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arXiv · 2608.21936

Quantum regularity of finite dimensional semisimple algebras

Yuri Bahturin·Lucio Centrone·Kauê Pereira
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Abstract

In this paper, we establish a necessary and sufficient criterion for a finite-dimensional semisimple algebra over an algebraically closed field of characteristic $0$ to admit a regular quantum commutative decomposition. We apply this characterization to group algebras and show that a finite group algebra admits such a decomposition if and only if the underlying group is a peak group, that is, a finite group whose irreducible character degrees have a greatest element with respect to the divisibility ordering. We investigate the structure of peak groups and establish several criteria for their solvability in terms of the prime divisors of their largest irreducible character degree. In particular, we show that several important classes of groups are peak groups, while supersolvability alone does not imply the peak property. Finally, we construct a non-solvable peak group within the class of Frobenius groups.

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Yuri Bahturin, Lucio Centrone, Kauê Pereira. 2026-08-22. Quantum regularity of finite dimensional semisimple algebras. https://arxiv.org/abs/2608.21936

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