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 , Δ , ε , Φ ) {\displaystyle {\mathcal {B_{A}}}=({\mathcal {A}},\Delta ,\varepsilon ,\Phi )} for which there exist α , β ∈ A {\displaystyle \alpha ,\beta \in {\mathcal {A}}} and a bijective antihomomorphism S (antipode) of A {\displaystyle {\mathcal {A}}} such that

∑ i S ( b i ) α c i = ε ( a ) α {\displaystyle \sum _{i}S(b_{i})\alpha c_{i}=\varepsilon (a)\alpha }
∑ i b i β S ( c i ) = ε ( a ) β {\displaystyle \sum _{i}b_{i}\beta S(c_{i})=\varepsilon (a)\beta }

for all a ∈ A {\displaystyle a\in {\mathcal {A}}} and where

Δ ( a ) = ∑ i b i ⊗ c i {\displaystyle \Delta (a)=\sum _{i}b_{i}\otimes c_{i}}

and

∑ i X i β S ( Y i ) α Z i = I , {\displaystyle \sum _{i}X_{i}\beta S(Y_{i})\alpha Z_{i}=\mathbb {I} ,}
∑ j S ( P j ) α Q j β S ( R j ) = I . {\displaystyle \sum _{j}S(P_{j})\alpha Q_{j}\beta S(R_{j})=\mathbb {I} .}

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

Φ = ∑ i X i ⊗ Y i ⊗ Z i {\displaystyle \Phi =\sum _{i}X_{i}\otimes Y_{i}\otimes Z_{i}}

and

Φ − 1 = ∑ j P j ⊗ Q j ⊗ R j . {\displaystyle \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