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
One of their key properties is that the corresponding category of modules forms a tensor category.
equipped with morphisms of algebras along with invertible elements
are called the comultiplication and counit,
are called the right and left unit constraints (resp.
is sometimes called the Drinfeld associator.
[1]: 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.
[1]: 368 Since many of the quasi-bialgebras that appear "in nature" have trivial unit constraints, ie.
[1]: 370 Note that a bialgebra is just a quasi-bialgebra with trivial unit and associativity constraints:
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.
is braided if it has a universal R-matrix, ie an invertible element
Finally we extend this by linearity to all of
[1]: 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
is also a quasi-bialgebra obtained by twisting
by F, which is called a twist or gauge transformation.
was a braided quasi-bialgebra with universal R-matrix
[1]: 376 However, the twist of a bialgebra is only in general a quasi-bialgebra.
Twistings fulfill many expected properties.
recovers the original quasi-bialgebra.
Twistings have the important property that they induce categorical equivalences on the tensor category of modules: Theorem: Let
Then the induced tensor functor
is a tensor category equivalence between
is an isomorphism of braided quasi-bialgebras, then the above induced functor is a braided tensor category equivalence.
[1]: 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.