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The Lie algebra structure of HH1(B) for blocks with a normal defect group

  • Markus Linckelmann
  • Jialin Wang
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Abstract

It is an open question whether the Lie algebra HH1(B) of a block of a finite group algebra kG over an algebraically closed field k of prime characteristic p is simple only if the block is nilpotent with an elementary abelian defect group of order at least $3$. We show this to be the case if B has a normal abelian defect group. Another open question is whether HH1(kSn) is a solvable Lie algebra, where Sn denotes a symmetric group on n letters. We give some general solvability criteria for HH1(B) of blocks B with a normal defect group, and, making use of results of Chuang-Kessar and Chuang-Rouquier, we deduce that if B is a block of a symmetric group algebra with an abelian defect group, then HH1(B) is solvable.

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Cite

BibTeXRIS
BibTeX
@misc{linckelmann2026algebra,
  title = {{The Lie algebra structure of $\mathrmHH^1(B)$ for blocks with a normal defect group}},
  author = {Markus Linckelmann and Jialin Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32673},
  eprint = {2610.32673},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32673},
}
RIS
TY  - UNPB
TI  - The Lie algebra structure of HH^1(B) for blocks with a normal defect group
AU  - Markus Linckelmann
AU  - Jialin Wang
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32673
UR  - https://arxiv.org/abs/2610.32673
AB  - It is an open question whether the Lie algebra HH^1(B) of a block of a finite group algebra kG over an algebraically closed field k of prime characteristic p is simple only if the block is nilpotent with an elementary abelian defect group of order at least $3$. We show this to be the case if B has a normal abelian defect group. Another open question is whether HH^1(kS_n) is a solvable Lie algebra, where S_n denotes a symmetric group on n letters. We give some general solvability criteria for HH^1(B) of blocks B with a normal defect group, and, making use of results of Chuang-Kessar and Chuang-Rouquier, we deduce that if B is a block of a symmetric group algebra with an abelian defect group, then HH^1(B) is solvable.
ER  -