Quasi-Hopf algebra

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 formula_1 for which there exist formula_2 and a bijective antihomomorphism "S" (antipode) of formula_3 such that
for all formula_6 and where 
and
where the expansions for the quantities formula_10and formula_11 are given by 
and
As for a quasi-bialgebra, the property of being quasi-Hopf is preserved under twisting. 
Usage.
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.