nLab quasi-Hopf algebra

Idea

General

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 $\mathbf{B}G$ of a finite group $G$ naturally has the structure of a Hopf algebra, the twisted groupoid convolution algebra of $\mathbf{B}G$ equipped with a 3-cocycle $c \colon \mathbf{B}G \to \mathbf{B}^3 U(1)$ 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 $(A,m,\eta)$ with a structure of not necessarily coassociative coalgebra $(A,\Delta,\epsilon)$, with multiplicative comultiplication $\Delta$ and counit $\epsilon$, and an invertible element $\phi \in A\otimes A\otimes A$ such that

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

$$(\Delta \otimes 1)\Delta(a) = \phi\left((1\otimes\Delta)\Delta(a)\right)\phi^{-1},\,\,\,\,\,\forall a\in A,$$

(ii) the following pentagon identity holds

$$(1\otimes 1\otimes\Delta)(\phi)(\Delta\otimes 1\otimes 1)(\phi) = (1\otimes\phi)(1\otimes\Delta\otimes 1)(\phi)(\phi\otimes 1)$$

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

$$(\epsilon\otimes A)\Delta(a) = a = (A\otimes\epsilon)\Delta(a), \,\,\,\,\,\,a\in A;$$

$$(A\otimes\epsilon\otimes A)\phi = 1.$$

It follows that $(\epsilon\otimes A\otimes A)\phi = 1 = (A\otimes A\otimes\epsilon)\phi$.

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

$$A \otimes (M\otimes N) \stackrel{\Delta\otimes M\otimes N}\longrightarrow (A\otimes A)\otimes(M\otimes N) \rightarrow (A\otimes M)\otimes (A\otimes N)\stackrel{\nu_M\otimes\nu_N}\longrightarrow M\otimes N$$

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

$$\Phi_{M,N,P}: (M\otimes N)\otimes P\stackrel\cong\longrightarrow M\otimes (N\otimes P)$$

$$(m\otimes n)\otimes p\mapsto \sum (\phi^1\triangleright m) \otimes ((\phi^2\triangleright n)\otimes (\phi^3\triangleright p))$$

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

$$(M\otimes\Phi_{N,P,Q})\Phi_{M,N\otimes P,Q}(\Phi_{M,N,P}\otimes Q)=\Phi_{M,N,P\otimes Q}\Phi_{M\otimes N,P,Q}$$

For this reason, $\phi$ 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 $(A, \Delta, \varepsilon, \phi)$ equipped with elements $\alpha,\beta \in A$ and an antiautomorhphism $S$ of $A$ (a suitable kind of antipode) such that:

$$\sum_i S(b_i)\alpha c_i = \varepsilon (a) \alpha, \sum_i b_i\beta S(c_i) = \varepsilon(a)\beta$$

for $a \in A$ with $\Delta(a) = \sum_i b_i \otimes c_i$ in Sweedler notation. Further we require:

$$\sum_i X_i\beta S(Y_i)\alpha Z_i = 1, \quad where \sum_i X_i \otimes Y_i\otimes Z_i = \phi,$$

$$\sum_j S(P_j)\alpha Q_j\beta S(R_j) =1, \quad where \sum_j P_j \otimes Q_j \otimes R_j = \phi^{-1}.$$

Twisting quasibialgebras by 2-cochains

The associator $\phi$ is a counital 3-cocycle in the sense of bialgebra cohomology theory of Majid. The 3-cocycle condition is the pentagon for $\phi$. 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 $n$-cochains.

Thus, for a bialgebra $A$, and fixed $n$, the $i$-th coface

$$\partial^i = id_{A^{\otimes (i-1)}}\otimes \Delta \otimes \id_{A^{\otimes (n-i)}} : A^{\otimes n}\to A^{\otimes (n+1)},$$

for $1\leq i\leq n$, and $\partial^0 = 1\otimes id_{A^{\otimes n}}$, $\partial^{n+1} = id_{A^{\otimes n}}\otimes 1$. For $F\in A^{\otimes n}$, Majid defines

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

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

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

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

Properties

Quasi-fiber functor

Recall from Tannaka duality that given a rigid monoidal category $C$ with a fiber functor $F: C\to RMod$ one can reconstruct a Hopf algebra $H$ via $H=End(F)$, so that in particular $C\cong Rep(H)$. While categories of the form $Rep(H_q )$ for $H_q$ a quasi-Hopf algebra do not admit a fiber functor unless $H_q$ is furthermore a Hopf algebra, they do admit a weaker notion called a quasi-fiber functor $F_q$.

Much as a fiber functor, a quasi-fiber functor $F_q$ is an exact, faithful functor equipped with a natural transformation

$$\mu: F(-)\otimes F(-)\to F(-\otimes -)$$

The main difference is that $\mu$ is not required to satisfy the associativity condition, so that $(F,\mu)$ does not describe a monoidal functor. The failure of $\mu$ to satisfy the associative condition is a direct reflection of the non-coassociativity of $H_q$. See Sections 5.1, 5.11, and 5.12 in EGNO 2016 for more.

Related concepts

• hopfish algebra

• quasitriangulated quasi-Hopf algebra

References

The notion was introduced in

• Vladimir Drinfel'd, Квазихопфовы алгебры, Algebra i Analiz 1 (1989), no. 6, 114–148, pdf; translation Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), no. 6, 1419–1457 MR1047964

The relation to Dijkgraaf-Witten theory appeared in

• Robbert Dijkgraaf, V. Pasquier, P. Roche, QuasiHopf algebras, group cohomology and orbifold models, Nucl. Phys. B Proc. Suppl. 18B (1990), 60-72; Quasi-quantum groups related to orbifold models, Modern quantum field theory (Bombay, 1990), 375–383, World Sci. 1991

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

• Gerhard Mack, Volker Schomerus, Quasi Hopf quantum symmetry in quantum theory, Nuclear Physics B 370:1 (1992) 185–230 doi

Recently a monograph appeared

• Daniel Bulacu, Stefaan Caenepeel, Florin Panaite, Freddy Van Oystaeyen, Quasi-Hopf algebras: a categorical approach, 544 pp., EMA 174 (2019)

Wikipedia article: Quasi-Hopf algebra

Other articles include

• В. Г. Дринфельд, О структуре квазитреугольных квазихопфовых алгебр, Функц. анализ и его прил. 26:1 (1992), 78–80, pdf; transl. V. G. Drinfeld, Structure of quasitriangular quasi-hopf algebras, Funct. Anal. Appl., 26:1 (1992), 63–65

• V. G. Drinfelʹd, О квазитреугольных квазихопфовых алгебрах и одной группе, тесно связанной с $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf {Q})$, Algebra i Analiz 2 (1990), no. 4, 149–181, pdf; translation On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf {Q})$, Leningrad Math. J. 2 (1991), no. 4, 829–860, MR1080203

• V. G. Drinfelʹd, Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations, Problems of modern quantum field theory (Alushta, 1989), 1–13, Res. Rep. Phys., Springer 1989.

• Shahn Majid, Quantum double for quasi-Hopf algebras, Lett. Math. Phys. 45 (1998), no. 1, 1–9, MR2000b:16077, doi, q-alg/9701002

• Peter Schauenburg, Hopf modules and the double of a quasi-Hopf algebra, Trans. Amer. Math. Soc. 354 (2002), 3349–3378 pdf

• M. Jimbo, H. Konno, S. Odake, J. Shiraishi, Quasi-Hopf twistors for elliptic quantum groups, Transformation Groups 4(4), 303–327 (1999) doi

• Ivan Kobyzev, Ilya Shapiro, A categorical approach to cyclic cohomology of quasi-Hopf algebras and Hopf algebroids, Applied Categorical Structures, 27:1 (2019) 85–109 doi

• L Frappat, D Issing, E Ragoucy, The quantum determinant of the elliptic algebra $\mathcal{A}_{q, p}(\widehat{gl}_N)$, J. Phys. A51:44, doi

See also Chapter 5, Sections 1, 11, and 12 of:

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik. Tensor categories. Vol. 205. American Mathematical Soc., 2016.

A complete classification of quasi-Hopf algebras of dimension 6:

• Daniel Bulacu, Matteo Misurati. Quasi-Hopf Algebras of Dimension $6$ (2024). (arXiv:2410.03476).