On existence of PI-exponent of algebras with involution

TL;DR

This paper investigates the growth of polynomial identities in algebras with involution, demonstrating exponential bounds, constructing examples with fractional PI-exponents, and showing that some algebras lack a PI-exponent.

Contribution

It establishes exponential bounds on $*$-codimensions, constructs algebras with fractional $*$-PI-exponents, and provides examples where the PI-exponent does not exist.

Findings

Growth of $*$-codimensions is exponentially bounded.

Existence of algebras with fractional $*$-PI-exponent.

Existence of infinite-dimensional algebras without a PI-exponent.

Abstract

We study polynomial identities of algebras with involution of nonassociative algebras over a field of characteristic zero. We prove that the growth of the sequence of $*$-codimensions of a finite-dimensional algebra is exponentially bounded. We construct a series of finite-dimensional algebras with fractional $*$-PI-exponent. We also construct a family of infinite-dimensional algebras $C_\alpha$ such that ${\rm exp}^*(C_\alpha)$ does not exist.