Quasi-Hopf algebra

A quasi-Hopf algebra is a generalization of a Hopf algebra, which was defined by the Russian mathematician Vladimir Drinfeld in 1989.

A "quasi-Hopf algebra" is a quasi-bialgebra $mathcal\{B\_A\}\; =\; (mathcal\{A\},\; Delta,\; varepsilon,\; Phi)$for which there exist and a bijective antihomomorphism "S" (antipode) of $mathcal\{A\}$ such that

: $sum\_i\; S(b\_i)\; alpha\; c\_i\; =\; varepsilon(a)\; alpha$:

for all $a\; in\; mathcal\{A\}$ and where

:$Delta(a)\; =\; sum\_i\; b\_i\; otimes\; c\_i$

and

::

where the expansions for the quantities $Phi$and $Phi^\{-1\}$ are given by

:$Phi\; =\; sum\_i\; X\_i\; otimes\; Y\_i\; otimes\; Z\_i$and:$Phi^\{-1\}=\; sum\_j\; P\_j\; otimes\; Q\_j\; otimes\; R\_j.$

As for a quasi-bialgebra, the property of being quasi-Hopf is preserved under twisting.

Usage

Quasi-Hopf algebras form the basis of 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. It provides a framework for solving two-dimensional integrable models by using the Quantum inverse scattering method.

ee also

* Quasitriangular Hopf algebra
* Quasi-triangular Quasi-Hopf algebra
* Ribbon Hopf algebra

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