quasi-Hopf algebra




The notion of a quasibialgebra generalizes that of a bialgebra Hopf algebra by introducing a nontrivial associativity coherence (Drinfeld 89) isomorphisms (representable by multiplication with an element in triple tensor product) into axioms; a quasi-Hopf algebra is a quasi-bialgebra with an antipode satisfying axioms which also involve nontrivial left and right unit coherences.

In particular, quasi-Hopf algebras may be obtained from ordinary Hopf algebras via twisting by a Drinfeld associator, i.e. a nonabelian bialgebra 3-cocycle.

Motivation from quantum field theory

Drinfel’d was motivated by study of monoidal categories in rational 2d conformal field theory (RCFT) as well as by an idea from Grothendieck‘s Esquisse namely the Grothendieck-Teichmüller tower and its modular properties. In RCFT, the monoidal categories appearing can be, by Tannaka reconstruction considered as categories of modules of Hopf algebra-like objects where the flexibility of associativity coherence in building a theory were natural thus leading to quasi-Hopf algebras.

A special case of the motivation in RCFT has a toy example of Dijkgraaf-Witten theory which can be quite geometrically explained. Namely, where the groupoid convolution algebra of the delooping groupoid BG\mathbf{B}G of a finite group GG naturally has the structure of a Hopf algebra, the twisted groupoid convolution algebra of BG\mathbf{B}G equipped with a 3-cocycle c:BGB 3U(1)c \colon \mathbf{B}G \to \mathbf{B}^3 U(1) is naturally a quasi-Hopf algebra. Since such a 3-cocycle is precisely the background gauge field of the 3d TFT called Dijkgraaf-Witten theory, and hence quasi-Hopf algebras arise there (Dijkgraaf-Pasquier-Roche 91).

Definition (Drinfeld)

A quasibialgebra is a unital associative algebra (A,m,η)(A,m,\eta) with a structure of not necessarily coassociative coalgebra (A,Δ,ϵ)(A,\Delta,\epsilon), with multiplicative comultiplication Δ\Delta and counit ϵ\epsilon, and an invertible element ϕAAA\phi \in A\otimes A\otimes A such that

(i) the coassociativity is modified by conjugation by ϕ\phi in the sense

(Δ1)Δ(a)=ϕ((1Δ)Δ(a))ϕ 1,aA, (\Delta \otimes 1)\Delta(a) = \phi\left((1\otimes\Delta)\Delta(a)\right)\phi^{-1},\,\,\,\,\,\forall a\in A,

(ii) the following pentagon identity holds

(11Δ)(ϕ)(Δ11)(ϕ)=(1ϕ)(1Δ1)(ϕ)(ϕ1) (1\otimes 1\otimes\Delta)(\phi)(\Delta\otimes 1\otimes 1)(\phi) = (1\otimes\phi)(1\otimes\Delta\otimes 1)(\phi)(\phi\otimes 1)

(iii) some identities involving unit η\eta and counit ϵ\epsilon hold:

(ϵA)Δ(a)=a=(Aϵ)Δ(a),aA; (\epsilon\otimes A)\Delta(a) = a = (A\otimes\epsilon)\Delta(a), \,\,\,\,\,\,a\in A;
(AϵA)ϕ=1. (A\otimes\epsilon\otimes A)\phi = 1.

It follows that (ϵAA)ϕ=1=(AAϵ)ϕ(\epsilon\otimes A\otimes A)\phi = 1 = (A\otimes A\otimes\epsilon)\phi.

The category of left AA-modules is a monoidal category, namely the coproduct is used to define the action of AA on the tensor product of modules (M,ν M)(M,\nu^M), (N,ν N)(N,\nu^N):

A(MN)ΔMN(AA)(MN)(AM)(AN)ν Mν NMN A \otimes (M\otimes N) \stackrel{\Delta\otimes M\otimes N}\longrightarrow (A\otimes A)\otimes(M\otimes N) \rightarrow (A\otimes M)\otimes (A\otimes N)\stackrel{\nu_M\otimes\nu_N}\longrightarrow M\otimes N

Using the Sweedler-like notation ϕ=ϕ 1ϕ 2ϕ 3\phi = \sum \phi^1\otimes \phi^2\otimes \phi^3, formulas

Φ M,N,P:(MN)PM(NP) \Phi_{M,N,P}: (M\otimes N)\otimes P\stackrel\cong\longrightarrow M\otimes (N\otimes P)
(mn)p(ϕ 1m)((ϕ 2n)(ϕ 3p)) (m\otimes n)\otimes p\mapsto \sum (\phi^1\triangleright m) \otimes ((\phi^2\triangleright n)\otimes (\phi^3\triangleright p))

define a natural transformation Φ\Phi and the pentagon for ϕ\phi yields the MacLane's pentagon for Φ\Phi understood as a new associator,

(MΦ N,P,Q)Φ M,NP,Q(Φ M,N,PQ)=Φ M,N,PQΦ MN,P,Q (M\otimes\Phi_{N,P,Q})\Phi_{M,N\otimes P,Q}(\Phi_{M,N,P}\otimes Q)=\Phi_{M,N,P\otimes Q}\Phi_{M\otimes N,P,Q}

For this reason, ϕ\phi is sometimes called the associator of the quasibialgebra. While it is due to Drinfeld, another variant of it, written as a formal power series and used in knot theory is often called the Drinfeld associator (see there).

A quasi-Hopf algebra is a quasibialgebra (A,Δ,ε,ϕ)(A, \Delta, \varepsilon, \phi) equipped with elements α,βA\alpha,\beta \in A and an antiautomorhphism SS of AA (a suitable kind of antipode) such that:

iS(b i)αc i=ε(a)α, ib iβS(c i)=ε(a)β\sum_i S(b_i)\alpha c_i = \varepsilon (a) \alpha, \sum_i b_i\beta S(c_i) = \varepsilon(a)\beta

for aAa \in A with Δ(a)= ib ic i\Delta(a) = \sum_i b_i \otimes c_i in Sweedler notation. Further we require:

iX iβS(Y i)αZ i=1,where iX iY iZ i=ϕ,\sum_i X_i\beta S(Y_i)\alpha Z_i = 1, \quad where \sum_i X_i \otimes Y_i\otimes Z_i = \phi,
jS(P j)αQ jβS(R j)=1,where jP jQ jR j=ϕ 1.\sum_j S(P_j)\alpha Q_j\beta S(R_j) =1, \quad where \sum_j P_j \otimes Q_j \otimes R_j = \phi^{-1}.

Twisting quasibialgebras by 2-cochains

The associator ϕ\phi is a counital 3-cocycle in the sense of bialgebra cohomology theory of Majid. The 3-cocycle condition is the pentagon for ϕ\phi. The abelian cohomology would add a coboundary of 2-cochain to get a cohomologous 3-cocycle. In nonabelian case, however, the twist by an invertible 2-cochain is done in a nonabelian way, described by Drinfeld and generalized by Majid to nn-cochains.

Thus, for a bialgebra AA, and fixed nn, the ii-th coface

i=id A (i1)Δid A (ni):A nA (n+1), \partial^i = id_{A^{\otimes (i-1)}}\otimes \Delta \otimes \id_{A^{\otimes (n-i)}} : A^{\otimes n}\to A^{\otimes (n+1)},

for 1in1\leq i\leq n, and 0=1id A n\partial^0 = 1\otimes id_{A^{\otimes n}}, n+1=id A n1\partial^{n+1} = id_{A^{\otimes n}}\otimes 1. For FA nF\in A^{\otimes n}, Majid defines

+F= ieven( iF), F= iodd( iF), \partial^+ F = \prod_{i\,\,\,\,even} (\partial^i F),\,\,\,\,\,\partial^- F = \prod_{i\,\,\,\,odd} (\partial^i F),

where the products are in the order of ascending ii. If FA nF\in A^{\otimes n} is a cochain then its coboundary is δF=( +F)( F 1)\delta F = (\partial^+ F)(\partial^- F^{-1}), which is automatically an (n+1)(n+1)-cochain. If FA nF \in A^{\otimes n} is an nn-cochain and ϕA (n+1)\phi\in A^{\otimes (n+1)} is an (n+1)(n+1)-cochain then one defines a cochain twist ϕ F\phi^F of ϕ\phi by FF by the formula

ϕ F=( +F)ϕ( F 1). \phi^F = (\partial^+ F)\phi(\partial^- F^{-1}).

Drinfeld proved that for n=2n=2 the following is true. Given a quasiabialgebra A=(A,m,η,Δ,ϵ,ϕ)A = (A,m,\eta,\Delta,\epsilon,\phi) and a 2-cochain FF, the data A F=(A,m,η,FΔ()F 1,ϵ,ϕ F)A^F = (A,m,\eta,F\Delta(-)F^{-1},\epsilon,\phi^F) is also a quasibialgebra. Furthermore, categories of modules AmodA-mod and A FmodA^F-mod are monoidally equivalent reflecting the idea that cohomologous cocycles lead to nonessential categorical effects. If (A,R)(A,R) is quasitriangular quasibialgebra then we can twist the R-element RHHR\in H\otimes H to R F=F 21RFR^F = F_{21} R F to obtain quasitriangular quasibialgebra (A F,R F)(A^F,R^F) and their braided monoidal categories of representations are braided monoidally equivalent.


The notion was introduced in

The relation to Dijkgraaf-Witten theory appeared in

and some arguments about the general relevance of quasi-Hopf algebras is in

Recently a monograph appeared

Wikipedia article: Quasi-Hopf algebra

Other articles include