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Contents

Idea

Gerstenhaber and Schack introduced a cohomology related to deformation theory of bialgebras. Teillefer has proven that this cohomology computes in fact the Ext-groups in certain abelian category of “tetramodules” over the bialgebra. A somewhat more systematic writeup of the proof is in the appendix Shoikhet 09.

Please distinguish the Gerstenhaber-Schack bialgebra cohomology from Shahn Majid ’s bialgebra cohomology which is (in full generality) nonabelian. In the$n$bialgebra cohomology? which is (in full generality) nonabelian and cohomologies associated to other categories of “modules” in bialgebra theory (Hopf modules, Yetter-Drinfeld modules etc.). In the $n$Lab we choose to use the term bialgebra cocycle unadorned for the Majid’s cocycles and always use Gerstenhaber-Schack cocycle/cohomology for the latter, as it is often referred nowdays.

References

• Murray Gerstenhaber, Samuel D. Schack, Bialgebra cohomology, deformations, and quantum groups, Proc. Nat. Acad. Sci. U.S.A. 87 (1990), no. 1, 478–481, MR90j:16062, article

• Rachel Taillefer, Injective Hopf bimodules, cohomologies of infinite dimensional Hopf algebras and graded-commutativity of the Yoneda product, J. Algebra 276 (2004), no. 1, 259–279, MR2005f:16067, doi

• Boris Shoikhet, Hopf algebras, tetramodules, and $n$-fold monoidal categories, arxiv/0907.3335

• Brian Parshall, Jian Pan Wang, On bialgebra cohomology. Algebra, groups and geometry, Bull. Soc. Math. Belg. Sér. A 42 (1990), no. 3, 607–642.