are given by and 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.
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.