Critical point (set theory)

In mathematics, quasi-bialgebras are a generalization of bialgebras: they were defined by the Ukrainian mathematician Vladimir Drinfeld in 1990.

A quasi-bialgebra $ℬ_{𝒜} = (𝒜, Δ, ε, Φ)$ is an algebra $𝒜$ over a field $𝔽$ of characteristic zero equipped with morphisms of algebras

Δ : 𝒜 \to 𝒜 \otimes 𝒜

ε : 𝒜 \to 𝔽

and an invertible element $Φ \in 𝒜 \otimes 𝒜 \otimes 𝒜$ such that the following are true

(i d \otimes Δ) \circ Δ (a) = Φ [(Δ \otimes i d) \circ Δ (a)] Φ^{- 1}, a \in 𝒜

[(i d \otimes i d \otimes Δ) (Φ)] [(Δ \otimes i d \otimes i d) (Φ)] = (1 \otimes Φ) [(i d \otimes Δ \otimes i d) (Φ)] (Φ \otimes 1)

(ε \otimes i d) \circ Δ = i d = (i d \otimes ε) \circ Δ

(i d \otimes ε \otimes i d) (Φ) = 1 \otimes 1.

The main difference between bialgebras and quasi-bialgebras is that for the latter comultiplication is no longer coassociative.

Twisting

Given a quasi-bialgebra, further quasi-bialgebras can be generated by twisting.

If $ℬ_{𝒜}$ is a quasi-bialgebra and $F \in 𝒜 \otimes 𝒜$ is an invertible element such that $(ε \otimes i d) F = (i d \otimes ε) F = 1$ , set

Δ^{'} (a) = F Δ (a) F^{- 1}, a \in 𝒜

Φ^{'} = (1 \otimes F) ((i d \otimes Δ) F) Φ ((Δ \otimes i d) F^{- 1}) (F^{- 1} \otimes 1) .

Then, the set $ℬ_{𝒜} = (𝒜, Δ^{'}, ε, Φ^{'})$ is also a quasi-bialgebra obtained by twisting $ℬ_{𝒜}$ by F, which is called a twist. Twisting by $F_{1}$ and then $F_{2}$ is equivalent to twisting by $F_{2} F_{1}$ .

Usage

Quasi-bialgebras form the basis of the study of quasi-Hopf algebras and further to 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.

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

Critical point (set theory)

Twisting

Usage

References

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