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Bertram, Wolfgang. "The geometry of Jordan and Lie structures /." Berlin [u.a.] : Springer, 2000. http://www.loc.gov/catdir/enhancements/fy0816/00066150-d.html.
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Ammar, Gregory, Christian Mehl, and Volker Mehrmann. "Schur-Like Forms for Matrix Lie Groups, Lie Algebras and Jordan Algebras." Universitätsbibliothek Chemnitz, 2005. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200501032.
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We describe canonical forms for elements of a classical Lie group of matrices under similarity transformations in the group. Matrices in the associated Lie algebra and Jordan algebra of matrices inherit related forms under these similarity transformations. In general, one cannot achieve diagonal or Schur form, but the form that can be achieved displays the eigenvalues of the matrix. We also discuss matrices in intersections of these classes and their Schur-like forms. Such multistructered matrices arise in applications from quantum physics and quantum chemistry.
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Barton, Christine H. "Magic squares of Lie algebras." Thesis, University of York, 2000. http://etheses.whiterose.ac.uk/10884/.
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Ricciardo, Antonio. "Lie algebras and triple systems." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amslaurea.unibo.it/8712/.
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This thesis is dedicated to the Tits-Kantor-Koecher (TKK) construction which establishes a bijective correspondence between unital Jordan algebras and shortly graded Lie algebras with Z-grading induced by an sl_2-triple. It is based on the observation that if g is a Lie algebra with a short Z-grading and f lies in g_1, then the formula ab=[[a,f],b] defines a structure of a Jordan algebra on g_{-1}. The TKK construction has been extended to Jordan triple systems and, more recently, to the so-called Kantor triple systems. These generalizations are studied in the thesis.
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Shlaka, Hasan Mohammed Ali Saeed. "Jordan-Lie inner ideals of finite dimensional associative algebras." Thesis, University of Leicester, 2018. http://hdl.handle.net/2381/42787.
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A subspace B of a Lie algebra L is said to be an inner ideal if [B, [B,L]] ⊆ B. Suppose that L is a Lie subalgebra of an associative algebra A. Then an inner ideal B of L is said to be Jordan-Lie if B2 = 0. In this thesis, we study Jordan-Lie inner ideals of finite dimensional associative algebras (with involution) and their corresponding Lie algebras over an algebraically closed field F of characteristic not 2 or 3. Let A be a finite dimensional associative algebra over F. Recall that A becomes a Lie algebra A(-) under the Lie bracket defined by [x,y] = xy - yx for all x,y ∈ A. Put A(0) = A(-) and A(k) = [A(k-1),A(k-1)] for all k ≥ 1. Let L be the Lie algebra A(k) (k ≥ 0). In the first half of this thesis, we prove that every Jordan-Lie inner ideal of L admits Levi decomposition. We get full classification of Jordan-Lie inner ideals of L satisfying a certain minimality condition. In the second half of this thesis, we study Jordan-Lie inner ideals of Lie subalgebras of finite dimensional associative algebras with involution. Let A be a finite dimensional associative algebra over F with involution * and let K(1) be the derived Lie subalgebra of the Lie algebra K of the skew-symmetric elements of A with respect to *. We classify * -regular inner ideals of K and K(1) satisfying a certain minimality condition and show that every bar-minimal * -regular inner ideal of K or K(1) is of the form eKe* for some idempotent e in A with e*e = 0. Finally, we study Jordan-Lie inner ideals of K(1) in the case when A does not have “small” quotients and show that they admit *-invariant Levi decomposition.
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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
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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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Chung, K.-W. "3-dimensional symplectic geometries and metasymplectic geometries." Thesis, University of York, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.235018.
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Silva, Diogo Diniz Pereira da Silva e. "Identidades graduadas em álgebras não-associativas." [s.n.], 2010. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306367.
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Orientador: Plamen Emilov KochloukovTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Neste trabalho apresentamos um estudo sobre identidades polinomiais graduadas em álgebras não associativas. Mais precisamente estudamos as identidades polinomiais graduadas da álgebra de Lie das matrizes de ordem 2 com traço zero com as três graduações naturais, a Z2-graduação, a Z2 _ Z2-graduação e a Z-graduação, neste caso conseguimos uma nova demonstração baseada em métodos elementares dos resultados de [27] que não se baseia em resultados da Teoria de Invariantes, estes resultados foram publicados em [30]. Estudamos também as identidades graduadas da álgebra de Jordan das matrizes simétricas de ordem 2, neste caso obtivemos bases para as identidades graduadas dessa álgebra de Jordan em todas as possíveis graduações, obtivemos também bases para as identidades fracas para os pares (Bn; Jn) e (B; J), onde Bn e B denotam as álgebras de Jordan de uma forma bilinear simétrica não degenerada nos espaços vetoriais Vn e V respectivamente, onde Vn tem dimensão n e V tem dimensão 1, esses resultados estão no artigo [29], aceito para publicação
Abstract: In this thesis we study graded identities in non associative algebras. Namely we study graded polynomial identities for the Lie algebra of the 2_2 matrices with trace zero with it's three natural gradings, the Z2-grading, the Z2_Z2-grading and the Z-grading, in this case we obtained a new proof of the results of [27] that doesn't involve use of Invariant Theory, this results were published in [30]. We also studied the graded identities of the Jordan algebra of the symmetric matrices of order two, we obtained basis for the graded identities of this Jordan algebra in all possible gradings, we also obtained basis for the weak identities of the pairs (Bn; Jn) and (B; J), where Bn and B are the Jordan algebras of a symmetric bilinear form in a the vector spaces Vn and V respectively, where Vn has dimension n and V has countable dimension, this results are in the article [29], accepted for publication
Doutorado
Álgebra Não-Comutativa
Doutor em Matemática
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Hidri, Samiha. "Formes bilinéaires invariantes sur les algèbres de Leibniz et les systèmes triples de Lie (resp. Jordan)." Thesis, Université de Lorraine, 2016. http://www.theses.fr/2016LORR0237/document.
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Dans cette thèse, on étudie la structure de quelques types d'algèbres (binaires et ternaires) munies d'une forme bilinéaire symétrique, non dégénérée et associative (ou invariante). La première partie de cette thèse est consacrée à l'étude des algèbres de Leibniz quadratiques. On montre que ces algèbres sont symétriques. De plus, on utilise la T*-extension et la double extension pour montrer plusieurs résultats sur ce type d'algèbres. Ensuite, on a remarqué que l'anti-commutativité du crochet de Lie donne naissance à de nouveaux types d'invariance pour les algèbres de Leibniz : L'invariance à gauche et l'invariance à droite. Alors, on s'est intéresse à l'étude des algèbres de Leibniz (gauche et droite) munies d'une forme bilinéaire symétrique, non dégénérée et invariante à gauche (et invariante à droite). On prouve que ces algèbres sont Lie admissibles. En second lieu, on s'intéresse aux systèmes triples de Lie et de Jordan. On débute la deuxième partie de cette thèse par la description inductive des systèmes triples de Lie quadratiques au moyen de la double extension. En plus, on introduit la T*extension des systèmes triples de Jordan pseudo-Euclidien. Finalement, on donne deux nouvelles caractérisations des systèmes triples de Jordan semi-simples parmi les systèmes triples de Jordan pseudo-EuclidiensIn this thesis, we study the stucture of several types of algebras endowed with Symmetric, non degenerate and invariant bilinear forms. In the first part, we study quadratic Leibniz algebras. First, we prove that these algebras are symmetric. Then, we use the T*-extension and the double extension to prove some properties of this type of Leibniz algebras. Besides, since we observe that the skew-symmetry of the Leibniz bracket gives rise to other types of invariance for a bilinear form on a Leibniz algebra: The left invariance and the right invariance. We focus on the study of left (resp. right) Leibniz algebras with symmetric, non degenerate and left (resp. right) invariant bilinear form. In particular, we prove that these algebras are Lie admissibles. The second part of this work is dedicated to the study of quadratic Lie triple systems and pseudo-euclidien Jordan triple systems. We start by giving an inductive description of quadratic Lie triple systems using double extension. Next, we introduce the T*-extension of Jordan triple systems. Finally, we give new caracterizations of semi-simple Jordan triple systems among pseudo-euclidian Jordan triple systems
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Pinzon, Daniel F. "VERTEX ALGEBRAS AND STRONGLY HOMOTOPY LIE ALGEBRAS." UKnowledge, 2006. http://uknowledge.uky.edu/gradschool_diss/382.
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Vertex algebras and strongly homotopy Lie algebras (SHLA) are extensively used in qunatum field theory and string theory. Recently, it was shown that a Courant algebroid can be naturally lifted to a SHLA. The 0-product in the de Rham chiral algebra has an identical formula to the Courant bracket of vector fields and 1-forms. We show that in general, a vertex algebra has an SHLA structure and that the de Rham chiral algebra has a non-zero l4 homotopy.
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Traustason, Gunnar. "Engel Lie algebras." Thesis, University of Oxford, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.334292.
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Eddy, Scott M. "Lie Groups and Lie Algebras." Youngstown State University / OhioLINK, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1320152161.
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Ben, Abdeljelil Amine. "Generalized Derivations of Ternary Lie Algebras and n-BiHom-Lie Algebras." Scholar Commons, 2019. https://scholarcommons.usf.edu/etd/7743.
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We generalize the results of Leger and Luks and other researchers about generalized derivations to the cases of ternary Lie algebras and n-BiHom Lie algebras. We investigate the derivations algebras of ternary Lie algebras induced from Lie algebras, we explore the subalgebra of quasi-derivations and give their properties. Moreover, we give a classification of the derivations algebras for low dimensional ternary Lie algebras.
For the class of n-BiHom Lie algebras, we study the algebras of generalized derivations and prove that the algebra of quasi-derivations can be embedded in the derivation algebra of a larger n-BiHom Lie algebra.
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He, Xiao. "W-algebras Associated to Truncated Current Lie Algebras." Doctoral thesis, Université Laval, 2018. http://hdl.handle.net/20.500.11794/30327.
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Étant donné une algèbre de Lie g semi-simple de dimension finie et un élément nilpotent non nul e 2 g, on peut construire plusieurs algèbres-W associées à (g; e). Parmi eux, l’algèbre-W affine est une algèbre vertex qui peut être réalisée comme une cohomologie semi-infinie d’une sous-algèbre nilpotente de ~g, où ~g est l’algèbre de Kac-Moody associée à g. L’algèbre-W finie est l’algèbre de Zhu de l’algèbre-W affine. Dans les constructions des algèbres-W, une forme bilinéaire non dégénérée invariante et une bonne Z-graduation de g jouent des rôles essentiels. Les algèbres de courants tronqués associées à g sont des quotients de l’algèbre de courants g C[t]. On peut montrer que: (1) des formes bilinéaires non dégénérées invariantes existent sur des algèbres de courants tronqués; (2) une bonne Z-graduation de g induit des bonnes Z-graduations des algèbres de courants tronqués. Alors, les constructions des algèbres-W fonctionnent bien dans le cas des algèbres de courants tronqués. Les résultats de cette thèse sont les suivants. Premièrement, nous introduisons les algèbres-W finies et affines associées aux algèbres de courants tronqués et nous généralisons certaines propriétés des algèbres-W associées aux algèbres de Lie semi-simples. Deuxièmement, nous developpons une version ajustée de la cohomologie semi-infinie, ce qui nous permet de définir les algèbres-W affines associées à des éléments nilpotents généraux d’une façon uniforme. À la fin, nous prouvons que les algèbres de Zhu de niveaux plus hauts d’une algèbre vertex conforme sont toutes isomorphes à des sous-quotients de son algèbre enveloppante universelle.Given a finite-dimensional semi-simple Lie algebra g and a non-zero nilpotent element e 2 g, one can construct various W-algebras associated to (g; e). Among them, the affine W-algebra is a vertex algebra which can be realized through semi-infinite cohomology, and the finite W-algebra is the Zhu algebra of the affineW-algebra. In the constructions ofW-algebras, a non-degenerate invariant bilinear form and a good Z-grading of g play essential roles. Truncated current Lie algebras associated to g are quotients of the current Lie algebra g C[t]. One can show that non-degenerate invariant bilinear forms exist on truncated current Lie algebras and a good Z-grading of g induces good Z-gradings of truncated current Lie algebras. The constructions of W-algebras can thus be adapted to the setting of truncated current Lie algebras. The main results of this thesis are as follows. First, we introduce finite and affine W-algebras associated to truncated current Lie algebras and generalize some properties of W-algebras associated to semi-simple Lie algebras. Second, we develop an adjusted version of semi-infinite cohomology, which helps us to define affine W-algebras associated to general nilpotent elements in a uniform way. Finally, we consider vertex operator algebras in general, and show that their higher level Zhu algebras are all isomorphic to subquotients of their universal enveloping algebras.
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Williams, Michael Peretzian. "Nilpotent N-Lie Algebras." NCSU, 2004. http://www.lib.ncsu.edu/theses/available/etd-02162004-083708/.
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In 1986, Kasymov introduced the concept of nilpotent $n$-Lie algebras, proved an analogue of Engel's Theorem and later proved an analog of Jacobson's refinement of Engel's Theorem. Despite these achievements, the subject of nilpotency in $n$-Lie algebras has not been examined in great detail in the literature since. We shall explore the concept of nilpotent $n$-Lie algebras by examining, and proving where possible, other classical nilpotent group theory and nilpotent Lie algebra results, in the $n$-Lie algebra setting.
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Levene, Rupert Howard. "Lie semigroup operator algebras." Thesis, Lancaster University, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.421841.
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Hillman, Rebecca Ann. "Relationship between Symmetric Brace Algebras and Pre-Lie Algebras." NCSU, 2005. http://www.lib.ncsu.edu/theses/available/etd-03292005-215238/.
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In this paper, we review the definitions of brace algebras, symmetric brace algebras, and pre-Lie algebras. We also look at a few examples of the calculations used in the brace algebra relations. We discuss the results of other mathematicians in these fields and where the topic of symmetric brace algebras is used. We then show a direct proof that a symmetric brace algebra is isomorphic to a pre-Lie algebra, using only the definitions. Showing that a symmetric brace algebra yields a pre-Lie algebra is very straightforward. However, the converse, a pre-Lie algebra yields a symmetric brace algebra, is not obvious. This paper uses an inductive proof to show the symmetry holds and that the brace is well-defined. Combinatorics plays a large factor in showing the brace is well-defined, along with identifying and classifying terms in a relation with m+n variables.
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Chopp, Mikaël. "Lie-admissible structures on Witt type algebras and automorphic algebras." Thesis, Metz, 2011. http://www.theses.fr/2011METZ020S/document.
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L’algèbre de Witt a été intensivement étudiée. Elle est présente dans de nombreux domaines des Mathématiques. Cette thèse est l’étude de deux généralisations de l’algèbre de Witt: les algèbres de type Witt et les algèbres de Krichever-Novikov. Dans une première partie on s’intéresse aux structures Lie-admissibles sur les algèbres de type Witt. On donne toutes les structures troisième-puissance associatives et flexibles Lie-admissibles sur ces algèbres. De plus, on étudie les formes symplectiques qui induisent un produit symétrique gauche. Dans une seconde partie on étudie les algèbres automorphes. Partant d’une surface de Riemann compacte quelconque, on considère l’action d’un sous-groupe fini du groupe des automorphismes de la surface sur des algèbres d’origines géométriques comme les algèbres de Krichever-Novikov. Plus précisément nous faisons le lien entre la sous-algèbre des éléments invariants sur la surface et l’algèbre sur la surface quotient. La structure presque-gradue des algèbres de Krichever-Novikov induit une presque-graduation sur ces sous-algèbres de certaines algèbres de Krichever- NovikovThe Witt algebra has been intensively studied and arise in many research fields in Mathematics. We are interested in two generalizations of the Witt algebra: the Witt type algebras and the Krichever-Novikov algebras. In a first part we study the problem of finding Lie-admissible structures on Witt type algebras. We give all third-power associative Lie-admissible structures and flexible Lie-admissible structures on these algebras. Moreover we study the symplectic forms which induce a graded left-symmetric product. In the second part of the thesis we study the automorphic algebras. Starting from arbitrary compact Riemann surfaces we consider the action of finite subgroups of the automorphism group of the surface on certain geometrically defined Lie algebras as the Krichever-Novikov type algebras. More precisely, we relate for G a finite subgroup of automorphism acting on the Riemann surface, the invariance subalgebras living on the surface to the algebras on the quotient surface under the group action. The almost-graded Krichever-Novikov algebras structure on the quotient gives in this way a subalgebra of a certain Krichever-Novikov algebra (with almost-grading) on the original Riemann surface
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Jamjoom, Fatmah B. "On the tensor products of JC-algebras and JW-algebras." Thesis, University of Reading, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.258348.
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Alajaji, Sami E. (Sami Emmanuel). "Central filtrations of Lie algebras." Thesis, McGill University, 1995. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=22714.
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Consider L to be a graded free Lie algebra over a principal ideal domain, and r a nonzero element of L such that its leading term s, i.e. its homogeneous component of highest order, is not a proper multiple. The main result we show in this thesis is that the graded ideal of leading terms of elements in R = (r) is equal to the ideal generated by the element s. As a consequence we prove that the center of L/R is trivial if the rank of the free Lie algebra L is greater than two.
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Alekseevsky, Dmitri, Peter W. Michor, Wolfgang Ruppert, and Peter Michor@esi ac at. "Extensions of Super Lie Algebras." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi980.ps.
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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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García, Butenegro Germán. "Hom-Lie algebras and deformations." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-43661.
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Document intends to re-establish Hom-Lie algebra theory for a wider class of morphisms on the underlying coefficient algebra. A look is taken into deformed Witt and Virasoro algebras and a new direction is taken into further quasi-Hom-Lie VIrasoro-type extensions for different Witt algebras.
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Yaseen, Hogar M. "Generalized root graded Lie algebras." Thesis, University of Leicester, 2018. http://hdl.handle.net/2381/42765.
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Let g be a non-zero finite-dimensional split semisimple Lie algebra with root system Δ. Let Γ be a finite set of integral weights of g containing Δ and {0}. We say that a Lie algebra L over F is generalized root graded, or more exactly (Γ,g)-graded, if L contains a semisimple subalgebra isomorphic to g, the g-module L is the direct sum of its weight subspaces Lα (α ∈ Γ) and L is generated by all Lα with α ̸= 0 as a Lie algebra. If g is the split simple Lie algebra and Γ = Δ∪{0} then L is said to be root-graded. Let g∼= sln and Θn = {0,±εi±ε j,±εi,±2εi | 1 ≤ i ̸= j ≤ n} where {ε1, . . . , εn} is the set of weights of the natural sln-module. Then a Lie algebra L is (Θn,g)-graded if and only if L is generated by g as an ideal and the g-module L decomposes into copies of the adjoint module, the natural module V, its symmetric and exterior squares S2V and ∧2V, their duals and the one dimensional trivial g-module. In this thesis we study properties of generalized root graded Lie algebras and focus our attention on (Θn, sln)-graded Lie algebras. We describe the multiplicative structures and the coordinate algebras of (Θn, sln)-graded Lie algebras, classify these Lie algebras and determine their central extensions.
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Roberts, Kieran. "Lie algebras and incidence geometry." Thesis, University of Birmingham, 2012. http://etheses.bham.ac.uk//id/eprint/3483/.
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An element \(\char{cmti10}{0x78}\) of a Lie algebra \(\char{cmmi10}{0x4c}\) over the field \(\char{cmmi10}{0x46}\) is extremal if [\(\char{cmti10}{0x78}\), [\(\char{cmti10}{0x78}\), \(\char{cmmi10}{0x4c}\)]] \(\subseteq\)\(\char{cmmi10}{0x46}\)\(\char{cmti10}{0x78}\). One can define the extremal geometry of \(\char{cmmi10}{0x4c}\) whose points \(\char{cmsy10}{0x45}\) are the projective points of extremal elements and lines \(\char{cmsy10}{0x46}\) are projective lines all of whose points belong to \(\char{cmsy10}{0x45}\). We prove that any finite dimensional simple Lie algebra \(\char{cmmi10}{0x4c}\) is a classical Lie algebra of type A\(_n\) if it satisfies the following properties: \(\char{cmmi10}{0x4c}\) contains no elements \(\char{cmti10}{0x78}\) such that [\(\char{cmti10}{0x78}\), [\(\char{cmti10}{0x78}\), \(\char{cmmi10}{0x4c}\)]] = 0, \(\char{cmmi10}{0x4c}\) is generated by its extremal elements and the extremal geometry \(\char{cmsy10}{0x45}\) of \(\char{cmmi10}{0x4c}\) is a root shadow space of type A\(_{n,(1,n)}\).
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Silva, Viviane Moretto da. "Algebras de Lie finitamente apresentaveis." [s.n.], 2005. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306934.
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Orientador: Dessislava Hristova KochloukovaDissertação (mestrado) - Universidade Estadual de Campinas. Instituto de Matematica, Estatistica e Computação Científica
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Resumo: Nesta dissertação de mestrado, estudamos propriedades de álgebras de Lie. As Álgebras de Lie têm grande importância nao somente na teoria de álgebras não associativas, elas surgem também em geometria, topologia e no estudo da teoria de grupos por exemplo. As definições e resultados básicos sobre álgebras de Lie estão inclusos no Capítulo 2. Para esta parte do trabalho, utilizamos os livros [1] e [2]. O nosso enfoque foi sobre álgebras universais envelopantes, mergulhando assim a álgebra de Lie em álgebras associativas (Seções 2.4, 2.5 e 2.6). O objetivo principal da dissertação foi estudar o artigo [4], ¿Finite presentation of abelian-by-finite dimensional Lie algebras¿, que classifica álgebras de Lie finitamente apresentáveis (no sentido de serem definidas por número finito de geradores e relações) que são extensões de ideal abeliano por álgebra de Lie de dimensão finita. Definimos álgebras de Lie livres na seção 2.7.Tratam-se de objetos na categoria de álgebras de Lie que satisfazem propriedade universal semelhante a definição de grupos livres. A classificação de álgebras de Lie que são extensões de ideal abeliano por álgebra de Lie de dimensão finita usa teoria de módulos Noetherianos. No Capítulo 1 incluímos resultados básicos sobre módulos, em particular estudamos módulos Noetherianos, não necessariamente sobre anéis comutativos (para este estudo utilizamos [9]), embora alguns resultados sejam válidos somente no caso onde o anel básico é comutativo (caso do Teorema da Base de Hilbert 1.31 no Capítulo 1). No final, nos Capítulos 3 e 4, explicamos de maneira bem minuciosa (com mais 6 detalhes que o original) o resultado principal de [4], que 'e apresentado na página 42: Proposicão 3.2: Seja L uma álgebra de Lie finitamente gerada sobre o corpo K. Suponha que L tenha um ideal abeliano A tal que L/A tem dimensão finita como espaço vetorial. Seja R álgebra universal envelopante de L/A. Suponha também que o quadrado tensorial A X A é finitamente gerado como R-módulo sobre a ação diagonal. Então L é finitamente apresentável. Os métodos da demonstração de 3.2 envolvem muitos cálculos com relações em L para mostrar que um conjunto finito E 'e suficiente para gerar todas as relações em L. Embora os cálculos sejam muitos, a técnica principal 'e a indução e a Identidade de Jacobi. A teoria de módulos Noetherianos também foi muito utilizada
Abstract: In this work we study the classification of finitely presented abelian-by-finite dimensional Lie algebras given in [4]. If L is a Lie algebra, an extension of an abelian ideal A by a finite dimensional Lie algebra L/A then L is finitely presented if and only if A X A is finitely generated as U(L/A)-module via the diagonal action, where U(L/A) is the universal enveloping algebra of L/A. We study in detail the result that finite generation of A X A over U(L/A) implies finite presentability of L
Mestrado
Matematica
Mestre em Matemática
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Carr, Andrew Nickolas. "Lie Algebras and Representation Theory." OpenSIUC, 2016. https://opensiuc.lib.siu.edu/theses/1988.
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Knibbeler, Vincent. "Invariants of automorphic Lie algebras." Thesis, Northumbria University, 2015. http://nrl.northumbria.ac.uk/23590/.
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Automorphic Lie Algebras arise in the context of reduction groups introduced in the late 1970s [35] in the field of integrable systems. They are subalgebras of Lie algebras over a ring of rational functions, denied by invariance under the action of a finite group, the reduction group. Since their introduction in 2005 [29, 31], mathematicians aimed to classify Automorphic Lie Algebras. Past work shows remarkable uniformity between the Lie algebras associated to different reduction groups. That is, many Automorphic Lie Algebras with nonisomorphic reduction groups are isomorphic [4, 30]. In this thesis we set out to find the origin of these observations by searching for properties that are independent of the reduction group, called invariants of Automorphic Lie Algebras. The uniformity of Automorphic Lie Algebras with nonisomorphic reduction groups starts at the Riemann sphere containing the spectral parameter, restricting the finite groups to the polyhedral groups. Through the use of classical invariant theory and the properties of this class of groups it is shown that Automorphic Lie Algebras are freely generated modules over the polynomial ring in one variable. Moreover, the number of generators equals the dimension of the base Lie algebra, yielding an invariant. This allows the definition of the determinant of invariant vectors which will turn out to be another invariant. A surprisingly simple formula is given expressing this determinant as a monomial in ground forms. All invariants are used to set up a structure theory for Automorphic Lie Algebras. This naturally leads to a cohomology theory for root systems. A first exploration of this structure theory narrows down the search for Automorphic Lie Algebras signicantly. Various particular cases are fully determined by their invariants, including most of the previously studied Automorphic Lie Algebras, thereby providing an explanation for their uniformity. In addition, the structure theory advances the classification project. For example, it clarifies the effect of a change in pole orbit resulting in various new Cartan-Weyl normal form generators for Automorphic Lie Algebras. From a more general perspective, the success of the structure theory and root system cohomology in absence of a field promises interesting theoretical developments for Lie algebras over a graded ring.
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Topley, Lewis William. "Centralisers in classical Lie algebras." Thesis, University of Manchester, 2013. https://www.research.manchester.ac.uk/portal/en/theses/centralisers-in-classical-lie-algebras(4138e280-d893-443e-b7f2-c30855dc82ee).html.
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In this thesis we shall discuss some properties of centralisers in classical Lie algebas and related structures. Let K be an algebraically closed field of characteristic p greater than or equal to 0. Let G be a simple algebraic group over K. We shall denote by g = Lie(G) the Lie algebra of G, and for x in g denote by g_x the centraliser. Our results follow three distinct but related themes: the modular representation theory of centralisers, the sheets of simple Lie algebras and the representation theory of finite W-algebras and enveloping algebras. When G is of type A or C and p > 0 is a good prime for G, we show that the invariant algebras S(g_x)^{G_x} and U(g_x)^{G_x} and polynomial algebras on rank g generators, that the algebra S(g_x)^{g_x} is generated by S(g_x)^p and S(g_x)^{G_x}, whilst U(g_x)^{g_x} is generated by U(g_x)^{G_x} and the p-centre, generalising a classical theorem of Veldkamp. We apply the latter result to confirm the first Kac-Weisfeiler conjecture for g_x, giving a precise upper bound for the dimensions of simple U(g_x)-modules. This allows us to characterise the smooth locus of the Zassenhaus variety in algebraic terms. These results correspond to an article, soon to appear in the Journal of Algebra. The results of the next chapter are particular to the case x nilpotent with G connected of type B, C or D in any characteristic good for G. Our discussion is motivated by the theory of finite W-algebras which shall occupy our discussion in the final chapter, although we make several deductions of independent interest. We begin by describing a vector space decomposition for [g_x g_x] which in turn allows us to give a formula for dim g_x^\ab where g_x^\ab := g_x / [g_x g_x]. We then concoct a combinatorial parameterisation of the sheets of g containing x and use it to classify the nilpotent orbits lying in a unique sheet. We call these orbits non-singular. Subsequently we give a formula for the maximal rank of sheets containing x and show that it coincides with dim g_x^\ab if and only if x is non-singular. The latter result is applied to show for any (not necessarily nilpotent) x in g lying in a unique sheet, that the orthogonal complement to [g_x g_x] is the tangent space to the sheet, confirming a recent conjecture. In the final chapter we set p = 0 and consider the finite W-algebra U(g,x), again with G of type B, C or D. The one dimensional representations are parameterised by the maximal spectrum of the maximal abelian quotient E = Specm U(g, x)^\ab and we classify the nilpotent elements in classical types for which E is isomorphic to an affine space A^d_K: they are precisely the non-singular elements of the previous chapter. The component group acts naturally on E and the fixed point space lies in bijective correspondence with the set of primitive ideals of U(g) for which the multiplicity of the correspoding primitive quotient is one. We call them multiplicity free. We show that this fixed point space is always an affine space, and calculate its dimension. Finally we exploit Skryabin's equivalence to study parabolic induction of multiplicity free ideals. In particular we show that every multiplicity free ideals whose associated variety is the closure of an induced orbit is itself induced from a completely prime primitive ideals with nice properties, generalising a theorem of Moeglin. The results of the final two chapters make up a part of a joint work with Alexander Premet.
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Nilsson, Jonathan. "Simple Modules over Lie Algebras." Doctoral thesis, Uppsala universitet, Algebra och geometri, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-283061.
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Simple modules are the elemental components in representation theory for Lie algebras, and numerous mathematicians have worked on their construction and classification over the last century. This thesis consists of an introduction together with four research articles on the subject of simple Lie algebra modules. In the introduction we give a light treatment of the basic structure theory for simple finite dimensional complex Lie algebras and their representations. In particular we give a brief overview of the most well-known classes of Lie algebra modules: highest weight modules, cuspidal modules, Gelfand-Zetlin modules, Whittaker modules, and parabolically induced modules. The four papers contribute to the subject by construction and classification of new classes of Lie algebra modules. The first two papers focus on U(h)-free modules of rank 1 i.e. modules which are free of rank 1 when restricted to the enveloping algebra of the Cartan subalgebra. In Paper I we classify all such modules for the special linear Lie algebras sln+1(C), and we determine which of these modules are simple. For sl2 we also obtain some additional results on tensor product decomposition. Paper II uses the theory of coherent families to obtain a similar classification for U(h)-free modules over the symplectic Lie algebras sp2n(C). We also give a proof that U(h)-free modules do not exist for any other simple finite-dimensional algebras which completes the classification. In Paper III we construct a new large family of simple generalized Whittaker modules over the general linear Lie algebra gl2n(C). This family of modules is parametrized by non-singular nxn-matrices which makes it the second largest known family of gl2n-modules after the Gelfand-Zetlin modules. In Paper IV we obtain a new class of sln+2(C)-modules by applying the techniques of parabolic induction to the U(h)-free sln+1-modules we constructed in Paper I. We determine necessary and sufficient conditions for these parabolically induced modules to be simple.
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MacDonald, Mark Lewis. "Cohomological invariants of Jordan algebras." Thesis, University of Cambridge, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.612273.
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Burroughs, Nigel John. "The quantisation of Lie groups and Lie algebras." Thesis, University of Cambridge, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.358486.
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Krook, Jonathan. "Overview of Lie Groups and Their Lie Algebras." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-275722.
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Intuitively, Lie groups are groups that are also smooth. The aim of this thesis is to describe how Lie groups are defined as smooth manifolds, and to look into their properties. To each Lie group there exists an associated vector space, which is called the Lie algebra of the Lie group. We will investigate what properties of a Lie group can be derived from its Lie algebra. As an application, we will characterise all unitary irreducible finite dimensional representations of the Lie group SO(3).Liegrupper kan ses som grupper som även är glatta. Målet med den här rapporten är att definiera Liegrupper som glatta mångfalder, och att undersöka några av liegruppernas egenskaper. Till varje Liegrupp kan man relatera ett vektorrum, som kallas Liegruppens Liealgebra. Vi kommer undersöka vilka egenskaper hos en Liegrupp som kan härledas från dess Liealgebra. Som tillämpning kommer vi karaktärisera alla unitära irreducibla ändligtdimensionella representationer av Liegruppen SO(3).
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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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Jakovljevic, Cvjetan, and University of Lethbridge Faculty of Arts and Science. "Conformal field theory and lie algebras." Thesis, Lethbridge, Alta. : University of Lethbridge, Faculty of Arts and Science, 1996, 1996. http://hdl.handle.net/10133/37.
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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.
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Dell'Arciprete, Alice. "Good gradings of simple Lie algebras." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amslaurea.unibo.it/17105/.
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This thesis aims to study the good gradings of simple finite-dimensional Lie algebras.
The Dynkin grading is given as an example of good Z-grading of a semisimple Lie algebra. The main properties of a good Z-grading of a semisimple Lie algebra are proved and all good Z-gradings of gl(n), the Lie algebra of all matrices of order n, are classified. Finally, we extend the definition of good gradings to Lie superalgebras and start studying the good gradings of the Cartan superalgebra W(n). The case of W(2) and W(3) are analyzed.
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Ellinas, D. "Studies in Lie and quantum algebras." Helsinki : Societas scientiarum Fennica, 1990. http://catalog.hathitrust.org/api/volumes/oclc/58497933.html.
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at, Andreas Cap@esi ac. "Graded Lie Algebras and Dynamical Systems." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1086.ps.
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Westrich, Quinton. "Lie Algebras in Braided Monoidal Categories." Thesis, Karlstad University, Faculty of Technology and Science, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-397.
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We begin by recalling some basic definitions from Lie algebra theory to motivate our subsequent transition to the more general setting of category theory. Next, we develop a relatively self-contained introduction to those areas of category theory needed for an understanding of what follows. Here we also motivate and introduce the graphical calculus notations. We then state the definitions of a braided commutator algebra, a braided Lie algebra, and a braided commutator Lie algebra. We proceed to show that color Lie algebras and Lie superalgebras are examples of braided Lie algebras. Thus, we are interested in examining color Lie algebras and Lie superalgebras in the generalized setting of braided Lie algebras. So we end by examining the representation theory of braided Lie algebras and braided commutator Lie algebras. In paricular, we find analogues of the adjoint representation, the tensor product representation, and the contragredient representation.
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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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Fletcher, Paul. "Lie algebras : infinite generalizations and deformations." Thesis, Durham University, 1990. http://etheses.dur.ac.uk/6187/.
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There are many applications of Lie algebras to theoretical physics. This thesis is a study of some new mathematical structures which also are applicable to current physical ideas. The structures studied are Lie algebras of infinite dimension and the deformations of Lie algebras known as quantum algebras. The approach is algebraic, although physical applications are indicated. Chapter 1 The mathematics of finite and infinite dimensional Lie algebras is reviewed, together with an indication of well established uses in physics. The terms and notation used in the rest of the thesis are introduced. Chapter 2 Explicit examples of new infinite dimensional algebras of a type related to the algebras of conformal transformations on arbitrary genus Riemann surfaces are given. The relationship of these algebras to the Virasoro algebra is discussed. Chapter 3 The sine algebra is introduced and its relationship to the Moyal bracket discussed. The finite Lie algebras are given in a trigonometric basis. The many applications of the Moyal algebra are reviewed. Chapter 4 An original proof of the uniqueness of the Moyal algebra is presented. It is shown that the Moyal bracket is the most general Lie bracket of functions of two variables, and thus that the underlying associative star product is unique. It follows that all 2-index Lie algebras correspond to the Moyal algebra in some basis. Chapter 5 Quantum deformations of Lie algebras, or quantum algebras, are introduced. The many deformations of su(2) are described and the associativity conditions are discussed. Some new higher dimensional and infinite dimensional quantum algebras are given. Chapter 6 Quantum groups are discussed as groups of transformations of the quantum plane. Higher dimensional quantum groups and quantum supergroups are also described.
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Parker, Mychelle. "Semisimple Subalgebras of Semisimple Lie Algebras." DigitalCommons@USU, 2020. https://digitalcommons.usu.edu/etd/7713.
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Let g be a Lie algebra. The subalgebra classification problem is to create a list of all subalgebras of g up to equivalence. The purpose of this thesis is to provide a software toolkit within the Differential Geometry package of Maple for classifying subalgebras of In particular the thesis will focus on classifying those subalgebras which are isomorphic to the Lie algebra sl(2) and those subalgebras of which have a basis aligned with the root space decomposition (regular subalgebras).
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Altassan, Alaa Abdullah. "Linear equations over free Lie algebras." Thesis, University of Manchester, 2013. https://www.research.manchester.ac.uk/portal/en/theses/linear-equations-over-free-liealgebras(6e29b286-1869-4207-b054-8baab98e70df).html.
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In this thesis, we study equations of the form $[x_1,u_1]+[x_2, u_2]+\ldots+[x_k,u_k]=0$ over a free Lie algebra $L$, where $k>1$ and the coefficients $u_1, u_2, \ldots,u_k$ belong to $L$. The starting point of this research is a paper [22], in which the authors embarked on a systematic study of very concrete linear equations over free Lie algebras. They focused on the given equations in the case where $k=2$. We generalise and develop a number of the results on equations with two variables to equations with an arbitrary number of indeterminates. Most of the results refer to the case where the coefficients coincide with the free generators of $L$. Throughout our research, we study some features of the solution space of these equations such as the homogenous structure and the fine homogenous structure. The main achievement in this work is that we give a detailed description of the solution space. Then we obtain explicit bases for some specific fine homogeneous components of the solution space, in particular, we give a basis for the "multilinear'' fine homogenous component. Moreover, we generalise earlier results on commutator calculus using the "language'' of free Lie algebras and apply them to determine the radical and the coordinate algebra of the solution space of the given equations.
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Pike, Jeffrey. "Quivers and Three-Dimensional Lie Algebras." Thesis, Université d'Ottawa / University of Ottawa, 2015. http://hdl.handle.net/10393/32398.
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We study a family of three-dimensional Lie algebras that depend on a continuous parameter. We introduce certain quivers and prove that idempotented versions of the enveloping algebras of the Lie algebras are isomorphic to the path algebras of these quivers modulo certain ideals in the case that the free parameter is rational and non-rational, respectively. We then show how the representation theory of the introduced quivers can be related to the representation theory of quivers of affine type A, and use this relationship to study representations of the family of Lie algebras of interest. In particular, though it is known that this particular family of Lie algebras consists of algebras of wild representation type, we show that if we impose certain restrictions on weight decompositions, we obtain full subcategories of the category of representations that are of finite or tame representation type.
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Wood, Lisa M. "ON THE SOLVABLE LENGTH OF ASSOCIATIVE ALGEBRAS, MATRIX GROUPS, AND LIE ALGEBRAS." NCSU, 2004. http://www.lib.ncsu.edu/theses/available/etd-10272004-164622/.
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Let A be an algebraic system with product a*b between elements a and b in A. It is of interest to compare the solvable length t with other invariants, for instance size, order, or dimension of A. Thus we ask, for a given t what is the smallest n such that there is an A of length t and invariant n. It is this problem that we consider for associative algebras, matrix groups, and Lie algebras. We consider A in each case to be subsets of (strictly) upper triangular n by n matrices. Then the invariant is n. We do these for the associative (Lie) algebras of all strictly upper triangular n by n matrices and for the full n by n upper triangular unipotent groups. The answer for n is the same in all cases. Then we restrict the problem to a fixed number of generators. In particular, using only 3 generators and we get the same results for matrix groups and Lie algebras as for the earlier problem. For associative algebras with 1 generator we also get the same result as the general associative algebra case. Finally we consider Lie algebras with 2 generators and here n is larger than in the general case. We also consider the problem of finding the dimension in the associative algebra, the general, and 3 generator Lie algebra cases.
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Shabanskaya, Anastasia V. "Classification of Six Dimensional Solvable Indecomposable Lie Algebras with a codimension one nilradical over ℝ." University of Toledo / OhioLINK, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=toledo1301590879.
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Cao, Mengyuan. "Representation Theory of Lie Colour Algebras and Its Connection with the Brauer Algebras." Thesis, Université d'Ottawa / University of Ottawa, 2018. http://hdl.handle.net/10393/38125.
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In this thesis, we study the representation theory of Lie colour algebras. Our strategy follows the work of G. Benkart, C. L. Shader and A. Ram in 1998, which is to use the Brauer algebras which appear as the commutant of the orthosymplectic Lie colour algebra when they act on a k-fold tensor product of the standard representation. We give a general combinatorial construction of highest weight vectors using tableaux, and compute characters of the irreducible summands in some borderline cases. Along the way, we prove the RSK-correspondence for tableaux and the PBW theorem for Lie colour algebras.
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Chang, Hao [Verfasser]. "Varieties of elementary Lie algebras / Hao Chang." Kiel : Universitätsbibliothek Kiel, 2016. http://d-nb.info/1105472140/34.
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Zusmanovich, Pasha. "Low-dimensional cohomology of current Lie algebras." Doctoral thesis, Stockholm : Department of Mathematics, Stockholm University, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-37768.
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