Deformation cohomology for Yetter-Drinfel'd modules and Hopf (bi)modules

Details Deformation cohomology for Yetter-Drinfel'd modules and Hopf (bi)modules Deformation cohomology for Yetter-Drinfel'd modules and Hopf (bi)modules

2000-06-07

arxiv.org/abs/math/0006048v1

Mathematics

Quantum Algebra

12 pages, Latex, no figures

Scientific paper

If A is a bialgebra over a field k and M, N are either left-right Yetter-Drinfel'd modules or left-right Hopf modules over A, we construct deformation cohomologies H^*(M,N) as total cohomologies of certain double complexes Y(M,N) and C(M,N), respectively. In both cases, H^1(M,N) is isomorphic to the group of equivalence classes of extensions of M by N in the corresponding category. In the Yetter-Drinfel'd case, H^*(k,k) is just the Gerstenhaber-Schack cohomology of the bialgebra A. If M, N are Hopf bimodules we construct a natural subbicomplex of the above C(M,N), yielding a cohomology theory for Hopf bimodules, similar to the one recently introduced by R. Taillefer.

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