On Regular Quantum Commutative Algebras
Yuri Bahturin, Lucio Centrone, Kau\^e Pereira

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 be an algebraically closed field of characteristic different from . We provide a positive solution to the Bahturin--Regev conjecture in the general finite-dimensional (non-graded) setting, assuming that 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.
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