The maximum dimension of a Lie nilpotent subalgebra of M_{n}(F) of index m

TL;DR

This paper establishes an upper bound on the dimension of Lie nilpotent subalgebras of matrix algebras over any field, generalizing classical results and providing explicit formulas for the bounds.

Contribution

It introduces a new upper bound for the dimension of Lie nilpotent subalgebras of matrix algebras, extending classical theorems to arbitrary fields and nilpotence indices.

Findings

The dimension of such subalgebras is at most M(m+1,n).

The bound M(m+1,n) is sharp and cannot be improved.

An explicit formula for M(m+1,n) is provided.

Abstract

The main result of this paper is the following: if F is any field and R is any F-subalgebra of the algebra of nxn matrices over F with Lie nilpotence index m, then the F-dimension of R is less or equal than M(m+1,n), where M(m+1,n) is the maximum of a certain expression of m+1 nonnegative integers and n. The case m=1 reduces to a classical theorem of Schur (1905), later generalized by Jacobson (1944) to all fields, which asserts that if F is an algebraically closed field of characteristic zero, and R is any commutative F-subalgebra of the full nxn matrix algebra over F, then the F-dimension of R is less or equal than ((n^2)/4)+1. Examples constructed from block upper triangular matrices show that the upper bound of M(m+1,n) cannot be lowered for any choice of m and n. An explicit formula for M(m+1,n) is also derived.