symmetric monoidal (∞,1)-category of spectra
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.
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 $(A,\Delta,\epsilon)$, 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
(ii) the following pentagon identity holds
(iii) some identities involving unit $\eta$ and counit $\epsilon$ hold:
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 $(M,\nu^M)$, $(N,\nu^N)$:
Using the Sweedler-like notation $\phi = \sum \phi^1\otimes \phi^2\otimes \phi^3$, formulas
define a natural transformation $\Phi$ and the pentagon for $\phi$ yields the MacLane's pentagon for $\Phi$ understood as a new associator,
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:
for $a \in A$ with $\Delta(a) = \sum_i b_i \otimes c_i$ in Sweedler notation. Further we require:
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
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
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
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.
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
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