Kleisli convolution representations and a Margolis--Sakurai version of the modular isomorphism problem

@misc{liu2026kleisli,
  title = {{Kleisli convolution representations and a Margolis--Sakurai version of the modular isomorphism problem}},
  author = {Yu-Zhe Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09525},
  eprint = {2609.09525},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09525},
}

TY  - UNPB
TI  - Kleisli convolution representations and a Margolis--Sakurai version of the modular isomorphism problem
AU  - Yu-Zhe Liu
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.09525
UR  - https://arxiv.org/abs/2609.09525
AB  - We introduce Kleisli convolution representations for groups, rings, and algebras. We show that their representation categories are equivalent to the usual ones for groups and finite-dimensional algebras, but for rings only recover modules whose underlying Abelian groups are free of finite rank. We also apply the Kleisli convolution representation to provide a partial answer to the Margolis--Sakurai version of the modular isomorphism problem: If G is a finite p-group with D_3(G)=1 and gcd(m,c_G!)=1, then 𝔽_p^mG≅𝔽_p^mH implies G≅ H, where c_G is the number of geometric connected components of Aut(𝔽̅_pG).
ER  -