Codimensions of algebras with pseudoautomorphism and their exponential growth

TL;DR

This paper studies the asymptotic behavior of $p$-codimensions in finite dimensional superalgebras with pseudoautomorphisms over a field of characteristic zero, confirming a conjecture that the $p$-exponent always exists as an integer.

Contribution

It proves the existence and integrality of the $p$-exponent for superalgebras with pseudoautomorphisms, and characterizes those with bounded exponential growth.

Findings

The $p$-exponent always exists and is an integer.

Confirmed Amitsur's conjecture in this setting.

Characterized algebras with exponential growth bounded by 2.

Abstract

Let $F$ be a fixed field of characteristic zero containing an element $i$ such that $i^2 = -1$. In this paper we consider finite dimensional superalgebras over $F$ endowed with a pseudoautomorphism $p$ and we investigate the asymptotic behaviour of the corresponding sequence of $p$-codimensions $c_n^p(A),$ $n=1,2, \ldots$. First we give a positive answer to a conjecture of Amitsur in this setting: the $p$-exponent $\exp^p(A) = \lim_{n \rightarrow \infty} \sqrt[n]{c_n^p(A)} $ always exists and it is an integer. In the final part we characterize the algebras whose exponential growth is bounded by $2$.