On Gerstenhaber's theorem for spaces of nilpotent matrices over a skew field

TL;DR

This paper generalizes Gerstenhaber's theorem to spaces of nilpotent matrices over a skew field, establishing an upper bound on their dimension and characterizing when equality holds.

Contribution

It extends classical results to skew fields, providing a new bound and characterization for nilpotent matrix spaces over such fields.

Findings

Dimension of nilpotent subspace bounded by q n(n-1)/2

Equality characterizes spaces similar to strictly upper-triangular matrices

Generalizes Gerstenhaber's theorem to skew fields

Abstract

Let K be a skew field, and K_0 be a subfield of the central subfield of K such that K has finite dimension q over K_0. Let V be a K_0-linear subspace of n by n nilpotent matrices with entries in K. We show that the dimension of V is bounded above by q n(n-1)/2, and that equality occurs if and only if V is similar to the space of all n by n strictly upper-triangular matrices over K. This generalizes famous theorems of Gerstenhaber and Serezhkin, which cover the special case K=K_0.