# 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 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*.^{[1]} 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]} Since many of the quasi-bialgebras that appear "in nature" have trivial unit constraints, ie. the definition may sometimes be given with this assumed.^{[1]} 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 .^{[1]}

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*.^{[1]} If was a braided quasi-bialgebra with universal R-matrix , then so is with universal R-matrix (using the notation from the above section).^{[1]} 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.^{[1]}

## 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 Heisenberg XXZ model in the framework of the Algebraic Bethe ansatz.

## See also

## References

- ↑
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## 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