nLab
Gerstenhaber-Schack cohomology

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Algebra

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 and cohomologies associated to other categories of “modules” in bialgebra theory (Hopf modules, Yetter-Drinfeld modules etc.). In the nnLab 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