Trace identities and almost polynomial growth

Antonio Ioppolo; Plamen Koshlukov; Daniela La Mattina

doi:10.1016/j.jpaa.2020.106501

paper

Trace identities and almost polynomial growth

arXiv:2012.10991 · doi:10.1016/j.jpaa.2020.106501

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.

14 pages