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bialgebra cocycle

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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 nnLab referred as Gerstenhaber-Schack cohomology.

Definition

Let (H,μ,η,Δ,ϵ)(H,\mu,\eta,\Delta,\epsilon) be a kk-bialgebra. Denote Δ i:B nB (n+1):=id B (i1)Δid B (ni+1)\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,,ni = 1,\ldots, n, and Δ 0:=1 Bid B n\Delta_0 := 1_B\otimes \id_B^{\otimes n}, Δ n:=id B n1 B\Delta_n := \id_B^{\otimes n}\otimes 1_B. Notice that for the compositions Δ iΔ j=Δ j+1Δ i\Delta_i\circ\Delta_j = \Delta_{j+1}\circ\Delta_i for iji\leq j.

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

χ=( i=0 ievenΔ iχ)( i=0 ioddΔ iχ 1)\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 χ=( +χ)( χ 1)\partial\chi = (\partial_+\chi)(\partial_-\chi^{-1}).

An invertible χH n\chi\in H^{\otimes n} is an nn-cocycle if χ=1\partial\chi = 1. The cocycle χ\chi is counital if for all ii, ϵ iχ=1\epsilon_i\chi=1 where ϵ i=id B i1ϵid B ni\epsilon_i =\id_B^{\otimes i-1}\otimes\epsilon\otimes\id_B^{\otimes n-i}.

Examples

Low dimensions

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

(1χ)(idΔ)χ=(χ1)(Δid)χ, (1\otimes\chi)(id\otimes\Delta)\chi = (\chi\otimes 1)(\Delta\otimes id)\chi,

which is counital if (ϵid)χ=(idϵ)χ=1(\epsilon\otimes id)\chi = (id\otimes\epsilon)\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 ϕH 3\phi\in H^{\otimes 3} reads:

(1ϕ)((idΔid)ϕ)(ϕ1)=((ididΔ)ϕ)((Δidid)ϕ) (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 HH.

For particular Hopf algebras

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

χ(b,c)χ(a,bc)=χ(a,b)χ(ab,c) \chi(b,c)\chi(a,b c) = \chi(a,b)\chi(a b,c)

and the condition for a 3-cocycle ϕ:G×G×Gk\phi:G\times G\times G\to k is

ϕ(b,c,d)ϕ(a,bc,d)ϕ(a,b,c)=ϕ(a,b,cd)ϕ(ab,c,d) \phi(b,c,d)\phi(a,b c,d)\phi(a,b,c) = \phi(a,b,c d)\phi(a b,c,d)

nn-cocycles can be in low dimensions twisted by (n1)(n-1)-cochains (I think it is in this context not know for hi dimensions), what gives an equivalence relation:

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

χ γ=( +γ)χ( γ 1)=(γγ)χΔγ 1 \chi^\gamma = (\partial_+\gamma)\chi(\partial_-\gamma^{-1})= (\gamma\otimes\gamma)\chi\Delta\gamma^{-1}

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

A dual theory

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

Instead of Δ i\Delta_i one uses multiplications i\cdot_i defined analogously ( i\cdot_i is the multiplication in ii-th place for 1in1\leq i\leq n and ψ 0=ϵψ\psi\circ\cdot_0 =\epsilon\otimes\psi, ψ n+1=ψϵ\psi\circ\cdot_{n+1} = \psi\otimes\epsilon). An nn-cochain on HH is a linear functional ψ:H nk\psi:H^{\otimes n}\to k, invertible in the convolution algebra. An nn-cochain ψ\psi on HH is a coboundary if

ψ=( i=0 evenψ i))( i=1 oddψ 1 i) \partial\psi = (\prod_{i=0}^{\mathrm{even}}\psi\circ \cdot_i))(\prod_{i=1}^{\mathrm{odd}}\psi^{-1}\circ\cdot_i)

If ψH\psi\in H then this condition reads

(ψ)(ab)=ψ(b (1))ψ(a (1))ψ 1(a (2)b (2)) (\partial\psi)(a\otimes b) = \sum \psi(b_{(1)})\psi(a_{(1)})\psi^{-1}(a_{(2)}b_{(2)})

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

(ψ)(abc)=ψ(b (1)c (1))ψ(a (1)b (2)c (2))ψ 1(a (2)b (3)c (3))ψ 1(a (3)b (4)) (\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 kGkG of a finite group then the cocycle conditions above can be obtained by a Hopf algebraic version of the kk-linear extension of the cocycle conditions for the group cohomology in the form appearing in Schreier’s theory of extensions.

However for all nn 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)