A quasi-Hopf algebra is a generalization of a Hopf algebra, which was defined by the Russian mathematician Vladimir Drinfeld in 1989.

A quasi-Hopf algebra is a quasi-bialgebra ${\displaystyle {\mathcal {B_{A}}}=({\mathcal {A}},\Delta ,\varepsilon ,\Phi )}$ for which there exist ${\displaystyle \alpha ,\beta \in {\mathcal {A}}}$ and a bijective antihomomorphism S (antipode) of ${\displaystyle {\mathcal {A}}}$ such that

${\displaystyle \sum _{i}S(b_{i})\alpha c_{i}=\varepsilon (a)\alpha }$

${\displaystyle \sum _{i}b_{i}\beta S(c_{i})=\varepsilon (a)\beta }$

for all ${\displaystyle a\in {\mathcal {A}}}$ and where

${\displaystyle \Delta (a)=\sum _{i}b_{i}\otimes c_{i}}$

and

${\displaystyle \sum _{i}X_{i}\beta S(Y_{i})\alpha Z_{i}=\mathbb {I} ,}$

${\displaystyle \sum _{j}S(P_{j})\alpha Q_{j}\beta S(R_{j})=\mathbb {I} .}$

where the expansions for the quantities ${\displaystyle \Phi }$and ${\displaystyle \Phi ^{-1}}$ are given by

${\displaystyle \Phi =\sum _{i}X_{i}\otimes Y_{i}\otimes Z_{i}}$

and

${\displaystyle \Phi ^{-1}=\sum _{j}P_{j}\otimes Q_{j}\otimes R_{j}.}$

As for a quasi-bialgebra, the property of being quasi-Hopf is preserved under twisting.

Quasi-Hopf algebras form the basis of the study of Drinfeld twists and the representations in terms of F-matrices associated with finite-dimensional irreducible representations of quantum affine algebra. F-matrices can be used to factorize the corresponding R-matrix. This leads to applications in Statistical mechanics, as quantum affine algebras, and their representations give rise to solutions of the Yang–Baxter equation, a solvability condition for various statistical models, allowing characteristics of the model to be deduced from its corresponding quantum affine algebra. The study of F-matrices has been applied to models such as the Heisenberg XXZ model in the framework of the algebraic Bethe ansatz. It provides a framework for solving two-dimensional integrable models by using the quantum inverse scattering method.

• Quasitriangular Hopf algebra
• Quasi-triangular quasi-Hopf algebra
• Ribbon Hopf algebra

• Vladimir Drinfeld, "Quasi-Hopf algebras", Leningrad Math J. 1 (1989), 1419-1457
• J. M. Maillet and J. Sanchez de Santos, Drinfeld Twists and Algebraic Bethe Ansatz, Amer. Math. Soc. Transl. (2) Vol. 201, 2000