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 $\alpha ,\beta \in {\mathcal {A}}$ and a bijective antihomomorphism S (antipode) of ${\mathcal {A}}$ 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 $a\in {\mathcal {A}}$ and where

\Delta (a)=\sum _{i}b_{i}\otimes c_{i}

and

\sum _{i}X_{i}\beta S(Y_{i})\alpha Z_{i}={\mathbb {I}},

\sum _{j}S(P_{j})\alpha Q_{j}\beta S(R_{j})={\mathbb {I}}.

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.

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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Quasi-Hopf algebra

Usage

See also

References