On Regular Quantum Commutative Algebras

TL;DR

This paper solves the Bahturin--Regev conjecture for finite-dimensional regular quantum commutative algebras over algebraically closed fields, providing criteria for group gradings on semisimple algebras.

Contribution

It offers a positive solution to the conjecture in the non-graded setting and establishes a criterion for realizing set-gradings as group gradings.

Findings

Solved the Bahturin--Regev conjecture in the finite-dimensional case.

Established a criterion for group gradings via regular quantum commutative decompositions.

Provided conditions under which set-gradings are group gradings.

Abstract

Let $K$ be an algebraically closed field of characteristic different from $2$. We provide a positive solution to the Bahturin--Regev conjecture in the general finite-dimensional (non-graded) setting, assuming that $\operatorname{char}(K)$ does not divide the quantum length of a minimal regular quantum commutative decomposition. Furthermore, we obtain a criterion, formulated in terms of regular quantum commutative decompositions, under which a set-grading on a semisimple associative algebra is realized as a group grading.