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.
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.:369-376 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.:368 Since many of the quasi-bialgebras that appear "in nature" have trivial unit constraints, ie. the definition may sometimes be given with this assumed.:370 Note that a bialgebra is just a quasi-bialgebra with trivial unit and associativity constraints: and .
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 .:371
Again, similar to the braided bialgebra case, this universal R-matrix satisfies (a non-associative version of) the Yang-Baxter equation:
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.:373 If was a braided quasi-bialgebra with universal R-matrix , then so is with universal R-matrix (using the notation from the above section).:376 However, the twist of a bialgebra is only 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.:375-376
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.
- ^ a b c d e f g h C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Springer-Verlag. ISBN 0387943706
- 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