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\section*{quasi-Hopf algebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{motivation_from_quantum_field_theory}{Motivation from quantum field theory}\dotfill \pageref*{motivation_from_quantum_field_theory} \linebreak
\noindent\hyperlink{definition_drinfeld}{Definition (Drinfeld)}\dotfill \pageref*{definition_drinfeld} \linebreak
\noindent\hyperlink{twisting_quasibialgebras_by_2cochains}{Twisting quasibialgebras by 2-cochains}\dotfill \pageref*{twisting_quasibialgebras_by_2cochains} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The notion of a quasibialgebra generalizes this of a [[bialgebra]][[Hopf algebra]] by introducing a nontrivial [[associativity]] [[coherence]] (\hyperlink{Drinfeld89}{Drinfeld 89}) isomorphisms (representable by multiplication with an element in triple tensor product) into axioms; a \emph{quasi-Hopf algebra} is a quasibialgebra 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 by twisting by a [[Drinfeld associator]], i.e. a nonabelian [[bialgebra cocycle|bialgebra 3-cocycle]].

\hypertarget{motivation_from_quantum_field_theory}{}\subsubsection*{{Motivation from quantum field theory}}\label{motivation_from_quantum_field_theory}

Drinfel'd was motivated by study of [[monoidal categories]] in [[rational CFT|rational]] 2d [[conformal field theory]] (RCFT) as well as by an idea from [[Grothendieck]]`s \emph{[[Esquisse d'un programme|Esquisse]]} namely the [[Grothendieck-Teichmüller tower]] and its modular properties. In RCFT, the [[monoidal categories]] appearing can be, by [[Tannaka duality|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 (\hyperlink{DijkgraafPasquierRoche}{Dijkgraaf-Pasquier-Roche 91}).

\hypertarget{definition_drinfeld}{}\subsection*{{Definition (Drinfeld)}}\label{definition_drinfeld}

A \textbf{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

\begin{displaymath}
(\Delta \otimes 1)\Delta(a) = \phi\left((1\otimes\Delta)\Delta(a)\right)\phi^{-1},\,\,\,\,\,\forall a\in A,
\end{displaymath}
(ii) the following pentagon identity holds

\begin{displaymath}
(1\otimes 1\otimes\Delta)(\phi)(\Delta\otimes 1\otimes 1)(\phi) =
(1\otimes\phi)(1\otimes\Delta\otimes 1)(\phi)(\phi\otimes 1)
\end{displaymath}
(iii) some identities involving unit $\eta$ and counit $\epsilon$ hold:

\begin{displaymath}
(\epsilon\otimes A)\Delta(a) = a = (A\otimes\epsilon)\Delta(a), \,\,\,\,\,\,a\in A;
\end{displaymath}
\begin{displaymath}
(A\otimes\epsilon\otimes A)\phi = 1.
\end{displaymath}
It follows that $(\epsilon\otimes\epsilon\otimes A)\phi
 = 1 = (A\otimes\epsilon\otimes\epsilon)\phi$.

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 $(M,\nu^M)$, $(N,\nu^N)$:

\begin{displaymath}
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
\end{displaymath}
Using the Sweedler-like notation $\phi = \sum \phi^1\otimes \phi^2\otimes \phi^3$, formulas

\begin{displaymath}
\Phi_{M,N,P}: (M\otimes N)\otimes P\stackrel\cong\longrightarrow M\otimes (N\otimes P)
\end{displaymath}
\begin{displaymath}
(m\otimes n)\otimes p\mapsto \sum (\phi^1\triangleright m)
\otimes ((\phi^2\triangleright n)\otimes (\phi^3\triangleright p))
\end{displaymath}
define a natural transformation $\Phi$ and the pentagon for $\phi$ yields the MacLane's pentagon for $\Phi$ understood as a new associator,

\begin{displaymath}
(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}
\end{displaymath}
For this reason, $\phi$ is sometimes called the associator of the quasibialgebra. While it is due 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 \textbf{quasi-Hopf algebra} is a quasibialgebra with a suitable notion of antipode.

\hypertarget{twisting_quasibialgebras_by_2cochains}{}\subsubsection*{{Twisting quasibialgebras by 2-cochains}}\label{twisting_quasibialgebras_by_2cochains}

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

\begin{displaymath}
\partial^i  = id_{A^{\otimes (i-1)}}\otimes \Delta \otimes \id_{A^{\otimes (n-i)}} : A^{\otimes n}\to A^{\otimes (n+1)},
\end{displaymath}
for $1\leq i\leq n$, and $\partial^0 = 1\otimes id_{A^{\otimes n}}$, $\partial^n = 
id_{A^{\otimes n}}\otimes 1$. For $F\in H^{\otimes n}$, Majid defines

\begin{displaymath}
\partial^+ F = \prod_{i\,\,\,\,even} (\partial^i F),\,\,\,\,\,\partial^- F = \prod_{i\,\,\,\,odd} (\partial^i F),
\end{displaymath}
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 $Fin A^{\otimes n}$ 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

\begin{displaymath}
\phi^F = (\partial^+ F)\phi(\partial^- F^{-1}).
\end{displaymath}
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 representations $A-mod$ 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.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[hopfish algebra]]


\item [[quasitriangulated quasi-Hopf algebra]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The notion was introduced in

\begin{itemize}%
\item [[Vladimir Drinfel'd]], \emph{ }, Algebra i Analiz \textbf{1} (1989), no. 6, 114--148, \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=aa&paperid=53&what=fullt&option_lang=rus}{pdf}; translation \emph{Quasi-Hopf algebras}, Leningrad Math. J. \textbf{1} (1990), no. 6, 1419--1457 \href{http://www.ams.org/mathscinet-getitem?mr=1047964}{MR1047964}

\end{itemize}
The relation to [[Dijkgraaf-Witten theory]] appeared in

\begin{itemize}%
\item [[Robbert Dijkgraaf]], V. Pasquier, P. Roche, \emph{QuasiHopf algebras, group cohomology and orbifold models}, Nucl. Phys. B Proc. Suppl. \textbf{18B} (1990), 60-72; \emph{Quasi-quantum groups related to orbifold models}, Modern quantum field theory (Bombay, 1990), 375--383, World Sci. 1991

\end{itemize}
and some arguments about the general relevance of quasi-Hopf algebras is in

\begin{itemize}%
\item Gerhard Mack, [[Volker Schomerus]], \emph{Quasi Hopf quantum symmetry in quantum theory}, Nuclear Physics B 370:1 (1992) 185--230 

\end{itemize}
Recently a monograph appeared

\begin{itemize}%
\item Daniel Bulacu, Stefaan Caenepeel, Florin Panaite, Freddy Van Oystaeyen, \emph{Quasi-Hopf algebras: a categorical approach}, 544 pp., EMA 174 (2019)

\end{itemize}
Wikipedia article: \href{https://en.wikipedia.org/wiki/Quasi-Hopf_algebra}{Quasi-Hopf algebra}

Other articles include

\begin{itemize}%
\item . . , \emph{    }, .    . \textbf{26}:1 (1992), 78--80, \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=768&what=fullt&option_lang=rus}{pdf}; transl. V. G. Drinfeld, \emph{Structure of quasitriangular quasi-hopf algebras}, Funct. Anal. Appl., 26:1 (1992), 63--65


\item V. G. Drinfeld, \emph{      ,    $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf {Q})$}, Algebra i Analiz 2 (1990), no. 4, 149--181, \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=aa&paperid=199&volume=2&year=1990&issue=4&fpage=149&what=fullt&option_lang=eng}{pdf}; translation \emph{On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf {Q})$}, Leningrad Math. J. \textbf{2} (1991), no. 4, 829--860, \href{http://www.ams.org/mathscinet-getitem?mr=1080203}{MR1080203}


\item V. G. Drinfeld, \emph{Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations}, Problems of modern quantum field theory (Alushta, 1989), 1--13, Res. Rep. Phys., Springer 1989.


\item [[Shahn Majid]], \emph{Quantum double for quasi-Hopf algebras}, Lett. Math. Phys. \textbf{45} (1998), no. 1, 1--9, \href{http://www.ams.org/mathscinet-getitem?mr=1631648}{MR2000b:16077}, \href{http://dx.doi.org/10.1023/A:1007450123281}{doi}, \href{http://arxiv.org/abs/q-alg/9701002}{q-alg/9701002}


\item Peter Schauenburg, \emph{Hopf modules and the double of a quasi-Hopf algebra}, Trans. Amer. Math. Soc. \textbf{354} (2002), 3349-3378 \href{http://www.ams.org/journals/tran/2002-354-08/S0002-9947-02-02980-X/S0002-9947-02-02980-X.pdf}{pdf}


\item M. Jimbo, H. Konno, S. Odake, J. Shiraishi, \emph{Quasi-Hopf twistors for elliptic quantum groups}, Transformation Groups 4(4), 303–327 (1999) \href{http://sci-hub.tw/10.1007/BF01238562}{doi}


\item Ivan Kobyzev, Ilya Shapiro, \emph{A categorical approach to cyclic cohomology of quasi-Hopf algebras and Hopf algebroids}, Applied Categorical Structures, \textbf{27}:1 (2019) 85–109 \href{https://doi.org/10.1007/s10485-018-9544-0}{doi}


\item L Frappat, D Issing, E Ragoucy, \emph{The quantum determinant of the elliptic algebra $\mathcal{A}_{q, p}(\widehat{gl}_N)$}, J. Phys. \textbf{A51}:44, \href{https://doi.org/10.1088/1751-8121/aae296}{doi}



\end{itemize}
[[!redirects quasibialgebra]] [[!redirects quasihopf algebra]] [[!redirects quasiHopf algebra]] [[!redirects quasi-bialgebra]] [[!redirects quasi-Hopf algebras]]



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