Trace identities and almost polynomial growth

TL;DR

This paper investigates trace identities in specific matrix algebras over characteristic zero fields, characterizing their trace codimensions and the growth behavior of trace varieties, revealing they grow either polynomially or exponentially.

Contribution

It provides a complete description of trace identities for certain 2x2 matrix algebras and characterizes the growth of trace varieties generated by finite dimensional algebras.

Findings

Trace on $D_2$ and $C_2$ are fully described.

Trace codimensions are analyzed for these algebras.

Varieties with trace growth are either polynomial or exponential.

Abstract

In this paper we study algebras with trace and their trace polynomial identities over a field of characteristic 0. We consider two commutative matrix algebras: $D_2$, the algebra of $2\times 2$ diagonal matrices and $C_2$, the algebra of $2 \times 2$ matrices generated by $e_{11}+e_{22}$ and $e_{12}$. We describe all possible traces on these algebras and we study the corresponding trace codimensions. Moreover we characterize the varieties with trace of polynomial growth generated by a finite dimensional algebra. As a consequence, we see that the growth of a variety with trace is either polynomial or exponential.