# nLab bialgebra cocycle

cohomology

### Theorems

#### Algebra

higher algebra

universal algebra

# 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 $\left(H,\mu ,\eta ,\Delta ,ϵ\right)$ be a $k$-bialgebra. Denote ${\Delta }_{i}:{B}^{\otimes n}\to {B}^{\otimes \left(n+1\right)}:={id}_{B}^{\otimes \left(i-1\right)}\otimes \Delta \otimes {id}_{B}^{\otimes \left(n-i+1\right)}$, 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

$\partial \chi =\left(\prod _{i=0}^{i\mathrm{even}}{\Delta }_{i}\chi \right)\left(\prod _{i=0}^{i\mathrm{odd}}{\Delta }_{i}{\chi }^{-1}\right)$\partial \chi = (\prod_{i=0}^{i \mathrm{ even}} \Delta_i\chi) (\prod_{i=0}^{i \mathrm{ odd}} \Delta_i \chi^{-1})

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

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

$\left(1\otimes \chi \right)\left(\mathrm{id}\otimes \Delta \right)\chi =\left(\chi \otimes 1\right)\left(\Delta \otimes \mathrm{id}\right)\chi ,$(1\otimes\chi)(id\otimes\Delta)\chi = (\chi\otimes 1)(\Delta\otimes id)\chi,

which is counital if $\left(ϵ\otimes \mathrm{id}\right)\chi =\left(\mathrm{id}\otimes ϵ\right)\chi =1$ (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:

$\left(1\otimes \varphi \right)\left(\left(\mathrm{id}\otimes \Delta \otimes \mathrm{id}\right)\varphi \right)\left(\varphi \otimes 1\right)=\left(\left(\mathrm{id}\otimes \mathrm{id}\otimes \Delta \right)\varphi \right)\left(\left(\Delta \otimes \mathrm{id}\otimes \mathrm{id}\right)\varphi \right)$(1\otimes\phi)((id\otimes\Delta\otimes id)\phi)(\phi\otimes 1) = ((id\otimes id\otimes\Delta)\phi)((\Delta\otimes id\otimes id)\phi)

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 $H=k\left(G\right)$ 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 $\chi :G×G\to k$ satisfying the cocycle condition

$\chi \left(b,c\right)\chi \left(a,bc\right)=\chi \left(a,b\right)\chi \left(ab,c\right)$\chi(b,c)\chi(a,b c) = \chi(a,b)\chi(a b,c)

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

$\varphi \left(b,c,d\right)\varphi \left(a,bc,d\right)\varphi \left(a,b,c\right)=\varphi \left(a,b,cd\right)\varphi \left(ab,c,d\right)$\phi(b,c,d)\phi(a,b c,d)\phi(a,b,c) = \phi(a,b,c d)\phi(a b,c,d)

$n$-cocycles can be in low dimensions twisted by $\left(n-1\right)$-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

${\chi }^{\gamma }=\left({\partial }_{+}\gamma \right)\chi \left({\partial }_{-}{\gamma }^{-1}\right)=\left(\gamma \otimes \gamma \right)\chi \Delta {\gamma }^{-1}$\chi^\gamma = (\partial_+\gamma)\chi(\partial_-\gamma^{-1})= (\gamma\otimes\gamma)\chi\Delta\gamma^{-1}

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 ${H}^{n}\left(L,k\right)$, 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

$\partial \psi =\left(\prod _{i=0}^{\mathrm{even}}\psi \circ {\cdot }_{i}\right)\right)\left(\prod _{i=1}^{\mathrm{odd}}{\psi }^{-1}\circ {\cdot }_{i}\right)$\partial\psi = (\prod_{i=0}^{\mathrm{even}}\psi\circ \cdot_i))(\prod_{i=1}^{\mathrm{odd}}\psi^{-1}\circ\cdot_i)

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

$\left(\partial \psi \right)\left(a\otimes b\right)=\sum \psi \left({b}_{\left(1\right)}\right)\psi \left({a}_{\left(1\right)}\right){\psi }^{-1}\left({a}_{\left(2\right)}{b}_{\left(2\right)}\right)$(\partial\psi)(a\otimes b) = \sum \psi(b_{(1)})\psi(a_{(1)})\psi^{-1}(a_{(2)}b_{(2)})

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

$\left(\partial \psi \right)\left(a\otimes b\otimes c\right)=\sum \psi \left({b}_{\left(1\right)}\otimes {c}_{\left(1\right)}\right)\psi \left({a}_{\left(1\right)}\otimes {b}_{\left(2\right)}{c}_{\left(2\right)}\right){\psi }^{-1}\left({a}_{\left(2\right)}\otimes {b}_{\left(3\right)}{c}_{\left(3\right)}\right){\psi }^{-1}\left({a}_{\left(3\right)}{b}_{\left(4\right)}\right)$(\partial\psi)(a\otimes b\otimes c) = \sum \psi(b_{(1)}\otimes c_{(1)})\psi(a_{(1)}\otimes b_{(2)}c_{(2)})\psi^{-1}(a_{(2)}\otimes b_{(3)}c_{(3)})\psi^{-1}(a_{(3)}b_{(4)})

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

Revised on January 19, 2012 16:52:39 by Zoran Škoda (161.53.130.104)