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 ${ℬ}_{𝒜}=(𝒜,\mathrm{\Delta },\epsilon ,\mathrm{\Phi })$ for which there exist $\alpha ,\beta \in 𝒜$ and a bijective antihomomorphism S (antipode) of $𝒜$ such that

$\sum _{i}S(b_i)\alpha c_i=\epsilon (a)\alpha$

$\sum _i{b}_i\beta S(c_i)=\epsilon (a)\beta$

for all $a\in 𝒜$ and where

$\mathrm{\Delta }(a)=\sum _i{b}_i\otimes c_i$

and

$\sum _i{X}_i\beta S(Y_i)\alpha Z_i=𝕀,$

$\sum _{j}S(P_j)\alpha Q_j\beta S(R_j)=𝕀.$

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

$\mathrm{\Phi }=\sum _i{X}_i\otimes Y_i\otimes Z_i$

and

${\mathrm{\Phi }}^{-1}=\sum _j{P}_j\otimes Q_j\otimes R_j.$

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.

See 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