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Shahn Majid has introduced a notion of bialgebra cocycles which as special cases comprise group cocycles, nonabelian Drinfel’d 2-cocycle and 3-cocycle, abelian Lie algebra cohomology and so on.
Besides this case, by “bialgebra cohomology” many authors in the literature mean the abelian cohomology (Ext-groups) in certain category of “tetramodules” over a fixed bialgebra, which will be in Lab referred as Gerstenhaber-Schack cohomology.
Let be a -bialgebra. Denote , for , and , . Notice that for the compositions for .
Let be an invertible element of . We define the coboundary by
This formula is symbolically also written as .
An invertible is an -cocycle if . The cocycle is counital if for all , where .
is a 1-cocycle iff it is invertible and grouplike i.e. (in particular it is counital). A 2-cocycle is an invertible element satisfying
which is counital if (in fact it is enough to require one out of these two counitality conditions). Counital 2-cocycle is hence the famous Drinfel'd twist.
The 3-cocycle condition for reads:
A counital 3-cocycle is the famous Drinfel’d associator appearing in CFT and quantum group theory. The coherence for monoidal structures can be twisted with the help of Drinfel’d associator; Hopf algebras reconstructing them appear then as quasi-Hopf algebras where the comultiplication is associative only up to twisting by a 3-cocycle in .
For particular Hopf algebras
If is a finite group and is the Hopf algebra of -valued functions on the group, then we recover the usual notions: e.g. the 2-cocycle is a function satisfying the cocycle condition
and the condition for a 3-cocycle is
-cocycles can be in low dimensions twisted by -cochains (I think it is in this context not know for hi dimensions), what gives an equivalence relation:
For example, if is a counital 2-cocycle, and a counital coboundary, then
is another 2-cocycle in . In particular, if we obtain that is a cocycle (that is every 2-coboundary is a cocycle).
A dual theory
In addition to cocycles “in” as above, Majid introduced a dual version – cocycles on . The usual Lie algebra cohomology , where is a -Lie algebra, is a special case of that dual construction.
Instead of one uses multiplications defined analogously ( is the multiplication in -th place for and , ). An -cochain on is a linear functional , invertible in the convolution algebra. An -cochain on is a coboundary if
If then this condition reads
and, for , the condition is
If one looks at the group algebra of a finite group then the cocycle conditions above can be obtained by a Hopf algebraic version of the -linear extension of the cocycle conditions for the group cohomology in the form appearing in Schreier’s theory of extensions.
However for all the Lie algebra cohomology also appears as a special case.
(to be completed later)
Shahn Majid, Cross product quantisation, nonabelian cohomology and twisting of Hopf algebras, in H.-D. Doebner, V.K. Dobrev, A.G. Ushveridze, eds., Generalized symmetries in Physics. World Sci. (1994) 13-41; (arXiv:hep.th/9311184)
Shahn Majid, Foundations of quantum group theory, Cambridge UP