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2-representation theory of the Taft algebra

  • Raphaël Paegelow
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Abstract

We study the $2$-representation theory of the Taft category, the tensor category of finite-dimensional modules over the Taft algebra. This category has two left cells. We first classify suitable transitive modules over its based representation ring. For the upper cell, we recall a construction by Etingof and Ostrik giving an infinite family of pairwise nonequivalent simple $2$-representations. For the bottom cell, we pass to the semisimplification of the Taft category and identify a distinguished fusion subcategory with an adjoint sl2 Verlinde category. Using algebra objects associated with ADET-diagrams and categorical Clifford theory, we classify the regular exact indecomposable module categories of this semisimplification. For these regular module categories, the split Grothendieck group is a complete invariant. Finally, we identify the two cell $2$-representations of the Taft category.

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Cite

BibTeXRIS
BibTeX
@misc{paegelow20262representation,
  title = {{2-representation theory of the Taft algebra}},
  author = {Raphaël Paegelow},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28534},
  eprint = {2610.28534},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28534},
}
RIS
TY  - UNPB
TI  - 2-representation theory of the Taft algebra
AU  - Raphaël Paegelow
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28534
UR  - https://arxiv.org/abs/2610.28534
AB  - We study the $2$-representation theory of the Taft category, the tensor category of finite-dimensional modules over the Taft algebra. This category has two left cells. We first classify suitable transitive modules over its based representation ring. For the upper cell, we recall a construction by Etingof and Ostrik giving an infinite family of pairwise nonequivalent simple $2$-representations. For the bottom cell, we pass to the semisimplification of the Taft category and identify a distinguished fusion subcategory with an adjoint sl_2 Verlinde category. Using algebra objects associated with ADET-diagrams and categorical Clifford theory, we classify the regular exact indecomposable module categories of this semisimplification. For these regular module categories, the split Grothendieck group is a complete invariant. Finally, we identify the two cell $2$-representations of the Taft category.
ER  -