On Jordanian Deformations of Lie algebras and Lie Superalgebras, Contraction Methods and Maps
On Jordanian Deformations of Lie algebras and Lie Superalgebras, Contraction Methods and Maps
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Date
2008-05-11
Auteurs
YANALLAH Abdelkader
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Résumé
Our main goal in this thesis is to present one important field among those of quantum groups and quantum deformations of Hopf algebras. In particular our purpose is to give the recent developments in the studies of Jordanian deformations of Lie algebras and superalgebras. We will be concerned by theirs algebraic structures and representations. We are also interested by theirs realizations and in particular with those obtained by contraction procedure from standard universal matrix Rq and leading the non standard universal matrix Rh. This presentation is located in the framework or the programm initiated by B. Abdesselam, A. Chakrabarti et al. [1]-[10]. In this series of papers the authors proposed a new scheme which had permitted the construction of the nonstandard version Uh(g) of the an enveloping (super)algebra U(g) by suitable contraction, from the corresponding standard ones Uq(g). The contraction was performed for all dimensions and for every case studied of Lie algebra or Lie superalgebra.The classical Lie algebras, that are examined, are all semisimple algebras sl(N), for arbitrary integer N 2 N, see [5] and [8] and also our review [8]. The case of superalgebras presented in this
thesis are related to our international publication [6]and [9] in the examples of Uh(osp(1|2)) and Uh(sl(N|1)) respectively. Recent additive formal proof were submitted for the case of Uh(sl(2|1)) [10]. The presentation in this thesis in general focuses on algebraic, coalgebraic structures and map realizations via contraction methods for cited Lie algebra and Lie superalgebra.
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Quantum Algebra, Hopf (Super)algebra, Jordanian deformation, Irredicible Representation, Contraction procedure, Map Realizations