Differential identities of matrix algebras

TL;DR

This paper investigates the differential identities of matrix algebras over fields of characteristic zero, identifying generators, calculating codimensions and cocharacters, and analyzing growth behavior under derivation actions.

Contribution

It determines a minimal set of generators for the differential identities of matrix algebras and analyzes their growth properties, extending understanding of differential algebra structures.

Findings

Identified 2 generators for the ideal of differential identities for $M_k(F)$.

Calculated exact differential codimensions and cocharacters.

Proved the differential algebra variety has almost polynomial growth for all $k",

Abstract

We study the differential identities of the algebra $M_k(F)$ of $k\times k$ matrices over a field $F$ of characteristic zero when its full Lie algebra of derivations, $L=\mbox{Der}(M_k(F))$, acts on it. We determine a set of 2 generators of the ideal of differential identities of $M_k(F)$ for $k\geq 2$. Moreover, we obtain the exact values of the corresponding differential codimensions and differential cocharacters. Finally we prove that, unlike the ordinary case, the variety of differential algebras with $L$-action generated by $M_k(F)$ has almost polynomial growth for all $k\geq 2$.