On systems of commuting matrices, Frobenius Lie algebras and Gerstenhaber's Theorem

Abstract

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.