symmetric monoidal (∞,1)-category of spectra
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 a Drinfeld associator, i.e. a nonabelian bialgebra 3-cocycle.
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).
A quasibialgebra is a unital associative algebra $(A,m,\eta)$ with a structure of not necessarily coassociative coalgebra $(A,\Delta,\epsilon)$ and an invertible element $\phi \in A\otimes A\otimes A$ such that
and some identities involving unit $\eta$ and counit $\epsilon$ hold.
A quasi-Hopf algebra is a quasibialgebra with a suitable notion of an antipode.
The notion was introduced in
The relation to Dijkgraaf-Witten theory appeared in
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