Quasi-bialgebra

In mathematics, quasi-bialgebras are a generalization of bialgebras, which were defined by the Ukrainian mathematician Vladimir Drinfeld 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 operations

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

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

:$(id\; otimes\; Delta)\; circ\; Delta(a)\; =\; Phi\; lbrack\; (Delta\; otimes\; id)\; circ\; Delta\; (a)\; brack\; Phi^\{-1\},\; a\; in\; mathcal\{A\}$:$lbrack\; (id\; otimes\; id\; otimes\; Delta)(Phi)\; brack\; lbrack\; (Delta\; otimes\; id\; otimes\; id)(Phi)\; brack\; =\; (1\; otimes\; Phi)\; lbrack\; (id\; otimes\; Delta\; otimes\; id)(Phi)\; brack\; (Phi\; otimes\; 1)$:$(varepsilon\; otimes\; id)\; circ\; Delta\; =\; id\; =\; (id\; otimes\; varepsilon)\; circ\; Delta$:$(id\; otimes\; varepsilon\; otimes\; id)(Phi)\; =\; 1.$

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

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

Then, the set $mathcal\{B\_A\}\; =\; (mathcal\{A\},\; Delta\; \text{'}\; ,\; varepsilon,\; Phi\; \text{'})$ 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 $F\_1F\_2$.

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