Quasi-bialgebra
In mathematics, quasi-bialgebras are a generalization of bialgebras: they were first defined by the Ukrainian mathematician Vladimir Drinfeld in 1990. A quasi-bialgebra differs from a bialgebra by having coassociativity replaced by an invertible element formula_1 which controls the non-coassociativity. One of their key properties is that the corresponding category of modules forms a tensor category. Definition. A quasi-bialgebra formula_2 is an algebra formula_3 over a field formula_4 equipped with morphisms of algebras along with invertible elements formula_7, and formula_8 such that the following identities hold: Where formula_13 and formula_14 are called the comultiplication and  counit, formula_15 and formula_16 are called the right and left unit constraints (resp.), and formula_17 is sometimes called the "Drinfeld associator". This definition is constructed so that the category formula_18 is a tensor category under the usual vector space tensor product, and in fact this can be taken as the definition instead of the list of above identities. Since many of the quasi-bialgebras that appear "in nature" have trivial unit constraints, ie. formula_19 the definition may sometimes be given with this assumed. Note that a bialgebra is just a quasi-bialgebra with trivial unit and associativity constraints: formula_19 and  formula_21. Braided quasi-bialgebras. A "braided quasi-bialgebra" (also called a "quasi-triangular quasi-bialgebra") is a quasi-bialgebra whose corresponding tensor category formula_18 is braided. Equivalently, by analogy with braided bialgebras, we can construct a notion of a "universal R-matrix" which controls the non-cocommutativity of a quasi-bialgebra. The definition is the same as in the braided bialgebra case except for additional complications in the formulas caused by adding in the associator. Proposition: A quasi-bialgebra formula_23 is braided if it has a "universal R-matrix", ie an invertible element formula_24 such that the following 3 identities  hold: Where, for every formula_28, formula_29 is the monomial with formula_30 in the formula_31th spot, where any omitted numbers correspond to the identity in that spot. Finally we extend this by linearity to all of formula_32. Again, similar to the braided bialgebra case, this universal R-matrix satisfies (a non-associative version of) the Yang–Baxter equation: Twisting. Given a quasi-bialgebra, further quasi-bialgebras can be generated by twisting (from now on we will assume formula_34) . If formula_35 is a quasi-bialgebra and formula_36 is an invertible element such that formula_37, set Then, the set formula_40 is also a  quasi-bialgebra obtained by twisting formula_41 by "F", which is called a "twist" or "gauge transformation". If formula_42 was a braided quasi-bialgebra with universal R-matrix formula_43 , then so is formula_44 with universal R-matrix formula_45 (using the notation from the above section). However, the twist of a bialgebra is only in general a quasi-bialgebra. Twistings fulfill many expected properties. For example, twisting by formula_46 and then formula_47 is equivalent to twisting by formula_48, and twisting by formula_49 then formula_50 recovers the original quasi-bialgebra. Twistings have the important property that they induce categorical equivalences on the tensor  category of modules: Theorem: Let formula_35, formula_52 be quasi-bialgebras, let formula_53 be the twisting of formula_52 by formula_49, and let there exist an isomorphism: formula_56. Then the induced tensor functor formula_57 is a tensor category equivalence between formula_58 and formula_59. Where formula_60. Moreover, if formula_61 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.