Exponential growth of codimensions of identities of algebras with unity

TL;DR

This paper investigates the exponential growth rates of codimensions of identities in unital algebras, showing that PI-exponents can vary widely and are densely distributed above 2.

Contribution

It constructs algebra series with precisely increasing exponential bases and demonstrates the density of finite-dimensional unital algebra exponents in [2, ∞).

Findings

PI-exponents of unital algebras can be any value greater than two.

The exponents of finite-dimensional unital algebras are dense in [2, ∞).

Constructed algebra series with exponential growth bases increasing by exactly one.

Abstract

The asymptotic behaviour is studied of exponentially bounded sequences of codimensions of identities of algebras with unity. A series of algebras is constructed for which the base of the exponential increases by exactly one when an outer unity is adjoined to the original algebra. We show that the PI-exponents of unital algebras can take any value greater than two, and the exponents of finite-dimensional unital algebras form a dense subset of the domain [2,$\infty$).