Cohomology theories of Hopf bimodules and cup-product. (English) Zbl 1078.16006
Summary: Given a Hopf algebra \(A\), there exist various cohomology theories for the category of Hopf bimodules over \(A\), introduced by M. Gerstenhaber and S. D. Schack, and by C. Ospel. We prove, when \(A\) is finite-dimensional, that they are all equal to the Ext functor on the module category of an associative algebra associated to \(A\), described by C. Cibils and M. Rosso. We also give an expression for a cup-product in the cohomology defined by C. Ospel, and prove that it corresponds to the Yoneda product of extensions.
MSC:
16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
18G60 | Other (co)homology theories (MSC2010) |
16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
57T05 | Hopf algebras (aspects of homology and homotopy of topological groups) |