论文标题
在通勤矩阵的系统中，弗罗贝尼乌斯说谎代数和格斯滕哈佛定理
On systems of commuting matrices, Frobenius Lie algebras and Gerstenhaber's Theorem
论文作者
论文摘要
这项工作涉及三个问题，即正方形矩阵的Lie代数的最大Abelian subgerbras（Masas）的分类，2步可解决的Frobenius Lie代数和Gerstenhaber定理的分类。令m和n为两个通勤的正方形矩阵，在代数封闭的场K中具有条目。然后，它们产生的关联交换k-algebra最多是n的尺寸。默里·格斯滕哈格（Murray Gerstenhaber）在1961年证明了这一结果。该物业的三个通勤矩阵的类似物仍然是一个悬而未决的问题，它的通勤矩阵的版本一般并非如此。在本文中，我们给出了足够的条件，可以满足该特性的满足，对于任何数量的通勤矩阵和任意场K。从关于2步可溶的Frobenius Lie代数的结构的讨论中得出了这种结果，并完全表征其相关的左对称性代数结构。我们讨论了2步可解决的frobenius Lie代数的分类，并表明它等同于正方形矩阵的lie代数的n维Masas，并承认与矩阵和矢量乘法相关的矛盾作用的开放轨道。在任何维度中讨论了许多示例，并且在低维度中提供了完整的分类列表。此外，在任何有限的维度中，我们对与非构型矩阵相对应的所有2步溶解的Frobenius Lie代数进行了完整分类。
This work relates to three problems, the classification of maximal Abelian subalgebras (MASAs) of the Lie algebra of square matrices, the classification of 2-step solvable Frobenius Lie algebras and the Gerstenhaber's Theorem. Let M and N be two commuting square matrices of order n with entries in an algebraically closed field K. Then the associative commutative K-algebra, they generate, is of dimension at most n. This result was proved by Murray Gerstenhaber in 1961. The analog of this property for three commuting matrices is still an open problem, its version for a higher number of commuting matrices is not true in general. In the present paper, we give a sufficient condition for this property to be satisfied, for any number of commuting matrices and arbitrary field K. Such a result is derived from a discussion on the structure of 2-step solvable Frobenius Lie algebras and a complete characterization of their associated left symmetric algebra structure. We discuss the classification of 2-step solvable Frobenius Lie algebras and show that it is equivalent to that of n-dimensional MASAs of the Lie algebra of square matrices, admitting an open orbit for the contragradient action associated to the multiplication of matrices and vectors. Numerous examples are discussed in any dimension and a complete classification list is supplied in low dimensions. Furthermore, in any finite dimension, we give a full classification of all 2-step solvable Frobenius Lie algebras corresponding to nonderogatory matrices.
