nLab
quasi-Hopf algebra

Idea

General

The notion of a quasi-Hopf algebra generalizes this of a Hopf algebra by weakening the associativity coherence (Drinfeld 89).

In particular, quasi-Hopf algebras may be obtained from ordinary Hopf algebras by twisting by Drinfel’d 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 idaa 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 $BG$ of a finite group $G$ naturally has the structure of a Hopf algebra, the twisted groupoid convolution algebra of $BG$ equipped with a 3-cocycle $c:BG\to B^{3}U(1)$ 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

A quasibialgebra is a unital associative algebra $(A,m,\eta )$ with a structure of not necessarily coassociative coalgebra $(A,\Delta ,ϵ)$ and an invertible element $\varphi \in A\otimes A\otimes A$ such that

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

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

and some identities involving unit $\eta$ and counit $ϵ$ hold.

A quasi-Hopf algebra is a quasibialgebra with a suitable notion of an antipode.

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

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 $\mathrm{Gal}(\overline{Q}/Q)$, 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 $\mathrm{Gal}(\overline{Q}/Q)$, 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