$\mathbb Z_2$-graded codimensions of unital algebras

TL;DR

This paper investigates the graded polynomial identities of certain nonassociative algebras constructed from infinite binary words, confirming that adjoining an external unit increases the graded PI-exponent by one when the grading group is cyclic of order two.

Contribution

It extends the understanding of PI-exponent behavior to graded identities in algebras built from combinatorial word constructions, specifically for cyclic group gradings of order two.

Findings

Confirmed the conjecture that adjoining a unit increases the graded PI-exponent by one.

Constructed examples of algebras with arbitrary real PI-exponent greater than one.

Extended previous results to the setting of graded identities with cyclic group grading.

Abstract

We study polynomial identities of nonassociative algebras constructed by using infinite binary words and their combinatorial properties. Infinite periodic and Sturmian words were first applied for constructing examples of algebras with arbitrary real PI-exponent greater than one. Later we used these algebras for confirmation of the conjecture that PI-exponent increases precisely by one after adjoining an external unit to a given algebra. Here we prove the same result for these algebras for graded identities and graded PI-exponent, provided that the grading group is cyclic of order two.