Combinatorics of binary words and codimensions of identities in left nilpotent algebras

TL;DR

This paper generalizes the combinatorial approach to polynomial identities in left nilpotent algebras, establishing the existence and exact value of PI-exponents for a broad class of binary words with subexponential complexity.

Contribution

It extends previous methods from periodic and Sturm words to more general binary words, enabling precise computation of PI-exponents in left nilpotent algebras.

Findings

Existence of PI-exponent for algebras constructed from binary words with subexponential complexity

Explicit calculation of the PI-exponent value

Generalization of combinatorial methods to a wider class of binary words

Abstract

Numerical characteristics of polynomial identities of left nilpotent algebras are examined. Previously, we came up with a construction which, given an infinite binary word, allowed us to build a two-step left nilpotent algebra with specified properties of the codimension sequence. However, the class of the infinite words used was confined to periodic words and Sturm words. Here the previously proposed approach is generalized to a considerably more general case. It is proved that for any algebra constructed given a binary word with subexponential function of combinatorial complexity, there exists a PI-exponent, and its precise value is computed.