , ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows from Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4 that there exists a point [λ] ∈ GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] such that [λ] is a critical point of type (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So the functional Fn : An → R attains its minimum value at [λ], and Fn([λ]) = 4 n by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Conversely, assume that Fn : An → R attains its minimum value at a point [λ] ∈ GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='[µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The first part of the proof implies that Mλ = cλI with cλ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' To prove µ is semisimple, it suffices to show that L = (λ, Cn) is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Consider the following orthogonal decompositions: (i) L = H ⊕ N, where N is the radical of λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (ii) N = V ⊕ Z, where Z = {A ∈ N : λ(A, N) = λ(N, A) = 0} is the annihilator of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Clearly, Z is an ideal of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' We have L = H ⊕ V ⊕ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Suppose that Z � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let {Hi}, {Vi}, {Zi} be an orthonormal basis of H, V, and Z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Put {Xi} = {Hi} ∪ {Vi} ∪ {Zi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any 0 � Z ∈ Z, by 12 HUI ZHANG AND ZAILI YAN hypothesis we have 0 > ⟨MλZ, Z⟩ =2 � ij |⟨λ(Xi, X j), Z⟩|2 − 2 � ij |⟨λ(Z, Xi), X j⟩|2 − 2 � ij |⟨λ(Xi, Z), X j⟩|2 =2 � ij � |⟨λ(Zi, H j), Z⟩|2 + |⟨λ(Hi, Z j), Z⟩|2� + α(Z) − 2 � ij |⟨λ(Z, Hi), Z j⟩|2 − 2 � ij |⟨λ(Hi, Z), Z j⟩|2, where α(Z) = 2 � ij |⟨λ(Yi, Y j), Z⟩|2 ≥ 0, {Yi} = {Hi} ∪ {Vi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This implies 0 > � k ⟨MλZk, Zk⟩ = � k α(Zk) ≥ 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So Z = 0, and consequently, N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Therefore L is a semisimple associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This completes the proof of theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In fact, by the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6, we know that if [µ] ∈ An for which there exists [λ] ∈ GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] such that Mλ is negative definite, then µ is a semisimple associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The maxima of Fn : An → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' We say that an algebra λ degenerates to µ, write as λ → µ if µ ∈ GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='λ, the closure of GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='λ with respect to the usual topology of Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The degeneration λ → µ is called direct degeneration if there are no nontrivial chains: λ → ν → µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The degeneration level of an algebra is the maximum length of chain of direct degenerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='8 ([3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' An n-dimensional associative algebra is of degeneration level one if and only if it is isomorphic to one of the following (1) µl: µl(X1, Xi) = Xi, i = 1, · · · , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (2) µr: µr(Xi, X1) = Xi, i = 1, · · · , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (3) µca: µs(X1, X1) = X2, where {X1, · · · , Xn} is a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The functional Fn : An → R attains its maximal value at a point [µ] ∈ Ln, n ≥ 3 if and only if µ is isomorphic to the commutative associative algebra µca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In such a case, Fn([µ]) = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Assume that Fn : An → R attains its maximal value at a point [µ] ∈ An, n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3, we know that [µ] is also a critical of Fn : PVn → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then it follows Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='5 that Fn|GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] also attains its minimum value at a point [µ] , consequently Fn|GL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] is a constant, so GL(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] = U(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [µ] (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4) THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 13 The relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4) implies that the only non-trivial degeneration of µ is 0 ([8, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1] ), conse- quently the degeneration level of µ is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It is easy to see that the critical points [µl], [µr] are both of type (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, n − 1), and [µca] is of type (3 < 5 < 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, n − 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4, we know Fn([µca]) = 20 > 4 = Fn([µl]) = Fn([µr]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So the theorem is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The structure for the critical points of Fn : An → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' In the following, we discuss the structure for an arbitrary critical points of Fn : An → R by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let [µ] be a critical point of Fn : An → R with Mµ = cµI + Dµ of type (k1 < · · · < kr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' d1, d2, · · · , dr), where cµ ∈ R and Dµ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Consider the orthogonal decomposition Cn = A− ⊕ A0 ⊕ A+, where A−, A0 and A+ denote the direct sum of eigenspaces of Dµ with eigenvalues smaller than zero, equal to zero and larger than zero, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then the following conditions hold: (i) ann(µ) ⊂ A+, where ann(µ) is the annihilator of µ (ii) A+ ⊂ N(µ), where N(µ) is the radical of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (iii) A− ⊂ (C(µ) ∩ N(µ)) \\ ann(µ), where C(µ) is the center of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (iv) (Lµ A − Rµ A)∗ ∈ Der(µ) for any A ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So the induced Lie algebra of A0 is reductive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For (i), assume that X ∈ ann(µ) and DµX = cX, then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='8) ⟨MµX, X⟩ = 2 � i, j |⟨µ(Xi, X j), X⟩|2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since Mµ = cµI + Dµ, then 0 ≤ ⟨MµX, X⟩ = (cµ + c)⟨X, X⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6 that c ≥ −cµ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This proves (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For (ii), it is an immediate consequence of (iii) by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Now, we prove (iii) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Assume that DµX = cX for some c < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since cLµ X = [Dµ, Lµ X], cRµ X = [Dµ, Rµ X], then c tr(Lµ X − Rµ X)(Lµ X − Rµ X)∗ = tr[Dµ, (Lµ X − Rµ X)](Lµ X − Rµ X)∗ = tr[Mµ, (Lµ X − Rµ X)](Lµ X − Rµ X)∗ = tr Mµ[(Lµ X − Rµ X), (Lµ X − Rµ X)∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Noting that (Lµ X − Rµ X) ∈ Der(µ), by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2 we have c tr(Lµ X − Rµ X)(Lµ X − Rµ X)∗ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 14 HUI ZHANG AND ZAILI YAN It follows that (Lµ X − Rµ X) = 0 since c < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So X ∈ C(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4, it is easy to see that X ∈ N(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Using (i), we conclude A− ⊂ (C(µ) ∩ N(µ)) \\ ann(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This proves (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For (iv), we first note that [Dµ, Lµ A] = Lµ DµA, [Dµ, Rµ A] = Rµ DµA, for any A ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' If A ∈ A0, we have [Dµ, Lµ A] = [Dµ, Rµ A] = 0, and so tr Mµ[(Lµ A − Rµ A), (Lµ A − Rµ A)∗] = tr(cµI + Dµ)[(Lµ A − Rµ A), (Lµ A − Rµ A)∗] = tr Dµ[(Lµ A − Rµ A), (Lµ A − Rµ A)∗] = tr[Dµ, (Lµ A − Rµ A)](Lµ A − Rµ A)∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2, it follows that (Lµ A − Rµ A)∗ ∈ Der(µ) since (Lµ A − Rµ A) ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This proves (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ In the sequel, we give a description of the critical points in terms of those which are nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let [λ] be a nilpotent critical point of Fm : Am → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Define L(λ) : = {Φ ∈ End(Cm) : Φ(λ(X, Y)) = λ(ΦX, Y)}, R(λ) : = {Ψ ∈ End(Cm) : Ψ(λ(X, Y)) = λ(X, ΨY)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, we set Γl = {Φ ∈ L(λ) : [Φ, Ψ] = 0, ∀Ψ ∈ R(λ)}, Γr = {Ψ ∈ R(λ) : [Φ, Ψ] = 0, ∀Φ ∈ L(λ)}, and Γ(λ) : = {(Φ, Ψ) ∈ Γl × Γr : λ(·, Φ(·)) = λ(Ψ(·), ·)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any (Φi, Ψi) ∈ Γ(λ), i = 1, 2, we define (Φ1, Ψ1)(Φ2, Ψ2) := (Φ1Φ2, Ψ2Ψ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then it follows that Γ(λ) is an associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Assume that S ⊂ Γ(λ) is a subalgebra such that (Φ∗, Ψ∗) ∈ S for any (Φ, Ψ) ∈ S, then S is a semisimple associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Note that S is an associative algebra of matrices, which are closed under conjugate transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Define an Hermitian inner product on S by ⟨H1, H2⟩ := tr H1H∗ 2 = tr Φ1Φ∗ 2 + tr Ψ1Ψ∗ 2, ∀Hi = (Φi, Ψi) ∈ S, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then it follows that ⟨HH1, H2⟩ = ⟨H1, H∗H2⟩, ⟨H1H, H2⟩ = ⟨H1, H2H∗⟩ for any H, H1, H2 ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let I be an ideal in S and I⊥ denote the orthogonal complement of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then it is easy to see that I⊥ is also an ideal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let S = R⊕N, where N is the radical of S and R = N⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows that R and N are both ideals of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, R is semisimple, and N is the annihilator of S (by considering the derived series).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Since S is an associative algebra of matrices which are closed under conjugate transpose, then HH∗ = 0 for any H ∈ N, hence H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So N = 0, and S is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 15 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let [λ] be a nilpotent critical point of Fm : Am → R with Mλ = cλI + Dλ of type (k2 < · · · < kr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' d2, · · · , dr), where cλ ∈ R and Dλ ∈ Der(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Assume that S ⊂ Γ(λ) is a subalgebra of dimension d1 such that (Φ∗, Ψ∗) ∈ S, [Dλ, Φ] = [Dλ, Ψ] = 0 for any (Φ, Ψ) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Consider the following semidirect sum µ = S ⋉ λ, where µ((Φ1, Ψ1) + X1, (Φ2, Ψ2) + X2) := (Φ1Φ2, Ψ2Ψ1) + Φ1(X2) + Ψ2(X1) + X1X2, for any (Φ1, Ψ1), (Φ2, Ψ2) ∈ S, X1, X2 ∈ Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then µ is an associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' If we extend the Hermitian inner product on Cm by setting ⟨H, K⟩ = − 2 cλ (tr LS HLS K∗ + tr HK∗), H, K ∈ S, then [µ] is a critical point of type (0, k2 < · · · < kr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' d1, d2, · · · , dr) for the functional Fn : An → R, where n = d1 + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any H = (Φ, Ψ) ∈ S, we have Lµ H = � LS H 0 0 Φ � , Rµ H = � RS H 0 0 Ψ � , where Lµ H, Rµ H (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' LS H, RS H) denote the left and right multiplication by H of the algebra µ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' S), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='11, we know that S is a semisimple associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then it follows that there is an orthonormal basis {Hi = (Φi, Ψi)} ⊂ S such that Φi∗ = −Φi, Ψi∗ = −Ψi, and Lµ Hi, Rµ Hi are skew-Hermitian for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let {Hi} ∪ {Xi} be an orthonormal basis of Cn = S ⊕ Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then for any H = (Φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Ψ) ∈ S and X ∈ Cm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' we have ⟨MµX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' H⟩ = −2 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' j ⟨µ(Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X j⟩⟨µ(Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' H),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X j⟩ − 2 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' j ⟨µ(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X j⟩⟨µ(H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X j⟩ = −2 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' j ⟨λ(Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X j⟩⟨Ψ(Xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X j⟩ − 2 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' j ⟨λ(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X j⟩⟨Φ(Xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X j⟩ = −2 � i ⟨λ(Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Ψ(Xi)⟩ − 2 � i ⟨λ(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Φ(Xi)⟩ = −2 tr Ψ∗Rλ X − 2 tr Φ∗Lλ X = −2 tr Rλ Ψ∗(X) − 2 tr Lλ Φ∗(X) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' where Lλ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Rλ X denote the left and right multiplication by X of the algebra λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' and the last two equalities follow from that λ is nilpotent and (Φ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Ψ∗) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, since Φi∗ = −Φi, Ψi∗ = −Ψi for 16 HUI ZHANG AND ZAILI YAN each i, then [Φi, Φi∗] = 0, [Ψi, Ψi∗] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='8) we have ⟨MµX, Y⟩ = 2 � i, j ⟨µ(Hi, X j), X⟩⟨µ(Hi, X j), Y⟩ + 2 � i, j ⟨µ(Xi, H j), X⟩⟨µ(Xi, H j), Y⟩ + 2 � i, j ⟨µ(Xi, X j), X⟩⟨µ(Xi, X j), Y⟩ − 2 � i, j ⟨µ(Hi, X), X j⟩⟨µ(Hi, Y), X j⟩ − 2 � i, j ⟨µ(Xi, X), X j⟩⟨µ(Xi, Y), X j⟩ − 2 � i, j ⟨µ(X, Hi), X j⟩⟨µ(Y, Hi), X j⟩ − 2 � i, j ⟨µ(X, Xi), X j⟩⟨µ(Y, Xi), X j⟩ = ⟨MλX, Y⟩ + 2 � i ⟨[Φi, Φi∗](X, Y⟩ + 2 � i ⟨[Ψi, Ψi∗](X), Y⟩ = ⟨MλX, Y⟩, for any X, Y ∈ Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Therefore Mµ|Cm = Mλ = cλI + Dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' On the other hand, noting that Lµ Hi and Rµ Hi are skew-Hermitian for each i, then for any H = (Φ, Ψ) ∈ S, we have ⟨MµH, H⟩ = 2 � i, j ⟨µ(Hi, H j), H⟩⟨µ(Hi, H j), H⟩ − 2 � i, j ⟨µ(Hi, H), H j⟩⟨µ(Hi, H), H j⟩ − 2 � i, j ⟨µ(Xi, H), X j⟩⟨µ(Xi, H), X j⟩ − 2 � i, j ⟨µ(H, Hi), H j⟩⟨µ(H, Hi), H j⟩ − 2 � i, j ⟨µ(H, Xi), X j⟩⟨µ(H, Xi), X j⟩ = −2(tr LS HLS H∗ + tr ΦΦ∗ + tr ΨΨ∗) = −2(tr LS HLS H∗ + tr HH∗) = cλ⟨H, H⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So Mµ = cµI + Dµ, where cµ = cλ, and Dµ = � 0 0 0 Dλ � ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let the notation be as Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' If (Lµ A)∗ ∈ {Lµ A : A ∈ A0} and (Rµ A)∗ ∈ {Rµ A : A ∈ A0} for any A ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then it follows from a similar proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='11 that A0 is a semisimple associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, the radical of [µ] corresponds to a critical point of type (k1 < · · · < ˆks < · · · < kr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' d1, · · · , ˆds, · · · , dr) by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='12, where ks = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Examples In this section, we classify the critical points of Fn : An → R for n = 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It shows that every 2-dimensional associative algebra is isomorphic to a critical point of F2, and there exists only THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 17 one 3-dimensional associative algebra which is not isomorphic to any critical point of F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Finally, based on the discussion in previous sections, we collect some natural and interesting questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For reader’s convenience, we recall the notation in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Let {e1, e2, · · · , en} be a basis of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Define the bilinear maps ψi, j k : Cn × Cn → Cn by ψi, j k (emen) = δi mδj nek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows that any algebra can be expressed in the form d = � ijk ck ijψi, j k , where ck ij ∈ C are the structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Two-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The classification of two-dimensional associative algebras can be found in [2, TABLE 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' We give the classification of the critical points of F2 : A2 → R as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Two-dimensional associative algebras, critical types and critical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Multiplication relation Critical type Critical value � d1 = ψ1,1 1 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1) 4 � d2 = ψ1,1 1 + ψ1,2 2 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1) 4 � d3 = ψ1,1 1 + ψ2,1 2 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1) 4 � d4 = ψ1,1 1 + ψ2,2 2 (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2) 2 � d5 = ψ1,1 2 (1 < 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1) 20 � d6 = ψ1,1 1 + ψ1,2 2 + ψ2,1 2 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1) 4 Indeed, endow these algebras with the Hermitian inner product ⟨·, ·⟩ so that {e1, e2} is an orthonormal basis, then it is easy to obtain TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For example, the multiplication relation of µ := (d6, ⟨·, ·⟩) is given by: e1e1 = e1, e1e2 = e2, e2e1 = e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' With respect to the given orthonormal basis {e1, e2}, the left and right multiplications of µ are represented by Lµ e1 = � 1 0 0 1 � , Lµ e2 = � 0 0 1 0 � , Rµ e1 = � 1 0 0 1 � , Rµ e2 = � 0 0 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='7) that Mµ = � −6 0 0 0 � Set cµ := tr M2 µ tr Mµ , then cµ = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows that Mµ = cµI + Dµ, where Dµ = � 0 0 0 6 � is clearly a derivation of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So [µ] is a critical point of F2 : A2 → R with the critical type (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1) and F2([µ]) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 18 HUI ZHANG AND ZAILI YAN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Three-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The complete classification of three-dimensional associative algebras can be found in [2, TABLE 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' The following table gives the classification of the critical points of F3 : A3 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Three-dimensional associative algebras, critical types and critical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Multiplication relation Critical type Critical value � d1 = ψ1,1 1 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d2 = ψ1,1 1 + ψ2,2 3 (0 < 1 < 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1, 1) 10 3 � d3 = ψ1,1 1 + ψ1,3 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d4 = ψ1,1 1 + ψ3,1 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d5 = ψ1,1 1 + ψ1,3 3 + ψ3,1 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d6 = ψ1,1 1 + ψ3,3 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, 1) 2 � d7 = ψ1,1 1 + ψ2,1 2 + ψ1,3 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d8 = ψ1,1 1 + ψ2,1 2 + ψ3,1 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d9 = ψ1,1 1 + ψ2,1 2 + ψ1,3 3 + ψ3,1 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d10 = ψ1,1 1 + ψ2,1 2 + ψ3,3 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, 1) 2 � d11 = ψ1,1 1 + ψ2,2 2 + ψ2,3 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, 1) 2 � d12 = ψ1,1 1 + ψ2,2 2 + ψ2,3 3 + ψ3,2 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, 1) 2 � d13 = ψ1,1 1 + ψ2,2 2 + ψ2,3 3 + ψ3,1 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, 1) 2 � d14 = ψ1,1 1 + ψ2,2 2 + ψ3,3 3 (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 3) 4 3 � d15 = ψ1,1 2 (3 < 5 < 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1, 1) 20 � d16 = ψ1,1 2 + ψ1,2 3 + ψ2,1 3 (1 < 2 < 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1, 1) 20 3 � d17 = ψ1,1 1 + ψ1,1 2 + ψ1,2 2 + ψ2,1 2 + ψ1,3 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d18 = ψ1,1 1 + ψ1,1 2 + ψ1,2 2 + ψ2,1 2 + ψ1,3 3 + ψ3,1 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d19 = ψ3,3 3 + ψ1,1 2 + ψ1,3 1 + ψ3,1 1 + ψ2,3 2 + ψ3,2 2 (0 < 1 < 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 1, 1) 10 3 � d20 = ψ1,1 1 + ψ1,2 2 + ψ1,3 3 (0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 1, 2) 4 � d21 = ψ1,1 3 + ψ1,2 3 − ψ2,1 3 − − � d22 = xψ1,2 3 + yψ2,1 3 (1 < 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, 1) 12 Indeed, endow the algebras with the Hermitian inner product ⟨·, ·⟩ so that {e1, e2, e3} is an orthonormal basis, it is easy to obtain all cases in TABLE II except for d2, d10, d11, d12, d13, d17, d18, d21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For the cases d2, d10, d11, d12, it follows from Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='4 and TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For the cases d13, d17, d18, it follows from [5] that d13 � U3 1, d17 � W3 10 and d18 � U3 0, where U3 1, W3 10 and U3 0 are defined by U3 1 : ψ1,1 1 + ψ3,3 1 + ψ1,2 2 + ψ2,1 2 + ψ2,3 2 + ψ1,3 3 + ψ3,1 3 − ψ3,2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' W3 10 : ψ1,2 1 + ψ2,1 1 + ψ2,2 2 + ψ2,3 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' U3 0 : ψ1,1 2 + ψ1,2 2 + ψ2,1 2 + ψ1,3 3 + ψ3,1 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Endow U3 1, W3 10 and U3 0 with the Hermitian inner product ⟨·, ·⟩ so that {e1, e2, e3} is an orthonormal basis, then it is easy to obtain the corresponding critical types and values for d13, d17, d18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' THE MOMENT MAP FOR THE VARIETY OF ASSOCIATIVE ALGEBRAS 19 In the sequel, we follow a similar procedure as in [6, 16] to classify all Hermitian inner products on d21, then show that for any Hermitian inner product ⟨·, ·⟩ on d21, (d21, ⟨·, ·⟩) cannot be a critical point of F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' First, note that the multiplication relation of d21 is given as follows: e1e1 = e3, e1e2 = e3, e2e1 = −e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Denote by ⟨·, ·⟩0 the Hermitian inner product on d21 such that {e1, e2, e3} is orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' With respect to this basis {e1, e2, e3}, the automorphism group of d21 is given by Aut(d21) = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed a 0 0 b a 0 c d a2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ⊂ GL(3, C), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1) where 0 � a ∈ C, and b, c, d ∈ C are arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any Hermitian inner product ⟨·, ·⟩ on d21, there exist k > 0 and φ ∈ Aut(d21) such that {aφe1, φe2, φe3} is orthonormal with respective to k⟨·, ·⟩, where a > 0 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It suffices to prove that U = {diag(a, 1, 1) : a > 0} ⊂ GL(3, C) is a set of representatives for the action C×Aut(d21) on M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=', the space of all Hermitian inner products on d21, which can be identified with the homogeneous space GL(3, C)/U(3) at the base point ⟨·, ·⟩0 ∈ M (see [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Indeed, since � g∈U C×Aut(d21) · g · U(3) = GL(3, C), it follows that U is a set of representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any Hermitian inner product ⟨·, ·⟩ on d21, we know that there exists g0 ∈ U such that ⟨·, ·⟩ ∈ (C×Aut(d21)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='⟨·, ·⟩0) Hence there exist c ∈ C×, φ ∈ Aut(d21) such that ⟨·, ·⟩ = (cφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='⟨·, ·⟩0) = (cφg0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='⟨·, ·⟩0) Put k = |c|2, then k⟨·, ·⟩ = k⟨(cφg0)−1(·), (cφg0)−1(·)⟩0 = kc−1¯c−1⟨(φg0)−1(·), (φg0)−1(·)⟩0 = ⟨(φg0)−1(·), (φg0)−1(·)⟩0 Since g0 ∈ U, then g0 = diag{a, 1, 1} for some a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' It follows that {aφe1, φe2, φe3} is orthonormal with respective to k⟨·, ·⟩ □ Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' For any Hermitian inner product ⟨·, ·⟩ on d21, (d21, ⟨·, ·⟩) can not be a critical point of F3 : A3 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 20 HUI ZHANG AND ZAILI YAN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Assume that ⟨·, ·⟩ is a Hermitian inner product on d21 such that (d21, ⟨·, ·⟩) is a critical point of F3 : A3 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Then the critical type is necessarily of (1 < 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, 1) by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Moreover, for the Hermitian inner product ⟨·, ·⟩ on d21, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1 we know that there exist k > 0 and φ ∈ Aut(d21) such that {x1 = aφe1, x2 = φe2, x3 = φe3} is orthonormal with respective to k⟨·, ·⟩, where a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' With respect to the basis {x1, x2, x3}, the multiplication relation of d21 is given as follows x1x1 = a2x3, x1x2 = ax3, x2x1 = −ax3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='7), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='6 and a straightforward calculation, it follows that the critical type is of (3a4 + 6a2 + 8, 5a4 + 10a2 + 8, 2(3a4 + 8a2 + 8)) which is never of type (1 < 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 2, 1) for any a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' This is a contradiction by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='1, and the proposition is therefore proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' By the previous discussion, we know that the critical types of Fn : An → R, n = 2, 3, are necessarily nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' So it is natural to ask the following question: Let [µ] ∈ An be a critical point of Fn : An → R with Mµ = cµI + Dµ for some cµ ∈ R and Dµ ∈ Der(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Are all the eigenvalues of Dµ necessarily nonnegative?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' On the other hand, it will be also interesting to construct or classify the critical points [µ] of Fn : An → R such that Dµ has negative eigenvalues if the above question does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' We note that 2-step nilpotent Lie algebras are automatically associative algebras, so it follows from [13, Example 1] that there exist associative algebras whose Nikolayevsky derivations do admit negative eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Statements and Declarations The authors declare that there is no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' References [1] B¨ohm, C.' metadata={'source': 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Springer-Verlag, New York, Heidelberg, Berlin, 1982 [16] Taketomi, Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Tamaru, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': On the nonexistence of left-invariant Ricci solitons a conjecture and examples, Transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Groups 23 (2018), 257–270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' [17] Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=': The moment map for the variety of 3-Lie algebras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 283 (2022), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' 11, Article ID 109683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' (Hui Zhang) School of Mathematics, Southeast University, Nanjing 210096, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content=' China Email address: 2120160023@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='nankai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='cn School of Mathematics and Statistics, Ningbo University, Ningbo, Zhejiang Province, 315211, People’s Republic of China Email address: yanzaili@nbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'} +page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFLT4oBgHgl3EQfrC9d/content/2301.12142v1.pdf'}