On existence of PI-exponents of unital algebras

TL;DR

This paper constructs a family of unital non-associative algebras demonstrating that the PI-exponent, a measure of polynomial identity growth, may not exist for certain algebras, challenging previous assumptions.

Contribution

It provides the first example of a unital algebra where the PI-exponent does not exist, showing new complexity in codimension growth behavior.

Findings

Constructed a family of unital non-associative algebras with specific PI-exponent properties

Showed that the PI-exponent does not exist for these algebras when alpha > 2

Demonstrated that the lower and upper PI-exponents can differ significantly

Abstract

We construct a family of unital non-associative algebras $\{T_\alpha\vert~ 2<\alpha\in\mathbb R\}$ such that $\underline{exp}(T_\alpha)=2$, whereas $\alpha\le\overline{exp}(T_\alpha)\le\alpha+1$. In particular, it follows that ordinary PI-exponent of codimension growth of algebra $T_\alpha$ does not exist for any $\alpha> 2$. This is the first example of a unital algebra whose PI-exponent does not exist.