Contents

Idea

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 $n$Lab referred as Gerstenhaber-Schack cohomology.

Definition

Let $(H,\mu ,\eta ,\Delta ,ϵ)$ be a $k$-bialgebra. Denote ${\Delta }_i:B^{\otimes n}\to B^{\otimes (n+1)}:={id}_B^{\otimes (i-1)}\otimes \Delta \otimes {id}_B^{\otimes (n-i+1)}$, for $i=1,\dots ,n$, and ${\Delta }_0:=1_B\otimes {id}_B^{\otimes n}$, ${\Delta }_n:={id}_B^{\otimes n}\otimes 1_B$. Notice that for the compositions ${\Delta }_i\circ {\Delta }_j={\Delta }_{j+1}\circ {\Delta }_i$ for $i\le j$.

Let $\chi$ be an invertible element of $H^{\otimes n}$. We define the coboundary $\partial \chi$ by

This formula is symbolically also written as $\partial \chi =({\partial }_{+}\chi )({\partial }_{-}{\chi }^{-1})$.

An invertible $\chi \in H^{\otimes n}$ is an $n$-cocycle if $\partial \chi =1$. The cocycle $\chi$ is counital if for all $i$, ${ϵ}_i\chi =1$ where ${ϵ}_i={id}_B^{\otimes i-1}\otimes ϵ\otimes {id}_B^{\otimes n-i}$.

Examples

Low dimensions

$\chi \in H$ is a 1-cocycle iff it is invertible and grouplike i.e. $\Delta \chi =\chi \otimes \chi$ (in particular it is counital). A 2-cocycle is an invertible element $\chi \in H^{\otimes 2}$ satisfying

which is counital if $(ϵ\otimes \mathrm{id})\chi =(\mathrm{id}\otimes ϵ)\chi =1$ (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 $\varphi \in H^{\otimes 3}$ 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 $H$.

For particular Hopf algebras

If $G$ is a finite group and $H=k(G)$ is the Hopf algebra of $k$-valued functions on the group, then we recover the usual notions: e.g. the 2-cocycle is a function $\chi :G\times G\to k$ satisfying the cocycle condition

and the condition for a 3-cocycle $\varphi :G\times G\times G\to k$ is

$n$-cocycles can be in low dimensions twisted by $(n-1)$-cochains (I think it is in this context not know for hi dimensions), what gives an equivalence relation:

For example, if $\chi \in H\otimes H$ is a counital 2-cocycle, and $\partial \gamma \in H$ a counital coboundary, then

is another 2-cocycle in $H\otimes H$. In particular, if $\chi =1$ we obtain that $\partial \gamma$ is a cocycle (that is every 2-coboundary is a cocycle).

A dual theory

In addition to cocycles “in” $H$ as above, Majid introduced a dual version – cocycles on $H$. The usual Lie algebra cohomology $H^n(L,k)$, where $L$ is a $k$-Lie algebra, is a special case of that dual construction.

Instead of ${\Delta }_i$ one uses multiplications ${\cdot }_i$ defined analogously (${\cdot }_i$ is the multiplication in $i$-th place for $1\le i\le n$ and $\psi \circ {\cdot }_0=ϵ\otimes \psi$, $\psi \circ {\cdot }_{n+1}=\psi \otimes ϵ$). An $n$-cochain on $H$ is a linear functional $\psi :H^{\otimes n}\to k$, invertible in the convolution algebra. An $n$-cochain $\psi$ on $H$ is a coboundary if

If $\psi \in H$ then this condition reads

and, for $\psi \in H\otimes H$, the condition is

If one looks at the group algebra $\mathrm{kG}$ of a finite group then the cocycle conditions above can be obtained by a Hopf algebraic version of the $k$-linear extension of the cocycle conditions for the group cohomology in the form appearing in Schreier’s theory of extensions.

However for all $n$ the Lie algebra cohomology also appears as a special case.

(to be completed later)

References

• 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