in general a quasi-bialgebra. Twistings fulfill many expected properties. For example, twisting by and then is equivalent to twisting by , and twisting by then recovers the original quasi-bialgebra. Twistings have the important property that they induce categorical equivalences on the tensor category of modules: Theorem: Let , be quasi-bialgebras, let be the twisting of by , and let there exist an isomorphism: . Then the induced tensor functor is a tensor category equivalence between and . Where . Moreover, if is an isomorphism of braided quasi-bialgebras, then the above induced functor is a braided tensor category equivalence. Usage Quasi-bialgebras form the basis of the study of quasi-Hopf algebras and further to 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 XXZ in the framework of the Algebraic Bethe ansatz. See also Bialgebra Hopf algebra Quasi-Hopf algebra References Further reading 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 Coalgebras Non-associative algebras