On systems of commuting matrices, Frobenius Lie algebras and 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 [Gerstenhaber, M.; On dominance and varieties of commuting matrices. Ann. of Math. (2) 73 (1961), 324-348]. Although 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, for any 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 (LSA) structure.