Growth of differential identities

TL;DR

This paper investigates the growth and structure of differential identities in associative algebras with derivations, including specific cases like upper triangular matrices and infinite-dimensional Grassmann algebras, revealing detailed algebraic behaviors.

Contribution

It provides a comprehensive analysis of differential identities and cocharacter sequences for algebras with derivations, including explicit descriptions for $UT_2$ and Grassmann algebras.

Findings

Differential codimensions can have polynomial growth.

Complete description of differential identities for $UT_2$ matrices.

Structure of differential identities for infinite-dimensional Grassmann algebra.

Abstract

In this paper we study the growth of the differential identities of some algebras with derivations, i.e., associative algebras where a Lie algebra $L$ (and its universal enveloping algebra $U(L)$) acts on them by derivations. In particular, we study in detail the differential identities and the cocharacter sequences of some algebras whose sequence of differential codimensions has polynomial growth. Moreover, we shall give a complete description of the differential identities of the algebra $UT_2$ of $2\times 2$ upper triangular matrices endowed with all possible action of a Lie algebra by derivations. Finally, we present the structure of the differential identities of the infinite dimensional Grassmann $G$ with respect to the action of a finite dimensional Lie algebra $L$ of inner derivations.