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 ${\mathcal{B}}_{\mathcal{𝒜}}=(\mathcal{𝒜},\mathrm{\Delta },\epsilon ,\mathrm{\Phi })$ is an algebra $\mathcal{𝒜}$ over a field $\mathbb{𝔽}$ of characteristic zero equipped with morphisms of algebras

$\mathrm{\Delta }:\mathcal{𝒜}\to \mathcal{𝒜}\mathcal{\otimes }\mathcal{𝒜}$

$\epsilon :\mathcal{𝒜}\to \mathbb{𝔽}$

and an invertible element $\mathrm{\Phi }\in \mathcal{𝒜}\mathcal{\otimes }\mathcal{𝒜}\mathcal{\otimes }\mathcal{𝒜}$ such that the following are true

$(id\otimes \mathrm{\Delta })\circ \mathrm{\Delta }(a)=\mathrm{\Phi }[(\mathrm{\Delta }\otimes id)\circ \mathrm{\Delta }(a)]{\mathrm{\Phi }}^{-1},a\in \mathcal{𝒜}$

$[(id\otimes id\otimes \mathrm{\Delta })(\mathrm{\Phi })][(\mathrm{\Delta }\otimes id\otimes id)(\mathrm{\Phi })]=(1\otimes \mathrm{\Phi })[(id\otimes \mathrm{\Delta }\otimes id)(\mathrm{\Phi })](\mathrm{\Phi }\otimes 1)$

$(\epsilon \otimes id)\circ \mathrm{\Delta }=id=(id\otimes \epsilon )\circ \mathrm{\Delta }$

$(id\otimes \epsilon \otimes id)(\mathrm{\Phi })=1\otimes 1.$

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 ${\mathcal{B}}_{\mathcal{𝒜}}$ is a quasi-bialgebra and $F\in \mathcal{𝒜}\mathcal{\otimes }\mathcal{𝒜}$ is an invertible element such that $(\epsilon \otimes id)F=(id\otimes \epsilon )F=1$, set

${\mathrm{\Delta }}^{\prime }(a)=F\mathrm{\Delta }(a)F^{-1},a\in \mathcal{𝒜}$

${\mathrm{\Phi }}^{\prime }=(1\otimes F)((id\otimes \mathrm{\Delta })F)\mathrm{\Phi }((\mathrm{\Delta }\otimes id)F^{-1})(F^{-1}\otimes 1).$

Then, the set ${\mathcal{B}}_{\mathcal{𝒜}}=(\mathcal{𝒜},{\mathrm{\Delta }}^{\prime },\epsilon ,{\mathrm{\Phi }}^{\prime })$ is also a quasi-bialgebra obtained by twisting ${\mathcal{B}}_{\mathcal{𝒜}}$ 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