published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

Cette thèse consiste en une étude des représentations de dimension finie de la super-algèbre de Lie gl(m,n). Dans le premier chapitre nous rappelons des résultats sur ces super-algèbres de Lie. Dans le second chapitre nous étudions les représentations simples de gl(2,2). Ces modules peuvent être obtenus comme quotient de modules induits dont on connaît les suites de composition, ce qui nous permet de calculer une formule des caractères finie explicite. Dans le troisième chapitre nous étudions les représentations d'une déformation quantique de l'algèbre enveloppante de gl(m,n). Nous rappelons tout d'abord la construction des bases cristallines pour les facteurs directs de puissances tensorielles de la représentation standard. Nous montrons, en affaiblissant la notion de cristal, l'existence de bases cristallines pour des modules qui ne sont pas semi-simpes, et nous donnons une méthode pour les construire. Le quatrième chapitre porte sur le dévissage du bloc maximalement atypique de la catégorie des représentations de dimension finie de gl(2,2). La connaissance de la sous-catégorie pleine des modules projectifs maximalement atypiques nous permet de reconstituer la catégorie. Nous étudions dans un premier temps les modules projectifs indécomposables et nous donnons leurs suites de Loewy. Puis dans un deuxième temps nous étudions leurs morphismes. Pour terminer nous formulons une conjecturons sur la composition de ces morphismes. / This thesis is a study of finite dimensional representations of the Lie superalgebra gl(m,n). In the first chapter we recall some results on these Lie superalgebra. In the second chapter we study the simple representations of gl(2.2). These modules can be obtained as quotient of some induced modules, the knowledge of the composition series of these modules allow us to compute an explicit finite character forumula for simple modules. In the third chapter we look at representations of a quantum deformation of the universal enveloping algebra of gl(m,n). We first recall the construction of crystal bases for the direct factors of a tensor power of the standard representation. We show by weakening the definition of crystal, that there exist crystal bases for non-semisimple modules, and we give a way to construct them. The fourth chapter focuses on the understanding of the maximaly atypical block of the category of finite dimensional representations of gl(2.2). Knowing the full subcategory of projective maximally atypical modules allows us to reconstruct the category. First, we study the projective indecomposable modules, and we compute their Loewy series. We then study their morphisms. Finally we make a conjecture on the composition of those morphisms.

Thesis. 1976. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / Microfiche copy available in Archives and Science. / Vita. / Bibliography: leaves 184-186. / by David Vogan. / Ph.D.

Thèse de doctorat : Mathématiques : Nancy 1 : 2008. / Titre provenant de l'écran-titre.

Thesis (M.Sc.)--Memorial University of Newfoundland, 2002. / Bibliography: leaves 42-43.

Thesis (M. Phil.)--University of Hong Kong, 2010. / Includes bibliographical references (p. 86-88). Also available in print.

Conformal field theories (CFTs) are intimately connected with Lie groups and their Lie algebras. Conformal symmetry is infinite-dimensional and therefore an infinite-dimensional algebra is required to describe it. This is the Virasoro algebra, which must be realized in any CFT. However, there are CFTs whose symmetries are even larger then Virasoro symmentry. We are particularly interested in a class of CFTs called Wess-Zumino-Witten (WZW) models. They have affine Lie algebras as their symmentry algebras. Each WZW model is based on a simple Lie group, whose simple Lie algebra is a subalgebra of its affine symmetry algebra. This allows us to discuss the dominant weight multiplicities of simple Lie algebras in light of WZW theory. They are expressed in terms of the modular matrices of WZW models, and related objects. Symmentries of the modular matrices give rise to new relations among multiplicities. At least for some Lie algebras, these new relations are strong enough to completely fix all multiplicities. / iv, 80 leaves : ill. ; 28 cm.

The Lie theory of extended groups applied to differential equations is arguably one of the most successful methods in the solution of differential equations. In fact, the theory unifies a number of previously unrelated methods into a single algorithm. However, as with all theories, there are instances in which it provides no useful information. Thus extensions and generalisations of the method (which classically employs only point and contact transformations) are necessary to broaden the class of equations solvable by this method. The most obvious extension is to generalised (or Lie-Backlund) symmetries. While a subset of these, called contact symmetries, were considered by Lie and Backlund they have been thought to be curiosities. We show that contact transformations have an important role to play in the solution of differential equations. In particular we linearise the Kummer-Schwarz equation (which is not linearisable via a point transformation) via a contact transformation. We also determine the full contact symmetry Lie algebra of the third order equation with maximal symmetry (y'''= 0), viz sp(4). We also undertake an investigation of nonlocal symmetries which have been shown to be the origin of so-called hidden symmetries. A new procedure for the determination of these symmetries is presented and applied to some examples. The impact of nonlocal symmetries is further demonstrated in the solution of equations devoid of point symmetries. As a result we present new classes of second order equations solvable by group theoretic means. A brief foray into Painleve analysis is undertaken and then applied to some physical examples (together with a Lie analysis thereof). The close relationship between these two areas of analysis is investigated. We conclude by noting that our view of the world of symmetry has been clouded. A more broad-minded approach to the concept of symmetry is imperative to successfully realise Sophus Lie's dream of a single unified theory to solve differential equations. / Thesis (Ph.D.)-University of Natal, 1995

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