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1
Bagnoli, Lucia. "Z-graded Lie superalgebras." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2017. http://amslaurea.unibo.it/14118/.
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This thesis investigates the role of filtrations and gradings in the study of Lie superalgebras. The main connections between the Z-grading of a Lie superalgebra and its structure are explained. As an example, the simplicity of the Lie superalgebras W(m,n) and S(m,n) is proved. Finally, the strongly symmetric gradings of length three and five of the Lie superalgebras W(m,n) and S(m,n) are classified.
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Furutsu, Hirotoshi. "Representations of Lie Superalgebras, II." 京都大学 (Kyoto University), 1991. http://hdl.handle.net/2433/168802.
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本文データは平成22年度国立国会図書館の学位論文(博士)のデジタル化実施により作成された画像ファイルを基にpdf変換したものであるKyoto University (京都大学)
0048
新制・課程博士
理学博士
甲第4695号
理博第1293号
新制||理||720(附属図書館)
UT51-91-E66
京都大学大学院理学研究科数学専攻
(主査)教授 平井 武, 教授 池部 晃生, 教授 岩崎 敷久
学位規則第5条第1項該当
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Palmieri, Riccardo. "Real forms of Lie algebras and Lie superalgebras." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amslaurea.unibo.it/9448/.
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In questa tesi abbiamo studiato le forme reali di algebre e superalgebre di Lie. Il lavoro si suddivide in tre capitoli diversi, il primo è di introduzione alle algebre di Lie e serve per dare le prime basi di questa teoria e le notazioni. Nel secondo capitolo abbiamo introdotto le algebre compatte e le forme reali. Abbiamo visto come sono correlate tra di loro tramite strumenti potenti come l'involuzione di Cartan e relativa decomposizione ed i diagrammi di Vogan e abbiamo introdotto un algoritmo chiamato "push the button" utile per verificare se due diagrammi di Vogan sono equivalenti. Il terzo capitolo segue la struttura dei primi due, inizialmente abbiamo introdotto le superalgebre di Lie con relativi sistemi di radici e abbiamo proseguito studiando le relative forme reali, diagrammi di Vogan e abbiamo introdotto anche qua l'algoritmo "push the button".
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Frisk, Anders. "On Stratified Algebras and Lie Superalgebras." Doctoral thesis, Uppsala : Department of Mathematics, Uppsala university, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-7781.
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Duong, Minh-Thanh. "A new invariant of quadratic lie algebras and quadratic lie superalgebras." Phd thesis, Université de Bourgogne, 2011. http://tel.archives-ouvertes.fr/tel-00673991.
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In this thesis, we defind a new invariant of quadratic Lie algebras and quadratic Lie superalgebras and give a complete study and classification of singular quadratic Lie algebras and singular quadratic Lie superalgebras, i.e. those for which the invariant does not vanish. The classification is related to adjoint orbits of Lie algebras o(m) and sp(2n). Also, we give an isomorphic characterization of 2-step nilpotent quadratic Lie algebras and quasi-singular quadratic Lie superalgebras for the purpose of completeness. We study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra and 2-step nilpotent pseudo-Euclidean Jordan algebras, in the same manner as it was done for singular quadratic Lie algebras and 2-step nilpotent quadratic Lie algebras. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7.
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Duong, Minh thanh. "A new invariant of quadratic lie algebras and quadratic lie superalgebras." Thesis, Dijon, 2011. http://www.theses.fr/2011DIJOS021/document.
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Dans cette thèse, nous définissons un nouvel invariant des algèbres de Lie quadratiques et des superalgèbres de Lie quadratiques et donnons une étude et classification complète des algèbres de Lie quadratiques singulières et des superalgèbres de Lie quadratiques singulières, i.e. celles pour lesquelles l’invariant n’est pas nul. La classification est en relation avec les orbites adjointes des algèbres de Lie o(m) et sp(2n). Aussi, nous donnons une caractérisation isomorphe des algèbres de Lie quadratiques 2-nilpotentes et des superalgèbres de Lie quadratiques quasi-singulières pour le but d’exhaustivité. Nous étudions les algèbres de Jordan pseudoeuclidiennes qui sont obtenues des extensions doubles d’un espace vectoriel quadratique par une algèbre d’une dimension et les algèbres de Jordan pseudo-euclidienne 2-nilpotentes, de la même manière que cela a été fait pour les algèbres de Lie quadratiques singulières et des algèbres de Lie quadratiques 2-nilpotentes. Enfin, nous nous concentrons sur le cas d’une algèbre de Novikov symétrique et l’étudions à dimension 7In this thesis, we defind a new invariant of quadratic Lie algebras and quadratic Lie superalgebras and give a complete study and classification of singular quadratic Lie algebras and singular quadratic Lie superalgebras, i.e. those for which the invariant does not vanish. The classification is related to adjoint orbits of Lie algebras o(m) and sp(2n). Also, we give an isomorphic characterization of 2-step nilpotent quadratic Lie algebras and quasi-singular quadratic Lie superalgebras for the purpose of completeness. We study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra and 2-step nilpotent pseudo-Euclidean Jordan algebras, in the same manner as it was done for singular quadratic Lie algebras and 2-step nilpotent quadratic Lie algebras. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7
7
Ma, Shuk-Chuen. "Conformal and Lie superalgebras related to the differential operators on the circle /." View Abstract or Full-Text, 2003. http://library.ust.hk/cgi/db/thesis.pl?MATH%202003%20MA.
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Thesis (Ph. D.)--Hong Kong University of Science and Technology, 2003.Includes bibliographical references (leaves 148-150). Also available in electronic version. Access restricted to campus users.
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Götz, Gerhard Markus. "Lie superalgebras and string theory in AdS3 x S3." Paris 6, 2006. http://www.theses.fr/2006PA066039.
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McSween, Alexandra. "Affine Oriented Frobenius Brauer Categories and General Linear Lie Superalgebras." Thesis, Université d'Ottawa / University of Ottawa, 2021. http://hdl.handle.net/10393/42342.
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To any Frobenius superalgebra A we associate an oriented Frobenius Brauer category and an affine oriented Frobenius Brauer categeory. We define natural actions of these categories on categories of supermodules for general linear Lie superalgebras gl_m|n(A) with entries in A. These actions generalize those on module categories for general linear Lie superalgebras and queer Lie superalgebras, which correspond to the cases where A is the ground field and the two-dimensional Clifford superalgebra, respectively. We include background on monoidal supercategories and Frobenius superalgebras and discuss some possible further directions.
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Gardini, Matteo. "Representations of semisimple Lie algebras." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amslaurea.unibo.it/9029/.
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La tesi è dedicata allo studio delle rappresentazioni delle algebre di Lie semisemplici su un campo algebricamente chiuso di caratteristica zero. Mediante il teorema di Weyl sulla completa riducibilità, ogni rappresentazione di dimensione finita di una algebra di Lie semisemplice è scrivibile come somma diretta di sottorappresentazioni irriducibili. Questo permette di poter concentrare l'attenzione sullo studio delle rappresentazioni irriducibili. Inoltre, mediante il ricorso all'algebra inviluppante universale si ottiene che ogni rappresentazione irriducibile è una rappresentazione di peso più alto. Perciò è naturale chiedersi quando una rappresentazione di peso più alto sia di dimensione finita ottenendo che condizione necessaria e sufficiente perché una rappresentazione di peso più alto sia di dimensione finita è che il peso più alto sia dominante. Immediata è quindi l'applicazione della teoria delle rappresentazioni delle algebre di Lie semisemplici nello studio delle superalgebre di Lie, in quanto costituite da un'algebra di Lie e da una sua rappresentazione, dove viene utilizzata la tecnica della Z-graduazione che viene utilizzata per la prima volta da Victor Kac nello studio delle algebre di Lie di dimensione infinita nell'articolo ''Simple irreducible graded Lie algebras of finite growth'' del 1968.
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Duff, Ana. "Derivations, invariant forms and the second homology group of orthosymplectic Lie superalgebras." Thesis, University of Ottawa (Canada), 2002. http://hdl.handle.net/10393/6346.
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We develop the description of the derivation algebras of orthosymplectic Lie super-algebras over supercommutative, associative superrings containing ½ and determine conditions under which the derivation algebra can be written as a semidirect product of the inner and the outer derivations. We then describe the supersymmetric invariant forms of the elementary orthosymplectic Lie superalgebra and determine the outer derivations which are skew with respect to a given supersymmetric invariant form. Finally, we describe the universal central extension and its centre, the second homology group, of the elementary orthosymplectic Lie superalgebra. The original motivation for this comes from the theory of extended affine Lie algebras. Specialized to the orthosymplectic Lie superalgebras representing the centreless cores of extended affine Lie algebras of type B and D, the above descriptions are the necessary and sufficient building blocks for the construction of an extended affine Lie algebra of type B and D from its centreless core.
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Geer, Nathan. "Link invariants, quantized superalgebras and the Kontsevich integral /." view abstract or download file of text, 2004. http://wwwlib.umi.com/cr/uoregon/fullcit?p3136414.
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Thesis (Ph. D.)--University of Oregon, 2004.Typescript. Includes vita and abstract. Includes bibliographical references (leaves 123-125). Also available for download via the World Wide Web; free to University of Oregon users.
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Kac, Victor G., Minoru Wakimoto, and kac@math mit edu. "Integrable Highest Weight Modules over Affine Superalgebras and Appell's." ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi920.ps.
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Dolan, Peter. "A Z2-graded generalization of Kostant's version of the Bott-Borel-Weil theorem /." view abstract or download file of text, 2007. http://proquest.umi.com/pqdweb?did=1400959341&sid=2&Fmt=2&clientId=11238&RQT=309&VName=PQD.
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Thesis (Ph. D.)--University of Oregon, 2007.Typescript. Includes vita and abstract. Includes bibliographical references (leaves 130-131). Also available for download via the World Wide Web; free to University of Oregon users.
15
Cummins, C. J. "Applications of S-function techniques to the representation theory of Lie superalgebras and symmetry breaking." Thesis, University of Southampton, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.374751.
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Sergeev, A., and mleites@matematik su se. "Enveloping Superalgebra $U(\frak o\frak s\frak p(1|2))$ and." J. Nonlinear Math. Phys. 8, no. 2 (2001) 1-27, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1025.ps.
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Grant, Jonathan William. "Diagrammatics for representation categories of quantum Lie superalgebras from skew Howe duality and categorification via foams." Thesis, Durham University, 2016. http://etheses.dur.ac.uk/11618/.
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In this thesis we generalise quantum skew Howe duality to Lie superalgebras in type A, and show how this gives a categorification of certain representation categories of $\mathfrak{gl}(m|n)$. In particular, we use skew Howe duality to describe a category of representations generated monoidally by the exterior powers of the fundamental representation. This description is in terms of MOY diagrams, with one additional local relation on $n+1$ strands. This generalises the $n=0$ case from Cautis, Kamnitzer and Morrison. Using this, we give a categorification of this category in terms of foams, which generalises that of Queffelec, Rose and Lauda in the case $n=0$. The Reshetikhin-Turaev procedure gives a knot polynomial associated to $\mathfrak{gl}(m|n)$, which is a specialisation of the HOMFLY polynomial $P(a,q)$ at $a=q^{m-n}$. For the case $n=0$, the polynomial can be described nicely in terms of MOY diagrams, and therefore is related strongly to skew Howe duality. This was used by Queffelec and Rose to define $\mathfrak{sl}(n)$ Khovanov-Rozansky homology by categorified skew Howe duality. For general $n$, the relationship is less nice, and skew Howe duality is not sufficient to describe a homology theory associated with $\mathfrak{gl}(m|n)$ from our approach. Part of the problem is that the representation category no longer contains duals of the fundamental representations, which means that although a braid has an image in this categorified representation category, it is not possible to close this braid in the same way that Queffelec and Rose do. However, the categorified representation category does give partial progress towards the problem of defining a quantum categorification of the Alexander polynomial.
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Olivetto, René [Verfasser], Kathrin [Akademischer Betreuer] Bringmann, and Sander [Akademischer Betreuer] Zwegers. "Harmonic Maass Forms, Jacobi Forms, and Applications to Lie Superalgebras / René Olivetto. Gutachter: Kathrin Bringmann ; Sander Zwegers." Köln : Universitäts- und Stadtbibliothek Köln, 2014. http://d-nb.info/1064693350/34.
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Olivetto, René Verfasser], Kathrin [Akademischer Betreuer] [Bringmann, and Sander [Akademischer Betreuer] Zwegers. "Harmonic Maass Forms, Jacobi Forms, and Applications to Lie Superalgebras / René Olivetto. Gutachter: Kathrin Bringmann ; Sander Zwegers." Köln : Universitäts- und Stadtbibliothek Köln, 2014. http://d-nb.info/1064693350/34.
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Jordan, Alex. "A super version of Zhu's theorem /." Connect to title online (Scholars' Bank) Connect to title online (ProQuest), 2008. http://hdl.handle.net/1794/8283.
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Thesis (Ph. D.)--University of Oregon, 2008.Typescript. Includes vita and abstract. Includes bibliographical references (leaves 40-41). Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.
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Calixto, Lucas Henrique 1989. "Super álgebras de funções." [s.n.], 2013. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306943.
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Orientador: Adriano Adrega de MouraDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
Made available in DSpace on 2018-08-22T08:28:52Z (GMT). No. of bitstreams: 1 Calixto_LucasHenrique_M.pdf: 1707951 bytes, checksum: a7576ec9f19a4faf6e8bd959192baeb8 (MD5) Previous issue date: 2013
Resumo: O principal objetivo dessa dissertação é explicar a classificação dos módulos irredutíveis de dimensão finita para qualquer super álgebra de funções sobre uma super álgebra de Lie básica. Os principais resultados dizem que um módulo irredutível de dimensão finita ou é uma representação de avaliação ou é um módulo de Kac para certo módulo de avaliação generalizado. Para chegar a tal objetivo, também fazemos uma revisão detalhada da classificação das super álgebras de Lie básicas
Abstract: The goal of this dissertation is to explain the classification of the irreducible finite-dimensional representations of a map superalgebra whose underlying simple Lie superalgebra is basic. The main result says that an irreducible finite-dimensional module is either an evaluation module or a Kac module associated to a certain generalized evaluation module. We also give a detailed review of the classification of the basic Lie superalgebras
Mestrado
Matematica
Mestre em Matemática
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Ha, Ngoc-Phu. "Théorie quantique des champs topologiques pour la superalgèbre de Lie sl(2/1)." Thesis, Lorient, 2018. http://www.theses.fr/2018LORIS505/document.
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Ce texte étudie le groupe quantique Uξ sl(2|1) associé à la superalgèbre de Lie sl(2|1) et une catégorie de ses représentations de dimension finie. L'objectif est de construire des invariants topologiques de 3-variétés en utilisant la notion de trace modifiée. D'abord nous prouvons que la H catégorie CH des modules de poids nilpotents sur Uξ sl(2|1) est enrubannée et qu'il existe une trace modifiée sur son idéal des modules projectifs. De plus CH possède une structure relativement G-prémodulaire ce qui est une condition suffisante pour construire un invariant de 3-variétés à la Costantino-Geer-Patureau. Cet invariant est le cœur d'une 1+1+1-TQFT (Topological Quantum Field Theory). D'autre part Hennings a proposé à partir d'une algèbre de Hopf de dimension finie une construction d’invariants qui dispense de considérer la catégorie de H l l ses représentations. Nous montrons que le groupe quantique déroulé Uξ sl(2|1)/(e1 , f1 ) possède une complétion qui est une algèbre de Hopf enrubannée topologique. Nous construisons un invariant de 3-variétés à la Hennings en utilisant cette structure algébrique, une transformation de Fourier discrète et la notion de G-intégrales. L'intégrale dans une algèbre de Hopf est centrale dans la construction de Hennings. La notion de trace modifiée dans une catégorie s'est récemment révélée être une généralisation des intégrales dans les algèbres de Hopf de dimension finie. Dans un contexte plus général d'algèbre de Hopf de dimension infinie nous prouvons la relation formulée entre la trace modifiée et la G -intégraleThis text studies the quantum group Uξ sl(2|1) associated with the Lie superalgebra sl(2|1) and a category of finite dimensional representations. The aim is to construct the topological invariants of 3-manifolds using the notion of modified trace. We first prove that the category CH of the nilpotent weight modules over Uξ sl(2|1) is ribbon and that there exists a modified trace on its ideal of projective modules. Furthermore, CH possesses a relative G-premodular structure which is a sufficient condition to construct an invariant of 3-manifolds of Costantino-Geer-Patureau type. This invariant is the heart of a 1+1+1-TQFT (Topological Quantum Field Theory). Next Hennings proposed from a finite dimensional Hopf algebra, a construction of invariants which does not require to consider the category of its representations. We show that the unrolled H l l quantum group Uξ sl(2|1)/(e1 , f1 ) has a completion which is a topological ribbon Hopf algebra. We construct an invariant of 3-manifolds of Hennings type using this algebraic structure, a discrete Fourier transform, and the notion of G-integrals. The integral in a Hopf algebra is central in the construction of Hennings. The notion of modified trace in a category has recently been revealed to be a generalization of the integrals in a finite dimensional Hopf algebra. In a more general context of infinite dimensional Hopf algebras we prove the relation formulated between the modified trace and the G-integral
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Kac, Victor G., Alexei Rudakov, and kac@math mit edu. "Representations of the Exceptional Lie Superalgebra E(3,6):." ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi921.ps.
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Muir, Neil John. "Polynomial representations of the general linear Lie superalgebra." Thesis, Queen Mary, University of London, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.437015.
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Kac, Victor G., Alexei Rudakov, and rudakov@math ntnu no. "Representations of the Exceptional Lie Superalgebra E(3,6): II. Four." ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi976.ps.
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RAVERA, LUCREZIA. "Group Theoretical Hidden Structure of Supergravity Theories in Higher Dimensions." Doctoral thesis, Politecnico di Torino, 2018. http://hdl.handle.net/11583/2700157.
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L'obiettivo della mia tesi di dottorato è quello di esplorare simmetrie nascoste nelle teorie di supergravità. In questa direzione ho affrontato diversi temi di ricerca: da un lato mi sono dedicata all'analisi della struttura di gauge nascosta in teorie di supergravità in varie dimensioni di spazio-tempo e allo studio di un particolare modello in quattro dimensioni in presenza di un bordo spaziale non banale; dall'altro lato, mi sono focalizzata sui legami algebrici tra superalgebre di rilevanza nel campo della supergravità. Nei primi tre capitoli della tesi vengono forniti un'introduzione generale e il background teorico necessario per una migliore comprensione degli argomenti trattati. In particolare, viene brevemente descritto l'approccio cosiddetto ''rheonomico'' (noto anche come approccio
geometrico) alle teorie di supergravità, nel quale le curvature vengono espresse in una base del superspazio. Questo approccio include il concetto di Algebre Differenziali Libere (un'estensione delle equazioni di Maurer-Cartan che coinvolge forme differenziali di grado maggiore), dal momento che le teorie di supergravità in dimensioni di spazio-tempo D ≥ 4 contengono potenziali di gauge descritti in termini di forme differenziali di ordine maggiore
di 1 associate a tensori antisimmetrici. Considerando la supergravità in undici dimensioni in questa formulazione geometrica, rivediamo anche come l'Algebra Differenziale Libera che descrive la teoria si può scrivere in modo del tutto equivalente in termini di una superalgebra ordinaria di 1-forme, introdotta per la prima volta in letteratura negli anni '80. Questa superalgebra nascosta nella teoria di supergravittà in undici dimensioni (che chiameremo
DF-algebra) include la cosiddetta M-algebra. Passiamo poi ai risultati originali della mia attività di ricerca durante il dottorato: partiamo dallo sviluppo della cosiddetta ''supergravità di AdS-Lorentz'' in D = 4 dimensioni di spazio-tempo, adottando l'approccio rheonomico e studiando i contributi di bordo alla teoria. Successivamente, ci concentriamo sull'analisi della struttura di gauge nascosta delle Algebre Differenziali Libere supersimmetriche e sulle peculiari simmetrie nascoste in questi modelli. Nello specifico, ci focalizziamo sulle superalgebre nascoste nelle teorie di super-
gravità in undici e in sette dimensioni, esplorando il ruolo fisico dei generatori fermionici nilpotenti che appaiono in modo naturale nelle superalgebre sopracitate. L'ultima parte della tesi è dedicata alla descrizione algebrica di (super)algebre e alla costruzione di nuove formulazioni analitiche del cosiddetto metodo di ''S-expansion". Il capitolo finale contiene un riassunto dei risultati dei miei studi di dottorato presentati nella tesi e alcuni possibili
sviluppi futuri. Nelle Appendici si possono trovare la notazione adottata, formule utili e calcoli dettagliati.The purpose of my PhD thesis is to investigate different group theoretical and geometrical aspects of supergravity theories. To this aim, several research topics are explored: On one side, the construction of supergravity models in diverse space-time dimensions, including the study of boundary contributions, and the disclosure of the hidden gauge structure of these theories; on the other side, the analysis of the algebraic links among different superalgebras related to supergravity theories. In the first three chapters, we give a general introduction and furnish the theoretical background necessary for a clearer understanding of the thesis. In particular, we recall the rheonomic (also called geometric) approach to supergravity theories, where the field curvatures are expressed in a basis of superspace. This includes the Free Differential Algebras framework (an extension of the Maurer-Cartan equations to involve higher-degree differential forms), since supergravity theories in D ≥ 4 space-time dimensions contain gauge potentials described by p-forms, of various p > 1, associated to p-index antisymmetric tensors. Considering D = 11 supergravity in this set up, we also review how the supersymmetric Free Differential Algebra describing the theory can be traded for an ordinary superalgebra of 1-forms, which was introduced for the first time in the literature in the '80s. This hidden superalgebra underlying D = 11 supergravity (which we will refer to as the DF-algebra) includes the so called M-algebra being, in particular, a spinor central extension of it. We then move to the original results of my PhD research activity: We start from the development of the so called AdS-Lorentz supergravity in D = 4 by adopting the rheonomic approach and discuss on boundary contributions to the theory. Subsequently, we focus on the analysis of the hidden gauge structure of supersymmetric Free Differential Algebras. More precisely, we concentrate on the hidden superalgebras underlying D = 11 and D = 7 supergravities, exploring the symmetries hidden in the theories and the physical role of the nilpotent fermionic generators naturally appearing in the aforementioned superalgebras. After that, we move to the pure algebraic and group theoretical description of (super)algebras, focusing on new analytic formulations of the so called S-expansion method. The final chapter contains the summary of the results of my doctoral studies presented in the thesis and possible future developments. In the Appendices, we collect notation, useful formulas, and detailed calculations.
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Germoni, Jérôme. "Représentations indécomposables des superalgèbres de Lie spéciales linéaires." Université Louis Pasteur (Strasbourg) (1971-2008), 1997. http://www.theses.fr/1997STR13014.
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Le probleme principal etudie dans cette these est la classification des representations indecomposables de dimension finie de sl(m, n), la superalgebre de lie des endormorphismes de supertrace nulle du superespace vectoriel complexe de dimension (m|n). On obtient une reponse complete, a savoir une parametrisation et une description explicite de nature combinatoire, lorsque m ou n vaut 1. Cette reponse s'etend aux representations dites simplement atypiques pour toute valeur de m et n. Par ailleurs, on montre que le probleme de classification de toutes les representations de sl(m, n) est de type sauvage lorsque m et n valent au moins 2, c'est-a-dire qu'il est au moins aussi complique que le probleme analogue pour l'algebre libre a deux generateurs
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Gruson, Caroline. "Sur les super groupes de Lie." Paris 7, 1993. http://www.theses.fr/1993PA077056.
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La première partie est une adaptation au cadre des super groupes de Lie du théorème du à Cartier qui assure que les groupes formels sont lisses en caractérisque zéro. La seconde partie donne une description des super groupes de Lie dits vraiment classiques comme groupes d'automorphismes des super algèbres semi-simples à involution, selon une méthode de Weil. La troisième partie est consacrée à l'étude de l'idéal définissant l'orbite d'un vecteur de plus haut poids d'une représentation simple de dimension finie d'une super algèbre de Lie basique classique complexe.
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Vignoli, Massimiliano. "Sistemi di radici." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2013. http://amslaurea.unibo.it/5926/.
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In questa tesi studiamo il ruolo dei sistemi di radici nella classificazione delle algebre di Lie e delle superalgebre di Lie. L'interesse per le superalgebre di Lie nasce nei primi anni '70 quando una parte dei fisici si convinse che sarebbe stato più utile e molto più chiaro riuscire ad avere uno schema di riferimento unitario in cui non dovesse essere necessario trattare separatamente particelle fisiche come bosoni e fermioni. Una teoria sistematica sulle superalgebre di Lie fu introdotta da V. Kac nel 1977 che diede la classificazione delle superalgebre di Lie semplici su un campo algebricamente chiuso.
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Carvalho, Maria Elisabete Félix Barreiro. "Quadratic Lie superalgebras and some problems on Lie bi-superalgebras." Doctoral thesis, 2007. http://hdl.handle.net/10316/7516.
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Chien-chou, Chen, and 陳建州. "Lie Superalgebras of General Linear Supergroups." Thesis, 1999. http://ndltd.ncl.edu.tw/handle/84890814496367257797.
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碩士國立成功大學
數學系
87
In this paper, we study the subspace of the dual of A(G) of a Lie supergroup (G,A) where G=GL(m/n,R). We prove the Lie superalgebra of the Lie supergroup (G,A) and Lie superalgebra containing left-invariant vector fields of A(G) are both isomorphic to gl(m/n,R).
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Chen, Chih-Whi, and 陳志瑋. "Representation theory of strange Lie superalgebras." Thesis, 2016. http://ndltd.ncl.edu.tw/handle/80153857523888242264.
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博士國立臺灣大學
數學研究所
104
In this dissertation, we study the representation theory of strange Lie superalgebras. It is divided into three parts. In the first part, we study categories of finite-dimensional modules over the periplectic Lie superalgebras $mathfrak{p}(n)$ and obtain a BGG type reciprocity. In particular, these categories have only finitely-many blocks. We also compute the characters for irreducible modules over periplectic Lie superalgebras of ranks $2$ and $3$, and obtain explicit description of the blocks for ranks $2$, $3$, and $4$. In the second part, we develop a reduction procedure which provides an equivalence from an arbitrary block of the BGG category for the queer Lie superalgebra $mathfrak{q}(n)$ to a block with weights in $Lambda_{{ell_1},s_{1}}(n_1) imes cdots imes Lambda_{{ell_k},s_{k}}(n_{k})$ (see, Theorem ef{FirstMainThm}) for a BGG category of finite direct sum of queer Lie superalgebras. The descriptions of blocks are given as well. We also establish equivalences between certain maximal parabolic subcategories for $mathfrak{q}(n)$ and blocks of atypicality-one of the category of finite-dimensional modules for $mathfrak{gl}(ell|n-ell)$, where $ell leq n$. In the third part, we establish a maximal parabolic version of the Kazhdan-Lusztig conjecture cite[Conjecture 5.10]{CKW} for the BGG category $mathcal{O}_{k,zeta}$ of $mathfrak{q}(n)$-modules of ``$pm zeta$-weights'', where $kleq n$ and $zetainCsetminushf $. As a consequence, the irreducible characters of these $mathfrak q(n)$-modules in this maximal parabolic category are given by the Kazhdan-Lusztig polynomials of type $A$ Lie algebras. As an application, closed character formulas for a class of $mathfrak q(n)$-modules resembling polynomial and Kostant modules of the general linear Lie superalgebras are obtained.
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CAFFI, Cristoforo. "Conformal embeddings in basic Lie superalgebras." Doctoral thesis, 2022. http://hdl.handle.net/11573/1634817.
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In this thesis we deal with some aspects concerning embeddings of regular subalgebras in basic Lie superalgebras. A first problem is to find a criterion to determine whether a regular subalgebra is maximal or not, similarly to what has been done for semisimple Lie algebras. We discovered that the the maximal equal rank subalgebras are obtained by taking all possible affine diagrams and deleting a node corresponding to a simple root with prime coefficient. The second and main purpose of the thesis is, for every maximal equal rank regular subalgebra g_0 in g, to compute numbers k such that we have a conformal embedding of the vertex algebra generated by g_0 in the affine vertex algebra of level k associated with g. We used the Adamovic-Perse criterion to compute all these levels and we noted that they rarely depend on the choice of g_0 in g. We also used the "fusion rule argument" to find some decompositions, as a g_0-module, of the simple vertex algebra associated with g of conformal level k.
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Moon, Dongho. "Schur-Weyl dualities for Lie superalgebras and Lie color algebras." 1998. http://catalog.hathitrust.org/api/volumes/oclc/40807018.html.
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Thesis (Ph. D.)--University of Wisconsin--Madison, 1998.Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 105-108).
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Bagci, Irfan. "Cohomology and support varieties for Lie superalgebras." 2009. http://purl.galileo.usg.edu/uga%5Fetd/bagci%5Firfan%5F200905%5Fphd.
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Thesis (Ph. D.)--University of Georgia, 2009.Directed by Daniel K. Nakano. Includes an article published in International mathematics research notices. For abstract see https://getd.libs.uga.edu/pdfs/bagci%5Firfan%5F200905%5Fphd.pdf. Includes bibliographical references.
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Mukherjee, Shantala. "Representations of nilpotent lie algebras and superalgebras." 2004. http://www.library.wisc.edu/databases/connect/dissertations.html.
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Padhan, Rudra Narayan. "On Isoclinism and Capability of Lie Superalgebras." Thesis, 2020. http://ethesis.nitrkl.ac.in/10227/1/2020_PhD_RNPadhan_515MA1002_OnIsoclinism.pdf.
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Many theorems and formulas of Lie superalgebras run quite parallel to Lie algebras, sometimes giving interesting results. So it is quite natural to extend the new concepts of Lie algebras immediately to Lie superalgebra case as the later type of algebras have wide applications in Physics, Mathematics and related theories. Isoclinism of Lie superalgebras and Schur multiplier has been defined and studied currently. The purpose of this thesis is to study more deeply the properties of isoclinism and the relation of Schur multiplier to capability. In this thesis it is shown that for finite dimensional Lie superalgebras of same dimension, the notion of isoclinism and isomorphism are equivalent. Furthermore, it is shown that covers of finite dimensional Lie superalgebras are isomorphic using the notion of isoclinism . For a Lie superalgebra L, the set of all superderivations of L whose image is contained in the center of L is known as central derivation of L and is denoted by SDerz(L). It is a subalgebra of superderivation algebra. This thesis presents the work on the central derivation of nilpotent Lie superalgebras which have nilindex 2. In particular, stem Lie superalgebras are characterized by their central derivations. Moreover, relation between SDerz(L) and stem Lie superalgebra is obtained for finite as well as infinite dimensional nilpotent Lie superalgebras with nilindex 2 and nonabelian finite dimensional nilpotent Lie superalgebra. In this thesis it is shown that distributive law holds for nonabelian tensor product of Lie superalgebras under certain direct sums. Thereby a rule for nonabelian exterior square of a Lie superalgebra is obtained. Capable Lie superalgebra is defined and then some characterization is given in this thesis. Specifically, it is proved that epicenter of a Lie superalgebra is equal to exterior square. All capable Lie superalgebras whose derived subalgebras have dimension at most one are classified. As an application to those results, it is shown that there exists at least one nonabelian nilpotent capable Lie superalgebra L of dimension a + b _ 3 where dim L = (a j b):
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Liu, Yi-Hsun, and 劉怡薰. "Extensions of modules over differentiably simple Lie superalgebras." Thesis, 1998. http://ndltd.ncl.edu.tw/handle/03786724925776008995.
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Yates, LA. "Quadratic superalgebras in mathematics and physics." Thesis, 2019. https://eprints.utas.edu.au/31387/1/Yates_whole_thesis.pdf.
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We introduce in this thesis a class of quadratic deformations of Lie superalgebras which we term quadratic superalgebras. These are finitely-generated algebras with a `Z`\(_{_2}\)-graded structure comprising an even and an odd part; the even part is an ordinary Lie algebra, the odd part is a module of the even part, and the anticommutator of two odd elements closes quadratically on the even generators. One motivation to study algebras with these structural properties is their arising in the observable algebra of gauge invariant fields in Hamiltonian lattice QCD [44] and the subsequent study of polynomial gl(n) superalgebras [45]. The present work both broadens the scope and extends the analysis of the latter. For this class of algebras we derive a Poincaré-Birkhoff-Witt theorem, including an explicit ordered basis, which we then employ as a means to investigate the structure of irreducible modules; these are analogous to Kac modules for Lie superalgebras. Further rationale to study quadratic superalgebras is due to the remarkable existence of zero-step modules; these are so-called atypical modules for which the entire irreducible module of the quadratic superalgebra consists of a single irreducible module of the even subalgebra.
In addition to their mathematical aspects, we investigate in this thesis an application of quadratic superalgebras in the context of space-time conformal supersymmetry. We show that the algebra of N = 1 space-time conformal supersymmetry, su(2; 2=1), arises as a contraction limit of a certain quadratic superalgebra. In this setting we exploit the existence of zero-step modules which, for a fixed parameter choice of the quadratic family under consideration, coincide with the massless positive energy unitary irreducible representations (in the standard classifcation of Mack) of the even subalgebra. For these massless particle multiplets the odd generators vanish identically and supersymmetry is carried (unbroken) without the accompaniment of superpartners. Thus, in the context of extended non-linear symmetry principles and their role in determining the spectrum of fundamental particles, we point out that there exist candidate algebraic structures which implement (extended) supersymmetric invariance while at the same time obviating the need for every particle of the standard model to be accompanied by a superpartner.
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(9115211), Chenliang Huang. "ON THE GAUDIN AND XXX MODELS ASSOCIATED TO LIE SUPERALGEBRAS." Thesis, 2020.
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We describe a reproduction procedure which, given a solution of the gl(m|n) Gaudin Bethe ansatz equation associated to a tensor product of polynomial modules, produces a family P of other solutions called the population. To a population we associate a rational pseudodifferential operator R and a superspace W of rational functions.
We show that if at least one module is typical then the population P is canonically identified with the set of minimal factorizations of R and with the space of full superflags in W. We conjecture that the singular eigenvectors (up to rescaling) of all gl(m|n) Gaudin Hamiltonians are in a bijective correspondence with certain superspaces of rational functions.
We establish a duality of the non-periodic Gaudin model associated with superalgebra gl(m|n) and the non-periodic Gaudin model associated with algebra gl(k).
The Hamiltonians of the Gaudin models are given by expansions of a Berezinian of an (m+n) by (m+n) matrix in the case of gl(m|n)
and of a column determinant of a k by k matrix in the case of gl(k). We obtain our results by proving Capelli type identities for both cases and comparing the results.
We study solutions of the Bethe ansatz equations of the non-homogeneous periodic XXX model associated to super Yangian Y(gl(m|n)).
To a solution we associate a rational difference operator D and a superspace of rational functions W. We show that the set of complete factorizations of D is in canonical bijection with the variety of superflags in W and that each generic superflag defines a solution of the Bethe ansatz equation. We also give the analogous statements for the quasi-periodic supersymmetric spin chains.
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Huang, Chenliang. "On the Gaudin and XXX models associated to Lie superalgebras." Thesis, 2020. http://hdl.handle.net/1805/23400.
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Indiana University-Purdue University Indianapolis (IUPUI)We describe a reproduction procedure which, given a solution of the gl(m|n) Gaudin Bethe ansatz equation associated to a tensor product of polynomial modules, produces a family P of other solutions called the population. To a population we associate a rational pseudodifferential operator R and a superspace W of rational functions. We show that if at least one module is typical then the population P is canonically identified with the set of minimal factorizations of R and with the space of full superflags in W. We conjecture that the singular eigenvectors (up to rescaling) of all gl(m|n) Gaudin Hamiltonians are in a bijective correspondence with certain superspaces of rational functions. We establish a duality of the non-periodic Gaudin model associated with superalgebra gl(m|n) and the non-periodic Gaudin model associated with algebra gl(k). The Hamiltonians of the Gaudin models are given by expansions of a Berezinian of an (m+n) by (m+n) matrix in the case of gl(m|n) and of a column determinant of a k by k matrix in the case of gl(k). We obtain our results by proving Capelli type identities for both cases and comparing the results. We study solutions of the Bethe ansatz equations of the non-homogeneous periodic XXX model associated to super Yangian Y(gl(m|n)). To a solution we associate a rational difference operator D and a superspace of rational functions W. We show that the set of complete factorizations of D is in canonical bijection with the variety of superflags in W and that each generic superflag defines a solution of the Bethe ansatz equation. We also give the analogous statements for the quasi-periodic supersymmetric spin chains.
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Pradhan, Sushree Sangeeta. "Review on Root System of Lie Superalgebras and Some Partial Results on Splints of A(m,n)." Thesis, 2015. http://ethesis.nitrkl.ac.in/7050/1/review_on_root__Pradhan_2015.pdf.
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Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. This dissertation deals with the splints of the root systems of Classical Lie superalgebra which can be seen as a generalisation of a Lie algebra to include a Z2 − grading. The term ’Splints’ is first coined by David A Richter which play an important role in determining the branching rules of a module over a complex semisimple Lie algebra. These results have been extended to classical Lie superalgebras which gave interesting results with regards to the graded algebras.
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Roeseler, Karsten. "Oktaven und Reduktionstheorie." Doctoral thesis, 2011. http://hdl.handle.net/11858/00-1735-0000-0006-B3F4-C.
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CHIEN-YI, MA. "Symmetric Tensors in Ortho-symplectic Lie Superalgebra of Dimension (4,4)." 2005. http://www.cetd.com.tw/ec/thesisdetail.aspx?etdun=U0001-0407200517394100.
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MA, CHIEN-YI, and 馬鑑一. "Symmetric Tensors in Ortho-symplectic Lie Superalgebra of Dimension (4,4)." Thesis, 2005. http://ndltd.ncl.edu.tw/handle/43066528074507216240.
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碩士國立臺灣大學
數學研究所
93
Ortho-symplectic Lie superalgebra osp can be realized as differential operators and homogeneous polynomial space is closed under its action, that is, homogeneous polynomial space is an osp-module. Our thesis is to study whether or not homogeneous polynomial space can be reduced to a direct sum of irreducible osp-modules. Our conclusion is for any odd homogeneous polynomial space, the answer is yes. For even, the answer is no in the case of degree 2, and therefore invalid for any even homogeneous polynomial space since it must contain a submodule isomorphic to degree 2 homogeneous polynomial space. However, a complete decomposition of arbitrary even homogeneous polynomial space has not been reached yet.
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BRILLI, DANIELE. "Pseudoalgebraic structures and representations of the exceptional Lie superalgebra E(5,10)." Doctoral thesis, 2021. http://hdl.handle.net/11573/1637612.
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The language of Lie pseudoalgebras is useful in giving finite description of infinite-dimensional Lie algebras and has proved to be a valuable tool in algebra and representation theory. In this thesis, we apply pseudoalgebraic techniques to the representation theory of the exceptional linearly compact Lie superalgebra E(5,10).
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Bertrand, Sébastien. "Extensions supersymétriques des équations structurelles des supervariétés plongées dans des superespaces." Thèse, 2017. http://hdl.handle.net/1866/20582.
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Zaimi, Meri. "Algèbres de Temperley-Lieb, Birman-Murakami-Wenzl et Askey-Wilson, et autres centralisateurs de U_q(sl_2)." Thesis, 2020. http://hdl.handle.net/1866/24381.
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Mémoire par articles.Ce mémoire contient trois articles reliés par l'idée sous-jacente d'une généralisation de la dualité de Schur-Weyl. L'objectif principal est d'obtenir une description algébrique du centralisateur de l'image de l'action diagonale de U_q(sl_2) dans le produit tensoriel de trois représentations irréductibles, lorsque q n'est pas une racine de l'unité. La relation entre une algèbre de Askey-Wilson étendue AW(3) et ce centralisateur est examinée à cet effet. Dans le premier article, les éléments du centralisateur de l'action de U_q(sl_2) dans son produit tensoriel triple sont définis à l'aide de la matrice R universelle de U_q(sl_2). Il est montré que ces éléments respectent les relations définissantes de AW(3). Dans le deuxième article, la matrice R universelle de la superalgèbre de Lie osp(1|2) est utilisée de manière similaire avec l'algèbre de Bannai-Ito BI(3). Dans ce cas, le formalisme de la matrice R permet de définir l'algèbre de Bannai-Ito de rang supérieur BI(n) comme le centralisateur de l'action de osp(1|2) dans son produit tensoriel n-fois. Le troisième article propose une conjecture qui établit un isomorphisme entre un quotient de AW(3) et le centralisateur de l'image de l'action diagonale de U_q(sl_2) dans le produit tensoriel de trois représentations irréductibles quelconques. La conjecture est prouvée pour plusieurs cas, et les algèbres de Temperley-Lieb, Birman-Murakami-Wenzl et Temperley-Lieb à une frontière sont retrouvées comme quotients de l'algèbre de Askey-Wilson.
This master thesis contains three articles related by the underlying idea of a generalization of the Schur-Weyl duality. The main objective is to obtain an algebraic description of the centralizer of the image of the diagonal action of U_q(sl_2) in the tensor product of three irreducible representations, when q is not a root of unity. The connection between a centrally extended Askey-Wilson algebra AW(3) and this centralizer is examined for this purpose. In the first article, the elements of the centralizer of the action of U_q(sl_2) in its threefold tensor product are defined with the help of the universal R-matrix of U_q(sl_2). These elements are shown to satisfy the defining relations of AW(3). In the second article, the universal R-matrix of the Lie superalgebra osp(1|2) is used in a similar fashion with the Bannai-Ito algebra BI(3). In this case, the formalism of the R-matrix allows to define the higher rank Bannai-Ito algebra BI(n) as the centralizer of the action of osp(1|2) in its n-fold tensor product. The third article proposes a conjecture that establishes an isomorphism between a quotient of AW(3) and the centralizer of the image of the diagonal action of U_q(sl_2) in the tensor product of any three irreducible representations. The conjecture is proved for several cases, and the Temperley-Lieb, Birman-Murakami-Wenzl and one-boundary Temperley-Lieb algebras are recovered as quotients of the Askey-Wilson algebra.
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Bergeron, Geoffroy. "Coefficients de Clebsch-Gordan de la super-algèbre osp(1|2)." Thèse, 2015. http://hdl.handle.net/1866/13477.
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Les fonctions génératrices des coefficients de Clebsch Gordan pour la superalgèbre de Lie osp(1|2) sont dérivées en utilisant deux approches. Une première approche généralise une méthode proposée par Granovskii et Zhedanov pour l'appliquer dans le cas de osp(1|2), une algèbre dont le coproduit est torsadé. Une seconde approche repose sur la réalisation de osp(1|2) en tant qu'algèbre dynamique d'un oscillateur parabosonique et utilise une équivalence dans cette réalisation entre le changements de coordonnées polaires à cartésiennes et le problème de Clebsch-Gordan. Un chapitre moins formel précède ces dérivations et présente comment le problème de Clebsch-Gordan s'interprète en tant que réalisation d'une algèbre de fusion. La notion abstraite de fusion est introduite, soulignant son importance en physique, pour en venir au cas particulier du problème de Clebsch-Gordan. Un survol du cas de l'algèbre osp(1|2) et de ses utilisations en physique mathématique conclut ce chapitre.The generating functions for the osp(1|2) Lie superalgebra Clebsch-Gordan coefficients are derived using two approaches. The first one consists of generalizing a method first proposed by Granovskii and Zhedanov to apply it to the case of osp(1|2), an algebra with a twisted coproduct. The second one is based on the realization of the osp(1|2) as the dynamical algebra for a parabosonic oscillator and used an equivalence in this realization between a change of basis from polar to cartesian coordinates and the Clebsch-Gordan problem. A less formal chapter precedes those derivations and present how the Clebsch-Gordan problem can be interpreted as a realization of a fusion algebra. The abstract notion of fusion is introduced, mentionning its importance in physics, and leads to the particular case of the Clebsch-Gordan problem. A brief review of the problem for the osp(1|2) algebra and its uses in mathematical physics concludes this chapter.