Gerstenhaber-schack and Hochschild cohomologies of Hopf algebras
Doc. Math., J. DMV 21, 955-986 (2016)
Summary: We show that the Gerstenhaber-Schack cohomology of a Hopf algebra determines its Hochschild cohomology, and in particular its Gerstenhaber-Schack cohomological dimension bounds its Hochschild cohomological dimension, with equality of the dimensions when the Hopf algebra is cosemisimple of Kac type. Together with some general considerations on free Yetter-Drinfeld modules over adjoint Hopf subalgebras and the monoidal invariance of Gerstenhaber-Schack cohomology, this is used to show that both Gerstenhaber-Schack and Hochschild cohomological dimensions of the coordinate algebra of the quantum permutation group are 3.
Mathematics Subject Classification
16T05, 16E40, 16E10
Hopf algebra, cohomology, cohomological dimension, Yetter-Drinfeld module