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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 divisibility. 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 show how to construct a peak group from any finite group.

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Cite

BibTeXRIS
BibTeX
@misc{bahturin2026quantum,
  title = {{Quantum regularity of finite dimensional semisimple algebras}},
  author = {Yuri Bahturin and Lucio Centrone and Kauê Pereira},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2608.21936},
  eprint = {2608.21936},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.21936},
}
RIS
TY  - UNPB
TI  - Quantum regularity of finite dimensional semisimple algebras
AU  - Yuri Bahturin
AU  - Lucio Centrone
AU  - Kauê Pereira
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2608.21936
UR  - https://arxiv.org/abs/2608.21936
AB  - 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 divisibility. 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 show how to construct a peak group from any finite group.
ER  -