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 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 is an algebra over a field equipped with morphisms of algebras along with invertible elements , and such that the following identities hold: Where and are called the comultiplication and counit, and are called the right and left unit constraints (resp.), and is sometimes called the Drinfeld associator. This definition is constructed so that the category 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. the definition may sometimes be given with this assumed. Note that a bialgebra is just a quasi-bialgebra with trivial unit and associativity constraints: and . Braided quasi-bialgebras A braided quasi-bialgebra (also called a quasi-triangular quasi-bialgebra) is a quasi-bialgebra whose corresponding tensor category 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 is braided if it has a universal R-matrix, ie an invertible element such that the following 3 identities hold: Where, for every , is the monomial with in the th spot, where any omitted numbers correspond to the identity in that spot. Finally we extend this by linearity to all of . 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 ) . If is a quasi-bialgebra and is an invertible element such that , set Then, the set is also a quasi-bialgebra obtained by twisting by F, which is called a twist or gauge transformation. If was a braided quasi-bialgebra with universal R-matrix , then so is with universal R-matrix (using the notation from the above section). However, the twist of a bialgebra is only