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

$\Delta :{\mathcal {A}}\rightarrow {\mathcal {A\otimes A}}$

$\varepsilon :{\mathcal {A}}\rightarrow \mathbb {F}$

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

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

$\lbrack (id\otimes id\otimes \Delta )(\Phi )\rbrack \ \lbrack (\Delta \otimes id\otimes id)(\Phi )\rbrack =(1\otimes \Phi )\ \lbrack (id\otimes \Delta \otimes id)(\Phi )\rbrack \ (\Phi \otimes 1)$

$(\varepsilon \otimes id)\circ \Delta =id=(id\otimes \varepsilon )\circ \Delta$

$(id\otimes \varepsilon \otimes id)(\Phi )=1\otimes 1.$

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

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

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

Then, the set ${\mathcal {B_{A}}}=({\mathcal {A}},\Delta ',\varepsilon ,\Phi ')$ is also a quasi-bialgebra obtained by twisting ${\mathcal {B_{A}}}$ 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