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

${\displaystyle \Delta :{\mathcal {A))\rightarrow {\mathcal {A\otimes A))}$

${\displaystyle \varepsilon :{\mathcal {A))\rightarrow \mathbb {F} }$

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

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

${\displaystyle \lbrack (id\otimes id\otimes \Delta )(\Phi )\rbrack \ \lbrack (\Delta \otimes id\otimes id)(\Phi )\rbrack =(1\otimes \Phi )\ \lbrack (id\otimes \Delta \otimes id)(\Phi )\rbrack \ (\Phi \otimes 1)}$

${\displaystyle (\varepsilon \otimes id)(\Delta a)=l^{-1}al,\qquad (id\otimes \varepsilon )\circ \Delta =r^{-1}ar,\quad \forall a\in {\mathcal {A))}$

${\displaystyle (id\otimes \varepsilon \otimes id)(\Phi )=r\otimes l^{-1}.}$

Where ${\displaystyle \Delta }$ and ${\displaystyle \epsilon }$ are called the comultiplication and counit, ${\displaystyle r}$ and ${\displaystyle l}$ are called the right and left unit constraints (resp.), and ${\displaystyle \Phi }$ is sometimes called the Drinfeld associator.^{[1]: 369–376} This definition is constructed so that the category ${\displaystyle {\mathcal {A))-Mod}$ 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.^[1]: 368 Since many of the quasi-bialgebras that appear "in nature" have trivial unit constraints, ie. ${\displaystyle l=r=1}$ the definition may sometimes be given with this assumed.^[1]: 370 Note that a bialgebra is just a quasi-bialgebra with trivial unit and associativity constraints: ${\displaystyle l=r=1}$ and ${\displaystyle \Phi =1\otimes 1\otimes 1}$.

Braided quasi-bialgebras

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

${\displaystyle (\Delta ^{op})(a)=R\Delta (a)R^{-1))$

${\displaystyle (id\otimes \Delta )(R)=(\Phi _{231})^{-1}R_{13}\Phi _{213}R_{12}(\Phi _{213})^{-1))$

${\displaystyle (\Delta \otimes id)(R)=(\Phi _{321})R_{13}(\Phi _{213})^{-1}R_{23}\Phi _{123))$

Where, for every ${\displaystyle a_{1}\otimes ...\otimes a_{k}\in {\mathcal {A))^{\otimes k))$, ${\displaystyle a_{i_{1}i_{2}...i_{n))}$ is the monomial with ${\displaystyle a_{j))$ in the ${\displaystyle i_{j))$th spot, where any omitted numbers correspond to the identity in that spot. Finally we extend this by linearity to all of ${\displaystyle {\mathcal {A))^{\otimes k))$.^[1]: 371

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

${\displaystyle R_{12}\Phi _{321}R_{13}(\Phi _{132})^{-1}R_{23}\Phi _{123}=\Phi _{321}R_{23}(\Phi _{231})^{-1}R_{13}\Phi _{213}R_{12))$^[1]: 372

Twisting

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

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

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

${\displaystyle \Phi '=(1\otimes F)\ ((id\otimes \Delta )F)\ \Phi \ ((\Delta \otimes id)F^{-1})\ (F^{-1}\otimes 1).}$

Then, the set ${\displaystyle ({\mathcal {A)),\Delta ',\varepsilon ,\Phi ')}$ is also a quasi-bialgebra obtained by twisting ${\displaystyle {\mathcal {B_{A))))$ by F, which is called a twist or gauge transformation.^[1]: 373 If ${\displaystyle ({\mathcal {A)),\Delta ,\varepsilon ,\Phi )}$ was a braided quasi-bialgebra with universal R-matrix ${\displaystyle R}$ , then so is ${\displaystyle ({\mathcal {A)),\Delta ',\varepsilon ,\Phi ')}$ with universal R-matrix ${\displaystyle F_{21}RF^{-1))$ (using the notation from the above section).^[1]: 376 However, the twist of a bialgebra is only in general a quasi-bialgebra. Twistings fulfill many expected properties. For example, twisting by ${\displaystyle F_{1))$ and then ${\displaystyle F_{2))$ is equivalent to twisting by ${\displaystyle F_{2}F_{1))$, and twisting by ${\displaystyle F}$ then ${\displaystyle F^{-1))$ recovers the original quasi-bialgebra.

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

Theorem: Let ${\displaystyle {\mathcal {B_{A))))$, ${\displaystyle {\mathcal {B_{A'))))$ be quasi-bialgebras, let ${\displaystyle {\mathcal {B'_{A'))))$ be the twisting of ${\displaystyle {\mathcal {B_{A'))))$ by ${\displaystyle F}$, and let there exist an isomorphism: ${\displaystyle \alpha :{\mathcal {B_{A))}\to {\mathcal {B'_{A'))))$. Then the induced tensor functor ${\displaystyle (\alpha ^{*},id,\phi _{2}^{F})}$ is a tensor category equivalence between ${\displaystyle {\mathcal {A'))-mod}$ and ${\displaystyle {\mathcal {A))-mod}$. Where ${\displaystyle \phi _{2}^{F}(v\otimes w)=F^{-1}(v\otimes w)}$. Moreover, if ${\displaystyle \alpha }$ is an isomorphism of braided quasi-bialgebras, then the above induced functor is a braided tensor category equivalence.^{[1]: 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.