Quasi-Hopf algebra explained

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

l{B _{A}} = ( l{A}, \Delta, \varepsilon, \Phi) for which there exist \alpha, \beta \in l{A} and a bijective antihomomorphism S (antipode) of l{A} such that\sum _{i} S(b _{i)} \alpha c _{i} = \varepsilon(a) \alpha \sum _{i} b _{i} \beta S(c _{i)} = \varepsilon(a) \beta

for all

a \in l{A} and where \Delta(a) = \sum _{i} b _{i} ⊗ 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

\Phi and \Phi ^{-1} are given by \Phi = \sum _{i} X _{i} ⊗ Y _{i} ⊗ Z _{i} and\Phi ^{-1} = \sum _{j} P _{j} ⊗ Q _{j} ⊗ 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