Quasi-bialgebra

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In mathematics, quasi-bialgebras are a generalization of bialgebras, which 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 operations

\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.

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

[edit] 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 F1 and then F2 is equivalent to twisting by F1F2.

[edit] 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.

[edit] 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
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