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Freeness and divisibility for right H-simple left H-comodule algebras over a pointed Hopf algebra H

  • Daisuke Nakamura
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Abstract

Let H be a pointed Hopf algebra and let A be a right H-simple left H-comodule algebra. We show that every relative (H,A)-Hopf module is free as an A-module and that this freeness characterizes the class of pointed Hopf algebras. We give a criterion for the category of relative (H,A)-Hopf modules to be semisimple. We also show that A can be embedded into a left H-comodule algebra of a specific form when H and A are ℕ0-graded. As a consequence, we prove that if H is finite-dimensional and Aco H=Bbbk, then A is finite-dimensional and dim A divides dim H.

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Cite

BibTeXRIS
BibTeX
@misc{nakamura2026freeness,
  title = {{Freeness and divisibility for right $H$-simple left $H$-comodule algebras over a pointed Hopf algebra $H$}},
  author = {Daisuke Nakamura},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.24039},
  eprint = {2607.24039},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.24039},
}
RIS
TY  - UNPB
TI  - Freeness and divisibility for right H-simple left H-comodule algebras over a pointed Hopf algebra H
AU  - Daisuke Nakamura
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2607.24039
UR  - https://arxiv.org/abs/2607.24039
AB  - Let H be a pointed Hopf algebra and let A be a right H-simple left H-comodule algebra. We show that every relative (H,A)-Hopf module is free as an A-module and that this freeness characterizes the class of pointed Hopf algebras. We give a criterion for the category of relative (H,A)-Hopf modules to be semisimple. We also show that A can be embedded into a left H-comodule algebra of a specific form when H and A are ℕ_0-graded. As a consequence, we prove that if H is finite-dimensional and A^co H=Bbbk, then A is finite-dimensional and dim A divides dim H.
ER  -