Cohomology of finite-dimensional pointed Hopf algebras

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2009
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Mastnak, M.
Pevtsova, J.
Witherspoon, S.
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Proceedings of the London Mathematical Society. 2009, 100(2), pp. 377-404. ISSN 0024-6115. Available under: doi: 10.1112/plms/pdp030
Zusammenfassung

We prove finite generation of the cohomology ring of any finite-dimensional pointed Hopf algebra, having abelian group of group-like elements, under some mild restrictions on the group order. The proof uses the recent classification by Andruskiewitsch and Schneider of such Hopf algebras. Examples include all of Lusztig's small quantum groups, whose cohomology was first computed explicitly by Ginzburg and Kumar, as well as many new pointed Hopf algebras. We also show that in general the cohomology ring of a Hopf algebra in a braided category is braided commutative. As a consequence we obtain some further information about the structure of the cohomology ring of a finite-dimensional pointed Hopf algebra and its related Nichols algebra.

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510 Mathematik
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ISO 690MASTNAK, M., J. PEVTSOVA, Peter SCHAUENBURG, S. WITHERSPOON, 2009. Cohomology of finite-dimensional pointed Hopf algebras. In: Proceedings of the London Mathematical Society. 2009, 100(2), pp. 377-404. ISSN 0024-6115. Available under: doi: 10.1112/plms/pdp030
BibTex
@article{Mastnak2009Cohom-12762,
  year={2009},
  doi={10.1112/plms/pdp030},
  title={Cohomology of finite-dimensional pointed Hopf algebras},
  number={2},
  volume={100},
  issn={0024-6115},
  journal={Proceedings of the London Mathematical Society},
  pages={377--404},
  author={Mastnak, M. and Pevtsova, J. and Schauenburg, Peter and Witherspoon, S.}
}
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