|
|
| Line 1: |
Line 1: |
| A '''quasi-Hopf algebra''' is a generalization of a [[Hopf algebra]], which was defined by the [[Russia]]n mathematician [[Vladimir Drinfeld]] in 1989.
| | Greetings! I am Myrtle Shroyer. I am a meter reader but I strategy on altering it. Playing baseball is the hobby he will by no means stop doing. His family members life in South Dakota but his wife desires them to move.<br><br>Look into my web site [http://www.articlecube.com/profile/Nydia-Lyons/532548 at home std testing] |
| | |
| A ''quasi-Hopf algebra'' is a [[quasi-bialgebra]] <math>\mathcal{B_A} = (\mathcal{A}, \Delta, \varepsilon, \Phi)</math>for which there exist <math>\alpha, \beta \in \mathcal{A}</math> and a [[bijection|bijective]] [[antihomomorphism]] ''S'' ([[Antipode_(algebra)|antipode]]) of <math>\mathcal{A}</math> such that
| |
| | |
| : <math>\sum_i S(b_i) \alpha c_i = \varepsilon(a) \alpha</math> | |
| : <math>\sum_i b_i \beta S(c_i) = \varepsilon(a) \beta</math>
| |
| | |
| for all <math>a \in \mathcal{A}</math> and where
| |
| | |
| :<math>\Delta(a) = \sum_i b_i \otimes c_i</math>
| |
| | |
| and
| |
| | |
| :<math>\sum_i X_i \beta S(Y_i) \alpha Z_i = \mathbb{I},</math>
| |
| :<math>\sum_j S(P_j) \alpha Q_j \beta S(R_j) = \mathbb{I}.</math>
| |
| | |
| where the expansions for the quantities <math>\Phi</math>and <math>\Phi^{-1}</math> are given by
| |
| | |
| :<math>\Phi = \sum_i X_i \otimes Y_i \otimes Z_i </math>
| |
| and
| |
| :<math>\Phi^{-1}= \sum_j P_j \otimes Q_j \otimes R_j. </math>
| |
| | |
| As for a [[quasi-bialgebra]], the property of being quasi-Hopf is preserved under [[quasi-bialgebra#Twisting|twisting]].
| |
| | |
| == Usage ==
| |
| | |
| Quasi-Hopf algebras form the basis of the study of [[Drinfeld twist]]s and the representations in terms of [[F-matrix|F-matrices]] associated with finite-dimensional irreducible [[representation theory|representations]] of [[quantum affine algebras|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 algebras|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]]. It provides a framework for solving two-dimensional [[integrable model]]s by using the [[Quantum inverse scattering method]].
| |
| | |
| ==See also==
| |
| * [[Quasitriangular Hopf algebra]]
| |
| * [[Quasi-triangular Quasi-Hopf algebra]]
| |
| * [[Ribbon Hopf algebra]]
| |
| | |
| == 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
| |
| | |
| [[Category:Coalgebras]]
| |
Greetings! I am Myrtle Shroyer. I am a meter reader but I strategy on altering it. Playing baseball is the hobby he will by no means stop doing. His family members life in South Dakota but his wife desires them to move.
Look into my web site at home std testing