Quasi-bialgebra

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In mathematics, quasi-bialgebras are a generalization of bialgebras, which were defined by the Ukrainian mathematician Vladimir Drinfel'd in 1990.

A quasi-bialgebra $\mathcal{B_A} = (\mathcal{A}, \Delta, \varepsilon, \Phi)$ is an algebra $\mathcal{A}$ over a field $\mathbb{F}$ of characteristic zero equipped with morphisms of algebras

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

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

and an invertible element $\Phi \in \mathcal{A \otimes A \otimes A}$ such that the following are true

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

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

$(\varepsilon \otimes id) \circ \Delta = id = (id \otimes \varepsilon) \circ \Delta$

$(id \otimes \varepsilon \otimes id)(\Phi) = 1 \otimes 1.$

The main difference between bialgebras and quasi-bialgebras is that for the latter comultiplication is no longer coassociative.

Twisting

Given a quasi-bialgebra, further quasi-bialgebras can be generated by twisting.

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

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

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

Then, the set $\mathcal{B_A} = (\mathcal{A}, \Delta ' , \varepsilon, \Phi ')$ is also a quasi-bialgebra obtained by twisting $\mathcal{B_A}$ by F, which is called a twist. Twisting by $F 1$ and then $F 2$ is equivalent to twisting by $F 1 F 2$ .

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

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

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