Quasi Hopf algebra - Alchetron, The Free Social Encyclopedia

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 ) for which there exist α , β ∈ A and a bijective antihomomorphism S (antipode) of A such that

∑ i S ( b i ) α c i = ε ( a ) α ∑ i b i β S ( c i ) = ε ( a ) β

for all a ∈ A and where

Δ ( a ) = ∑ i b i ⊗ c i

and

∑ i X i β S ( Y i ) α Z i = I , ∑ j S ( P j ) α Q j β S ( R j ) = I .

where the expansions for the quantities Φ and Φ − 1 are given by

Φ = ∑ i X i ⊗ Y i ⊗ Z i

and

Φ − 1 = ∑ 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.

References

Quasi-Hopf algebra Wikipedia

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