Proceedings of the London Mathematical Society Advance Access published online on August 9, 2009
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdp030
© 2009 London Mathematical Society
Cohomology of finite-dimensional pointed Hopf algebras
Department of Mathematics and Computer Science
Saint Mary's University
Halifax
NS B3H 3C3
Canada
Department of Mathematics
University of Washington
Seattle, WA 98195
USA
julia@math.washington.edu
Mathematisches Institut der Universität München
Theresienstr. 39
80333 München
Germany
schauenburg@math.lmu.de
Department of Mathematics
Texas A&M University
College Station, TX 77843
USA
sjw@math.tamu.edu
Received 29 January 2009.
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.
2000 Mathematics Subject Classification 16E40, 16W30.
The first author was supported by an NSERC postdoctoral fellowship. The second author was partially supported by the NSF grants DMS-0629156 and DMS-0800940. The third author was supported by Deutsche Forschungsgemeinschaft through a Heisenberg Fellowship. The last author was partially supported by NSF grants DMS-0443476 and DMS-0800832, and NSA grant H98230-07-1-0038.