# 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 $\Phi$ 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 $\mathcal{B_A} = (\mathcal{A}, \Delta, \varepsilon, \Phi,l,r)$ is an algebra $\mathcal{A}$ over a field $\mathbb{F}$ equipped with morphisms of algebras

$\Delta : \mathcal{A} \rightarrow \mathcal{A \otimes A}$
$\varepsilon : \mathcal{A} \rightarrow \mathbb{F}$

along with invertible elements $\Phi \in \mathcal{A \otimes A \otimes A}$, and $r,l \in A$ such that the following identities hold:

$(id \otimes \Delta) \circ \Delta(a) = \Phi \lbrack (\Delta \otimes id) \circ \Delta (a) \rbrack \Phi^{-1}, \quad \forall a \in \mathcal{A}$
$\lbrack (id \otimes id \otimes \Delta)(\Phi) \rbrack \ \lbrack (\Delta \otimes id \otimes id)(\Phi) \rbrack = (1 \otimes \Phi) \ \lbrack (id \otimes \Delta \otimes id)(\Phi) \rbrack \ (\Phi \otimes 1)$
$(\varepsilon \otimes id)(\Delta a) = l^{-1} a l, \qquad (id \otimes \varepsilon) \circ \Delta = r^{-1} a r, \quad \forall a \in \mathcal{A}$
$(id \otimes \varepsilon \otimes id)(\Phi) = 1 \otimes 1.$

Where $\Delta$ and $\epsilon$ are called the comultiplication and counit, $r$ and $l$ are called the right and left unit constraints (resp.), and $\Phi$ is sometimes called the Drinfeld associator[1]. This definition is constructed so that the category $\mathcal{A}-Mod$ 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[2]. Since many of the quasi-bialgebras that appear "in nature" have trivial unit constraints, ie. $l=r=1$ the definition may sometimes be given with this assumed[3]. Note that a bialgebra is just a quasi-bialgebra with trivial unit and associativity constraints: $l=r=1$ and $\Phi=1 \otimes 1 \otimes 1$.

## Braided Quasi-Bialgebras

A braided quasi-bialgebra (also called a quasi-triangular quasi-bialgebra) is a quasi-bialgebra whose corresponding tensor category $\mathcal{A}-Mod$ 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 $(\mathcal{A},\Delta,\epsilon,\Phi,l,r)$ is braided if it has a universal R-matrix, ie an invertible element $R \in \mathcal{A \otimes A}$ such that the following 3 identities hold:

$(\Delta^{op})(a)=R \Delta(a) R^{-1}$
$(id \otimes \Delta)(R)=(\Phi_{231})^{-1} R_{13} \Phi_{213} R_{12} (\Phi_{213})^{-1}$
$(\Delta \otimes id)(R)=(\Phi_{321}) R_{13} (\Phi_{213})^{-1} R_{23} \Phi_{123}$

Where, for every $a_1 \otimes ... \otimes a_k \in \mathcal{A}^{\otimes k}$, $a_{i_1 i_2 ... i_n}$ is the monomial with $a_j$ in the $i_j$th spot, where any omitted numbers correspond to the identity in that spot. Finally we extend this by linearity to all of $\mathcal{A}^{\otimes k}$[4].

Again, similar to the braided bialgebra case, this universal R-matrix satisfies (a non-associative version of) the Yang-Baxter equation:

$R_{12}\Phi_{321}R_{13}(\Phi_{132})^{-1}R_{23}\Phi_{123}=\Phi_{321}R_{23}(\Phi_{231})^{-1}R_{13}\Phi_{213}R_{12}$[5]

## Twisting

Given a quasi-bialgebra, further quasi-bialgebras can be generated by twisting (from now on we will assume $r=l=1$) .

If $\mathcal{B_A}$ is a quasi-bialgebra and $F \in \mathcal{A \otimes A}$ is an invertible element such that $(\varepsilon \otimes id) F = (id \otimes \varepsilon) F = 1$, set

$\Delta ' (a) = F \Delta (a) F^{-1}, \quad \forall a \in \mathcal{A}$
$\Phi ' = (1 \otimes F) \ ((id \otimes \Delta) F) \ \Phi \ ((\Delta \otimes id)F^{-1}) \ (F^{-1} \otimes 1).$

Then, the set $(\mathcal{A}, \Delta ' , \varepsilon, \Phi ')$ is also a quasi-bialgebra obtained by twisting $\mathcal{B_A}$ by F, which is called a twist or gauge transformation[6]. If $(\mathcal{A},\Delta,\varepsilon, \Phi)$ was a braided quasi-bialgebra with universal R-matrix $R$ , then so is $(\mathcal{A},\Delta',\varepsilon, \Phi ')$ with universal R-matrix $F_{21}RF^{-1}$ (using the notation from the above section)[7]. However, the twist of a bialgebra is only in general a quasi-bialgebra. Twistings fulfill many expected properties. For example, twisting by $F_1$ and then $F_2$ is equivalent to twisting by $F_2F_1$, and twisting by $F$ then $F^{-1}$ recovers the original quasi-bialgebra.

Twistings have the important property that they induce categorical equivalences on the tensor category of modules:

Theorem: Let $\mathcal{B_A}$, $\mathcal{B_{A'}}$ be quasi-bialgebras, let $\mathcal{B'_{A'}}$ be the twisting of $\mathcal{B_{A'}}$ by $F$, and let there exist an isomorphism: $\alpha:\mathcal{B_A} \to \mathcal{B'_{A'}}$. Then the induced tensor functor $(\alpha^*,id,\phi_2^F)$ is a tensor category equivalence between $\mathcal{A'}-mod$ and $\mathcal{A}-mod$. Where $\phi_2^F(v \otimes w)=F^{-1}(v \otimes w)$. Moreover, if $\alpha$ is an isomorphism of braided quasi-bialgebras, then the above induced functor is a braided tensor category equivalence[8].

## 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.

## References

1. ^ C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Pg. 369 Springer-Verlag. ISBN 0387943706
2. ^ C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Pg. 368 Springer-Verlag. ISBN 0387943706
3. ^ C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Pg. 370 Springer-Verlag. ISBN 0387943706
4. ^ C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Pg. 371 Springer-Verlag. ISBN 0387943706
5. ^ C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Pg. 372 Springer-Verlag. ISBN 0387943706
6. ^ C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Pg. 373 Springer-Verlag. ISBN 0387943706
7. ^ C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Pg. 376 Springer-Verlag. ISBN 0387943706
8. ^ C. Kassel. "Quantum Groups". Graduate Texts in Mathematics Pg. 375,376 Springer-Verlag. ISBN 0387943706