# 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:
    $(id\otimes \Delta )\circ \Delta (a)=\Phi \lbrack (\Delta \otimes id)\circ \Delta (a)\rbrack \Phi ^{-1},\quad \forall a\in {\mathcal {A}}$
    $\lbrack (id\otimes id\otimes \Delta )(\Phi )\rbrack \ \lbrack (\Delta \otimes id\otimes id)(\Phi )\rbrack =(1\otimes \Phi )\ \lbrack (id\otimes \Delta \otimes id)(\Phi )\rbrack \ (\Phi \otimes 1)$
    $(\varepsilon \otimes id)(\Delta a)=l^{-1}al,\qquad (id\otimes \varepsilon )\circ \Delta =r^{-1}ar,\quad \forall a\in {\mathcal {A}}$
    $(id\otimes \varepsilon \otimes id)(\Phi )=r\otimes l^{-1}.$
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.$^{:369–376}$ 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.$^{:368}$ 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.$^{:370}$ 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 ${\mathcal {A}}^{\otimes k}$.$^{:371}$
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}$$^{:372}$

## 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.$^{:373}$ 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).$^{:376}$ 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 $F_{2}F_{1}$, 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.$^{:375–376}$

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