- Quasi-bialgebra
In
mathematics , quasi-bialgebras are a generalization ofbialgebra s, which were defined by the Ukrainian mathematicianVladimir Drinfeld in1990 .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} ightarrow mathcal{A otimes A}:varepsilon : mathcal{A} ightarrow 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) brack Phi^{-1}, a in mathcal{A}:lbrack (id otimes id otimes Delta)(Phi) brack lbrack (Delta otimes id otimes id)(Phi) brack = (1 otimes Phi) lbrack (id otimes Delta otimes id)(Phi) brack (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
bialgebra s and quasi-bialgebras is that for the lattercomultiplication is no longercoassociative .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_1F_2.
Usage
Quasi-bialgebras form the basis of the study of
quasi-Hopf algebra s and further to the study ofDrinfeld twist s 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 correspondingR-matrix .This leads to applications instatistical mechanics , as quantum affine algebras, and their representations give rise to solutions of theYang-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 theHeisenberg XXZ model in the framework of the AlgebraicBethe 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
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