Gerstenhaber-Schack and Hochschild Cohomologies of Hopf Algebras

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.

2010 Mathematics Subject Classification: 16T05, 16E40, 16E10

Keywords and Phrases: Hopf algebra, cohomology, cohomological dimension, Yetter-Drinfeld module

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