Dissertations / Theses: 'Lie groups'

1

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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2

Ahluwalia, Kanwardeep Singh. "Lie bialgebras and Poisson lie groups." Thesis, University of Cambridge, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.388758.

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3

pl, tomasz@uci agh edu. "A Lie Group Structure on Strict Groups." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1076.ps.

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4

Harkins, Andrew. "Combining lattices of soluble lie groups." Thesis, University of Newcastle Upon Tyne, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.341777.

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5

Öhrnell, Carl. "Lie Groups and PDE." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-420706.

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6

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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7

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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8

Ray, Jishnu. "Iwasawa algebras for p-adic Lie groups and Galois groups." Thesis, Université Paris-Saclay (ComUE), 2018. http://www.theses.fr/2018SACLS189/document.

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Un outil clé dans la théorie des représentations p-adiques est l'algèbre d'Iwasawa, construit par Iwasawa pour étudier les nombres de classes d'une tour de corps de nombres. Pour un nombre premier p, l'algèbre d'Iwasawa d'un groupe de Lie p-adique G, est l'algèbre de groupe G complétée non-commutative. C'est aussi l'algèbre des mesures p-adiques sur G. Les objets provenant de groupes semi-simples, simplement connectés ont des présentations explicites comme la présentation par Serre des algèbres semi-simples et la présentation de groupe de Chevalley par Steinberg. Dans la partie I, nous donnons une description explicite des certaines algèbres d'Iwasawa. Nous trouvons une présentation explicite (par générateurs et relations) de l'algèbre d'Iwasawa pour le sous-groupe de congruence principal de tout groupe de Chevalley semi-simple, scindé et simplement connexe sur Z_p. Nous étendons également la méthode pour l'algèbre d'Iwasawa du sous-groupe pro-p Iwahori de GL (n, Z_p). Motivé par le changement de base entre les algèbres d'Iwasawa sur une extension de Q_p nous étudions les représentations p-adiques globalement analytiques au sens d'Emerton. Nous fournissons également des résultats concernant la représentation de série principale globalement analytique sous l'action du sous-groupe pro-p Iwahori de GL (n, Z_p) et déterminons la condition d'irréductibilité. Dans la partie II, nous faisons des expériences numériques en utilisant SAGE pour confirmer heuristiquement la conjecture de Greenberg sur la p-rationalité affirmant l'existence de corps de nombres "p-rationnels" ayant des groupes de Galois (Z/2Z)^t. Les corps p-rationnels sont des corps de nombres algébriques dont la cohomologie galoisienne est particulièrement simple. Ils sont utilisés pour construire des représentations galoisiennes ayant des images ouvertes. En généralisant le travail de Greenberg, nous construisons de nouvelles représentations galoisiennes du groupe de Galois absolu de Q ayant des images ouvertes dans des groupes réductifs sur Z_p (ex GL (n, Z_p), SL (n, Z_p ), SO (n, Z_p), Sp (2n, Z_p)). Nous prouvons des résultats qui montrent l'existence d'extensions de Lie p-adiques de Q où le groupe de Galois correspond à une certaine algèbre de Lie p-adique (par exemple sl(n), so(n), sp(2n)). Cela répond au problème classique de Galois inverse pour l'algèbre de Lie simple p-adique
A key tool in p-adic representation theory is the Iwasawa algebra, originally constructed by Iwasawa in 1960's to study the class groups of number fields. Since then, it appeared in varied settings such as Lazard's work on p-adic Lie groups and Fontaine's work on local Galois representations. For a prime p, the Iwasawa algebra of a p-adic Lie group G, is a non-commutative completed group algebra of G which is also the algebra of p-adic measures on G. It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups as noticed by Clozel. In Part I, we lay the foundation by giving an explicit description of certain Iwasawa algebras. We first find an explicit presentation (by generators and relations) of the Iwasawa algebra for the principal congruence subgroup of any semi-simple, simply connected Chevalley group over Z_p. Furthermore, we extend the method to give a set of generators and relations for the Iwasawa algebra of the pro-p Iwahori subgroup of GL(n,Z_p). The base change map between the Iwasawa algebras over an extension of Q_p motivates us to study the globally analytic p-adic representations following Emerton's work. We also provide results concerning the globally analytic induced principal series representation under the action of the pro-p Iwahori subgroup of GL(n,Z_p) and determine its condition of irreducibility. In Part II, we do numerical experiments using a computer algebra system SAGE which give heuristic support to Greenberg's p-rationality conjecture affirming the existence of "p-rational" number fields with Galois groups (Z/2Z)^t. The p-rational fields are algebraic number fields whose Galois cohomology is particularly simple and they offer ways of constructing Galois representations with big open images. We go beyond Greenberg's work and construct new Galois representations of the absolute Galois group of Q with big open images in reductive groups over Z_p (ex. GL(n, Z_p), SL(n, Z_p), SO(n, Z_p), Sp(2n, Z_p)). We are proving results which show the existence of p-adic Lie extensions of Q where the Galois group corresponds to a certain specific p-adic Lie algebra (ex. sl(n), so(n), sp(2n)). This relates our work with a more general and classical inverse Galois problem for p-adic Lie extensions

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9

Jimenez, William. "Riemannian submersions and Lie groups." College Park, Md. : University of Maryland, 2005. http://hdl.handle.net/1903/2648.

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Thesis (Ph. D.) -- University of Maryland, College Park, 2005.
Thesis research directed by: Mathematics. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.

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10

Hindeleh, Firas Y. "Tangent and Cotangent Bundles, Automorphism Groups and Representations of Lie Groups." University of Toledo / OhioLINK, 2006. http://rave.ohiolink.edu/etdc/view?acc_num=toledo1153933389.

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11

Snopçe, Ilir. "Lie methods in pro-p groups." Diss., Online access via UMI:, 2009.

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12

Groves, Daniel. "Problems in Lie rings and groups." Thesis, University of Oxford, 2000. http://ora.ox.ac.uk/objects/uuid:4b5479ad-30ac-4ad6-98a3-51484095868b.

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We construct a Lie relator which is not an identical Lie relator. This is the first known example of a non-identical Lie relator. Next we consider the existence of torsion in outer commutator groups. Let L be a free Lie ring. Suppose that 1 < i ≤ j ≤ 2i and i ≤ k ≤ i + j + 1. We prove that L/[L^j, Lⁱ, L^k<./em>] is torsion free. Also, we prove that if 1 < i ≤ j ≤ 2i and j ≤ k ≤ l ≤ i + j then L/[L^j, Lⁱ, L^k, L^l] is torsion free. We then prove that the analogous groups, namely F/[γ_j(F),γ_i(F),γ_k(F)] and F/[γ_j(F),γ_i(F),γ_k(F),γ_l(F)] (under the same conditions for i, j, k and i, j, k, l respectively), are residually nilpotent and torsion free. We prove the existence of 2-torsion in the Lie rings L/[L^j, Lⁱ, L^k] when 1 ≤ k < i,j ≤ 5, and thus show that our methods do not work in these cases. Finally, we consider the order of finite groups of exponent 8. For m ≥ 2, we define the function T(m,n) by T(m,1) = m and T(m,k + 1) = m^T(m,k). We prove that if G is a finite m-generator group of exponent 8 then |G| ≤ T(m, 7⁴⁷¹), improving upon the best previously known bound of T(m, 8⁸⁸).

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13

Reynolds, Matthew. "Signalizers in groups of Lie type." Thesis, University of Warwick, 2013. http://wrap.warwick.ac.uk/57753/.

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14

Elkasapy, Abdelrhman. "Word Maps on Compact Lie Groups." Doctoral thesis, Universitätsbibliothek Leipzig, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-184066.

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15

Chern, Shikai. "Dirac Induction for Unimodular Lie Groups /." The Ohio State University, 1996. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487933648648503.

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16

Bou-Rabee, Nawaf Marsden Jerrold E. Marsden Jerrold E. "Hamilton-Pontryagin integrators on Lie groups /." Diss., Pasadena, Calif. : California Institute of Technology, 2007. http://resolver.caltech.edu/CaltechETD:etd-06052007-153115.

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17

Santacruz, Camilo Andres Angulo. "A cohomology theory for Lie 2-algebras and Lie 2-groups." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-15022019-084657/.

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In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and Lie groups in that their second groups classify extensions. We use this fact together with an adapted van Est map to prove the integrability of Lie 2-algebras anew.
Nesta tese, nós introduzimos uma nova teoria de cohomologia associada às 2-álgebras de Lie e uma nova teoria de cohomologia associada aos 2-grupos de Lie. Prova-se que estas teorias de cohomologia estendem as teorias de cohomologia clássicas de álgebras de Lie e grupos de Lie em que os seus segundos grupos classificam extensões. Finalmente, usaremos estos fatos junto com um morfismo de van Est adaptado para encontrar uma nova prova da integrabilidade das 2-álgebras de Lie.

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18

Sale, Andrew W. "The length of conjugators in solvable groups and lattices of semisimple Lie groups." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:ea21dab2-2da1-406a-bd4f-5457ab02a011.

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The conjugacy length function of a group Γ determines, for a given a pair of conjugate elements u,v ∈ Γ, an upper bound for the shortest γ in Γ such that uγ = γv, relative to the lengths of u and v. This thesis focuses on estimating the conjugacy length function in certain finitely generated groups. We first look at a collection of solvable groups. We see how the lamplighter groups have a linear conjugacy length function; we find a cubic upper bound for free solvable groups; for solvable Baumslag--Solitar groups it is linear, while for a larger family of abelian-by-cyclic groups we get either a linear or exponential upper bound; also we show that for certain polycyclic metabelian groups it is at most exponential. We also investigate how taking a wreath product effects conjugacy length, as well as other group extensions. The Magnus embedding is an important tool in the study of free solvable groups. It embeds a free solvable group into a wreath product of a free abelian group and a free solvable group of shorter derived length. Within this thesis we show that the Magnus embedding is a quasi-isometric embedding. This result is not only used for obtaining an upper bound on the conjugacy length function of free solvable groups, but also for giving a lower bound for their L_p compression exponents. Conjugacy length is also studied between certain types of elements in lattices of higher-rank semisimple real Lie groups. In particular we obtain linear upper bounds for the length of a conjugator from the ambient Lie group within certain families of real hyperbolic elements and unipotent elements. For the former we use the geometry of the associated symmetric space, while for the latter algebraic techniques are employed.

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19

Hindeleh, Firas. "Tangent and cotangent bundles automorphism groups and representations of Lie groups /." See Full Text at OhioLINK ETD Center (Requires Adobe Acrobat Reader for viewing), 2006. http://www.ohiolink.edu/etd/view.cgi?acc_num=toledo1153933389.

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Thesis (Ph.D.)--University of Toledo, 2006.
Typescript. "A dissertation [submitted] as partial fulfillment of the requirements of the Doctor of Philosophy degree in Mathematics." Bibliography: leaves 79-82.

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20

Wickramasekara, Sujeewa, and sujeewa@physics utexas edu. "On the Representations of Lie Groups and Lie Algebras in Rigged Hilbert." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi994.ps.

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21

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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22

Lupi, Giulia. "Kernel approximations in Lie groups and application to group-invariant CNN." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/23905/.

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In questa tesi viene studiata un'equazione di convezione-diffusione-erosione introdotta in problemi di image processing. In particolare, si cercano approssimazioni dei nuclei per l'equazione di diffusione e per l'equazione di erosione. Per fare tali approssimazioni si é utilizzato il metodo della parametrice per l'equazione di diffusione, mentre il nucleo dell'equazione di erosione viene trovato a partire dal nucleo dell'equazione di diffusione attraverso la trasformata di Cramér-Fuorier.

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23

Ebert, Svend. "Wavelets on Lie groups and homogeneous spaces." Doctoral thesis, Technische Universitaet Bergakademie Freiberg Universitaetsbibliothek "Georgius Agricola", 2011. http://nbn-resolving.de/urn:nbn:de:bsz:105-qucosa-78988.

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Within the past decades, wavelets and associated wavelet transforms have been intensively investigated in both applied and pure mathematics. They and the related multi-scale analysis provide essential tools to describe, analyse and modify signals, images or, in rather abstract concepts, functions, function spaces and associated operators. We introduce the concept of diffusive wavelets where the dilation operator is provided by an evolution like process that comes from an approximate identity. The translation operator is naturally defined by a regular representation of the Lie group where we want to construct wavelets. For compact Lie groups the theory can be formulated in a very elegant way and also for homogeneous spaces of those groups we formulate the theory in the theory of non-commutative harmonic analysis. Explicit realisation are given for the Rotation group SO(3), the k-Torus, the Spin group and the n-sphere as homogeneous space. As non compact example we discuss diffusive wavelets on the Heisenberg group, where the construction succeeds thanks to existence of the Plancherel measure for this group. The last chapter is devoted to the Radon transform on SO(3), where the application on diffusive wavelets can be used for its inversion. The discussion of a variational spline approach provides criteria for the choice of points for measurements in concrete applications.

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24

Reyes, Carrion Ramon. "Some special geometries defined by Lie groups." Thesis, University of Oxford, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.357592.

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25

Zerbes, Sarah. "Selmer groups over p-adic Lie extensions." Thesis, University of Cambridge, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.613792.

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26

Maclean, Craig David. "Analytical motion planning on 3D-Lie groups." Thesis, University of Strathclyde, 2014. http://oleg.lib.strath.ac.uk:80/R/?func=dbin-jump-full&object_id=24388.

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In this thesis optimal analytical controls derived using the framework of geometric control theory are used to plan motions for real engineering systems, and their performance assessed. Beginning with the nonholonomic wheeled robot on the 3-D Special Euclidean group, the co-ordinate free Maximum Principle of optimal control is used to create a kinematic motion planner via parametric optimisation. The reachable sets of the planner are investigated, and an obstacle avoidance framework is created and demonstrated. The practical applicability of this motion planner is assessed in both the unit speed and arbitrary speed cases. Subsequently, the natural motions of an axisymmetric and asymmetric rigid body are derived in convenient quaternion form, which is particularly suitable for practical implementation. The initial angular velocities of the rigid body are parametrically optimised to produce attitude reference tracks for a small spacecraft. It is shown through comparison with a standard proportional-derivative controller that the natural motions require lower accumulated torque to track. Additional testing of the axisymmetric references, which are comprised of simple trigonometric functions, found that the references can be utilised in a "bang-off-bang" manner in low disturbance environments to save on computation and control effort. Next, the general solution to the optimal control problem of a rigid body constrained to spin at a constant rate is derived. This takes the form of a Weierstrass elliptic function which is impractical for motion planning. However, a particular case is identified that is comprised of trigonometric functions and utilised to plan efficient motions for a spinning solar sail and compared in simulation to a pure spin benchmark. The geometric references are found to limit the spacecraft body rates, while tracking requires lower accumulated torque in most cases. Finally, an actuator study is carried out to identify the technology requirements for reference tracking for a spinning solar sail.

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27

Melcher, Tai Alexis. "Hypoelliptic heat kernel inequalities on lie groups /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2004. http://wwwlib.umi.com/cr/ucsd/fullcit?p3129940.

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28

Günther, Janne-Kathrin. "The C*-algebras of certain Lie groups." Thesis, Université de Lorraine, 2016. http://www.theses.fr/2016LORR0118/document.

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Dans la présente thèse de doctorat, les C*-algèbres des groupes de Lie connexes réels nilpotents de pas deux et du groupe de Lie SL(2,R) sont caractérisées. En outre, comme préparation à une analyse de sa C*-algèbre, la topologie du spectre du produit semi-direct U(n) x H_n est décrite, où H_n dénote le groupe de Lie de Heisenberg et U(n) le groupe unitaire qui agit sur H_n par automorphismes. Pour la détermination des C*-algèbres de groupes, la transformation de Fourier à valeurs opérationnelles est utilisée pour appliquer chaque C*-algèbre dans l'algèbre de tous les champs d'opérateurs bornés sur son spectre. On doit trouver les conditions que satisfait l'image de cette C*-algèbre sous la transformation de Fourier et l'objectif est de la caractériser par ces conditions. Dans cette thèse, il est démontré que les C*-algèbres des groupes de Lie connexes réels nilpotents de pas deux et la C*-algèbre de SL(2,R) satisfont les mêmes conditions, des conditions appelées «limites duales sous contrôle normique». De cette manière, ces C*-algèbres sont décrites dans ce travail et les conditions «limites duales sous contrôle normique» sont explicitement calculées dans les deux cas. Les méthodes utilisées pour les groupes de Lie nilpotents de pas deux et pour le groupe SL(2,R) sont très différentes l'une de l'autre. Pour les groupes de Lie nilpotents de pas deux, on regarde leurs orbites coadjointes et on utilise la théorie de Kirillov, alors que pour le groupe SL(2,R), on peut mener les calculs plus directement
In this doctoral thesis, the C*-algebras of the connected real two-step nilpotent Lie groups and the Lie group SL(2,R) are characterized. Furthermore, as a preparation for an analysis of its C*-algebra, the topology of the spectrum of the semidirect product U(n) x H_n is described, where H_n denotes the Heisenberg Lie group and U(n) the unitary group acting by automorphisms on H_n. For the determination of the group C*-algebras, the operator valued Fourier transform is used in order to map the respective C*-algebra into the algebra of all bounded operator fields over its spectrum. One has to find the conditions that are satisfied by the image of this C*-algebra under the Fourier transform and the aim is to characterize it through these conditions. In the present thesis, it is proved that both the C*-algebras of the connected real two-step nilpotent Lie groups and the C*-algebra of SL(2,R) fulfill the same conditions, namely the “norm controlled dual limit” conditions. Thereby, these C*-algebras are described in this work and the “norm controlled dual limit” conditions are explicitly computed in both cases. The methods used for the two-step nilpotent Lie groups and the group SL(2,R) are completely different from each other. For the two-step nilpotent Lie groups, one regards their coadjoint orbits and uses the Kirillov theory, while for the group SL(2,R) one can accomplish the calculations more directly

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Günther, Janne-Kathrin. "The C*-algebras of certain Lie groups." Electronic Thesis or Diss., Université de Lorraine, 2016. http://www.theses.fr/2016LORR0118.

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Dans la présente thèse de doctorat, les C*-algèbres des groupes de Lie connexes réels nilpotents de pas deux et du groupe de Lie SL(2,R) sont caractérisées. En outre, comme préparation à une analyse de sa C*-algèbre, la topologie du spectre du produit semi-direct U(n) x H_n est décrite, où H_n dénote le groupe de Lie de Heisenberg et U(n) le groupe unitaire qui agit sur H_n par automorphismes. Pour la détermination des C*-algèbres de groupes, la transformation de Fourier à valeurs opérationnelles est utilisée pour appliquer chaque C*-algèbre dans l'algèbre de tous les champs d'opérateurs bornés sur son spectre. On doit trouver les conditions que satisfait l'image de cette C*-algèbre sous la transformation de Fourier et l'objectif est de la caractériser par ces conditions. Dans cette thèse, il est démontré que les C*-algèbres des groupes de Lie connexes réels nilpotents de pas deux et la C*-algèbre de SL(2,R) satisfont les mêmes conditions, des conditions appelées «limites duales sous contrôle normique». De cette manière, ces C*-algèbres sont décrites dans ce travail et les conditions «limites duales sous contrôle normique» sont explicitement calculées dans les deux cas. Les méthodes utilisées pour les groupes de Lie nilpotents de pas deux et pour le groupe SL(2,R) sont très différentes l'une de l'autre. Pour les groupes de Lie nilpotents de pas deux, on regarde leurs orbites coadjointes et on utilise la théorie de Kirillov, alors que pour le groupe SL(2,R), on peut mener les calculs plus directement
In this doctoral thesis, the C*-algebras of the connected real two-step nilpotent Lie groups and the Lie group SL(2,R) are characterized. Furthermore, as a preparation for an analysis of its C*-algebra, the topology of the spectrum of the semidirect product U(n) x H_n is described, where H_n denotes the Heisenberg Lie group and U(n) the unitary group acting by automorphisms on H_n. For the determination of the group C*-algebras, the operator valued Fourier transform is used in order to map the respective C*-algebra into the algebra of all bounded operator fields over its spectrum. One has to find the conditions that are satisfied by the image of this C*-algebra under the Fourier transform and the aim is to characterize it through these conditions. In the present thesis, it is proved that both the C*-algebras of the connected real two-step nilpotent Lie groups and the C*-algebra of SL(2,R) fulfill the same conditions, namely the “norm controlled dual limit” conditions. Thereby, these C*-algebras are described in this work and the “norm controlled dual limit” conditions are explicitly computed in both cases. The methods used for the two-step nilpotent Lie groups and the group SL(2,R) are completely different from each other. For the two-step nilpotent Lie groups, one regards their coadjoint orbits and uses the Kirillov theory, while for the group SL(2,R) one can accomplish the calculations more directly

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30

Chahbazian, Clément. "Particle Filtering on Lie Groups : Application to Navigation." Electronic Thesis or Diss., université Paris-Saclay, 2023. http://www.theses.fr/2023UPAST069.

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L'estimation Bayésienne est une discipline importante dans un grand nombre de domaines scientifiques et techniques. Elle se base sur le théorème de Bayes qui permet d'associer une observation avec une connaissance a priori sur un évènement ou un paramètre. Cependant, ce théorème ne peut pas être résolu analytiquement en présence de fortes non-linéarités et de nombreuses méthodes ont été développées pour le traiter numériquement. Parmi elles, les filtres particulaires représentent les densités de probabilités avec un nuage de particules. Cette approche permet de traiter des problèmes fortement non-linéaires avec un cadre générique. Cependant, les filtres particulaires présentent des défis, tels que l'étape de ré-échantillonnage, la résolution de problèmes de grande dimension, ainsi que la charge calculatoire. En outre, des études portant sur des algorithmes d'estimation dans les groupes de Lie ont démontré l'intérêt de ces approches sur de nombreux aspects. En effet, représenter les variables d'estimation sur les groupes de Lie permet d'utiliser les propriétés algébriques et géométriques de ces espaces et amène à une gestion naturelle des incertitudes. Ainsi, les filtres obtenus présentent une amélioration de leur précision et de leur robustesse par rapport aux approches classiques. Cette thèse porte sur le domaine nouveau du filtrage particulaire dans les groupes de Lie. Elle établit donc un ensemble filtres particulaires résolvant le théorème de Bayes dans les groupes de Lie en se focalisant sur différents aspects de ces algorithmes, tels que l'étape de ré-échantillonnage, la représentation des particules, ou encore une nouvelle borne inférieure d'erreur. Les méthodes proposées sont appliquées à la navigation de systèmes autonomes qui ont besoin d'algorithmes robustes pour estimer leur état (position, vitesse, attitude) afin d'effectuer leur contrôle et leur guidage. Une centrale inertielle (IMU) est généralement utilisée pour exécuter la fonction de navigation. Cependant, ces capteurs dérivent et doivent être régulièrement mis à jour à l'aide de mesures de capteurs supplémentaires et d'un filtre de navigation assurant la fusion de données. Ainsi, les algorithmes proposés dans la thèse sont testés sur des scénarios de navigation exigeants, et ont démontré un gain significatif en précision et en robustesse par rapport aux méthodes classiques
Bayesian estimation is an important discipline in many scientific and technical domains. It is based on Bayes' theorem, which allows to associate an observation with an a priori knowledge about an event or a parameter. However, this theorem cannot be solved analytically in the case of strong non-linearities. Thus, many methods were developed to address this problem numerically. Among them, particle filters represent probability densities with a cloud of particles. This allows to solve strongly nonlinear problems with a generic approach. However, particle filters present several challenges, such as the resampling step, the resolution of high-dimensional problems, and the computational load. Moreover, studies on estimation algorithms in Lie groups demonstrated the interest of these approaches in many aspects. Indeed, representing the estimation variables on Lie groups allows the use of algebraic and geometric properties of these spaces and leads to a natural handling of uncertainties. Thus, filters on Lie groups show improved accuracy and robustness compared to conventional approaches. This thesis focuses on the new field of particle filtering on Lie groups. It establishes a class of particle filters solving Bayes' theorem on Lie groups by focusing on different aspects of these algorithms, such as the resampling step, the particle representation, and the lower error bound. Furthermore, the proposed methods are applied to the navigation of autonomous systems that need robust algorithms to estimate their state (position, velocity, attitude) in order to perform their control and guidance. An inertial measurement unit (IMU) is usually used to complete the navigation function. However, these sensors drift and need to be frequently updated with aiding sensor measurements, which requires a navigation filter for data fusion. Thus, the algorithms presented in this thesis are tested on challenging navigation scenarios, and demonstrated a significant gain in accuracy and robustness compared to conventional methods

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31

Stefanicki, Tomasz. "On subalgebras of free Lie algebras and on the Lie algebra associated to the lower central series of a group." Thesis, McGill University, 1987. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=63885.

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32

Giroux, Yves. "Degenerate enveloping algebras of low-rank groups." Thesis, McGill University, 1986. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=74026.

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33

Semple, James Fraser. "Completion of restricted Lie algebras and collapsing groups." Thesis, University of Oxford, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.316874.

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34

Baarnhielm, Jonas Henrik Ambjorn. "Algorithmic problems in twisted groups of lie type." Thesis, Queen Mary, University of London, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.498313.

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35

Biggs, James D. "Integrable Hamiltonian systems on six dimensional Lie groups." Thesis, University of Reading, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.443934.

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36

Ayala, V., and Hacibekiroglu A. Kara. "Examples of linear control systems on Lie groups." Pontificia Universidad Católica del Perú, 2013. http://repositorio.pucp.edu.pe/index/handle/123456789/95402.

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37

Sanders, Paul Jonathon. "Prime-power Lie algebras and finite p-groups." Thesis, University of Warwick, 1994. http://wrap.warwick.ac.uk/107557/.

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In this thesis we use the Lie ring functors of Magnus and Lazard to investigate finite p-groups which possess either a cyclic subgroup of small index, or whose derived subgroup has exponent dividing p. The class of groups we consider is sufficiently large to include completely 11 of the 15 families of groups of order p7 (p > 7), partitioned by Hall’s type invariants. Some information on the other 4 families is also derived. Motivated by the work of Burnside [3] and Miller [21] concerning the stable behaviour of groups of order pn and exponent pn-2 for p and n sufficiently large, we investigate the existence of stability for the number of isomorphism classes of groups of order pn and exponent pn-f as p and n vary, with f an arbitrary fixed integer greater than 2. To facilitate this, we allow finitely many specified primes to be excluded for each / at the outset. The approach is to then consider the corresponding problem for pn—element nilpotent Lie rings and use the Lie ring functors to recover a solution for the groups. A non-trivial step immediately arises in showing the existence of an initial set of excluded primes which are sufficient to enable the Lie ring functors to be invoked since they only apply when the prime is greater than the nilpotency class (of both the groups and Lie rings). This step is dealt with and explored in chapters 2 and 3. In the main theorem of chapter 4 we show that for f > 3 the number of isomorphism classes of groups of order pn and exponent pn-f is independent of n for n sufficiently large and p not one of the excluded primes (which depend only on f). The excluded primes ensure regularity holds for such groups and the proof of this theorem yields precise stability results in terms of the corresponding type invariants. The method of proof shows explicitly how to construct the corresponding Lie rings, and in chapter 5 we utilise this procedure to produce a formula for the number of groups of order pn and exponent p"—3 where p > 5 and n > 7. The precise stability results of chapter 4 enable us to reduce some of the calculations to the known classification of groups of order p5 (p > 5). On the other hand, in chapter 6 we use the Lie ring functors to solve a restricted form of a conjecture of J. Moody [22] by exhibiting, for a prime p greater than or equal to the positive integer n, a natural, but not functorial, one-to-one correspondence between isomorphism classes of finite groups of order pn whose derived subgroup has exponent dividing p, and isomorphism classes of nilpotent Fp[T]/(T")—Lie algebras L of Fp—dimension n in which T[L,L] = 0. By viewing such an algebra as a nilpotent Fp —Lie algebra equipped with a nilpotent element of its centroid one obtains a “formula” for the number of such groups. This applies, in particular, to the groups of order p7 since the 7-dimensional nilpotent Fp —Lie algebras are known from [30] (and the Magnus Lazard functors) for p > 7. In view of current interest in these groups, we conclude with a summary of the information contained in this thesis on groups of order p7.

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38

Wahlström, Josefin. "An Introduction to Kleinian Geometry via Lie Groups." Thesis, Uppsala universitet, Algebra och geometri, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-422749.

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39

Damers, Julien. "Lie groups applied to localisation of mobile robots." Electronic Thesis or Diss., Brest, École nationale supérieure de techniques avancées Bretagne, 2022. http://www.theses.fr/2022ENTA0007.

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Dans le cadre du développement des énergies renouvelables, les activités offshores liées aux parcs éoliens ont grandement augmenté en nombre. Ainsi les coûts de maintenance et de surveillance de telles structures aussi bien en hommes qu'en moyens se sont accrus et cette tendance s'amplifiera sûrement au cours des prochaines années. Ceci a encouragé le développement des véhicules sous-marins autonomes (AUV). Ceux-ci sont encore très coûteux du fait des capteurs onéreux qu'ils embarquent. Ils sont donc encore rares, ce qui ne suffit pas à combler les actuels besoins. Ainsi le développement d'AUVs à bas-coût est un sujet de recherche en forte croissance. C'est dans ce champ de recherche que cette thèse vient s'inscrire. Un des problèmes principaux rencontré en robotique sous-marine est la localisation de l'engin. C'est la résolution de celui-ci qui nous servira d’objectif tout au long de ce travail. Pour ce faire, une nouvelle méthode d'intégration garantie, robuste aux incertitudes sur les conditions initiales est présentées dans ce manuscrit. Celle-ci se base sur l'utilisation des symétries de Lie appliquées aux équations différentielles. Nous la présenterons en premier lieu sur un problème théorique simple avant de l’appliquer sur le problème de localisation d’un robot sous-marin
With the development of offshore activities, the costs of maintenance and monitoring of offshore plants in terms of crew members, boats, and money have greatly increased and are still growing dramatically. This encouraged the development of autonomous underwater vehicles (AUV). These are still very expensive because of the numerous high-end sensors they need to embark on to accomplish their missions. Thus their number is relatively low. Therefore research is made to develop low-cost AUVs that could be produced in a larger amount to perform the same missions. This thesis comes within the scope of this research field. One of the main problems when dealing with AUVs is the localisation of the vehicle which will be the one addressed throughout this work. To tackle it, we present a new guaranteed integration method, which is more robust to the uncertainties on the initial condition than the ones currently available, based on Lie symmetries. This method is first presented through different simple theoretical examples. We then apply it to a localisation problem in a robotic context

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40

Assmann, Björn. "Applications of Lie methods to computations with polycyclic groups." Thesis, St Andrews, 2007. http://hdl.handle.net/10023/435.

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41

Caprace, Pierre-Emmanuel. ""Abstract" homomorphisms of split Kac-Moody groups." Doctoral thesis, Universite Libre de Bruxelles, 2005. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/210962.

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Cette thèse est consacrée à une classe de groupes, appelés groupes de Kac-Moody, qui généralise de façon naturelle les groupes de Lie semi-simples, ou plus précisément, les groupes algébriques réductifs, dans un contexte infini-dimensionnel. On s'intéresse plus particulièrement au problème d'isomorphismes pour ces groupes, en vue d'obtenir un analogue infini-dimensionnel de la célèbre théorie des homomorphismes 'abstraits' de groupes algébriques simples, due à Armand Borel et Jacques Tits.

Le problème d'isomorphismes qu'on étudie s'avère être un cas particulier d'un problème plus général, qui consiste à caractériser les homomorphismes de groupes algébriques vers les groupes de Kac-Moody, dont l'image est bornée. Ce problème peut à son tour s'énoncer comme un problème de rigidité pour les actions de groupes algébriques sur les immeubles, via l'action naturelle d'un groupe de Kac-Moody sur une paire d'immeubles jumelés. Les résultats partiels, relatifs à ce problème de rigidité, que nous obtenons, nous permettent d'apporter une solution complète au problème d'isomorphismes pour les groupes de Kac-Moody déployés.

En particulier, on obtient un résultat de dévissage pour les automorphismes de ces objets. Celui-ci fournit à son tour une description complète de la structure du groupe d'automorphismes d'un groupe de Kac-Moody déployé sur un corps de caractéristique~$0$.

Nos arguments permettent également de traiter de façon analogue certaines formes anisotropes de groupes de Kac-Moody complexes, appelées formes unitaires. On montre en particulier que la topologie Hausdorff naturelle que portent ces formes est un invariant de leur structure de groupe abstrait. Ceci généralise un résultat bien connu de H. Freudenthal pour les groupes de Lie compacts.

Enfin, l'on s'intéresse aux homomorphismes de groupes de Kac-Moody à image fini-dimensionnelle, et l'on démontre la non-existence de tels homomorphismes à noyau central, lorsque le domaine est un groupe de Kac-Moody de type indéfini sur un corps infini. Ceci réduit un problème ouvert, dit problème de linéarité pour les groupes de Kac-Moody, au cas de corps de base finis.
Doctorat en sciences, Spécialisation mathématiques
info:eu-repo/semantics/nonPublished

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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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43

Yu, Jun. "Symmetric subgroups of automorphism groups of compact simple Lie algebras /." View abstract or full-text, 2009. http://library.ust.hk/cgi/db/thesis.pl?MATH%202009%20YU.

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44

Ando, Hiroshi. "Polish Groups of Finite Type and Their Lie Algebras." 京都大学 (Kyoto University), 2012. http://hdl.handle.net/2433/157737.

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45

Azam, Saeid. "Extended affine Lie algebras and extended affine Weyl groups." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp02/NQ27440.pdf.

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46

King, Jeremy David. "Finite presentability of Lie algebras and pro-p groups." Thesis, University of Cambridge, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.364385.

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Welsh, Trevor Alan. "Young tableaux and modules of groups of Lie algebras." Thesis, University of Southampton, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.332783.

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Bien, Frédéric Vincent. "Spherical D̳-modules and representations of reductive lie groups." Thesis, Massachusetts Institute of Technology, 1986. http://hdl.handle.net/1721.1/101303.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1986.
MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE.
The underscored character in the title transcriptions is a script D on the title page.
Bibliography: leaves 89-91.
by Frédéric Vincent Bien.
Ph.D.

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O'Dea, Fergus Rae. "Harmonic maps into Lie groups, integrable systems and supersymmetry /." Full text (PDF) from UMI/Dissertation Abstracts International, 2000. http://wwwlib.umi.com/cr/utexas/fullcit?p3004350.

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50

Garcia-Naranjo, Luis Constantino. "Almost Poisson Brackets for Nonholonomic Systems on Lie Groups." Diss., The University of Arizona, 2007. http://hdl.handle.net/10150/195845.

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We present a geometric construction of almost Poisson brackets for nonholonomic mechanical systems whose configuration space is a Lie group G. We study the so-called LL and LR systems where the kinetic energy defines a left invariant metric on G and the constraints are invariant with respect to left (respectively right) translation on G.For LL systems, the equations on the momentum phase space, T*G, can be left translated onto g*, the dual space of the Lie algebra g. We show that the reduced equations on g* can be cast in Poisson form with respect to an almost Poisson bracket that is obtained by projecting the standard Lie-Poisson bracket onto the constraint space.For LR systems, we use ideas of semidirect product reduction to transfer the equations on T*G into the dual Lie algebra, s*, of a semidirect product. This provides a natural Lie algebraic setting for the equations of motion commonly found in the literature. We show that these equations can also be cast in Poisson form with respect to an almost Poisson bracket that is obtained by projecting the Lie-Poisson structure on s* onto a constraint submanifold.In both cases the constraint functions are Casimirs of the bracket and are satisfied automatically. Our construction is a natural generalization of the classical ideas of Lie-Poisson and semidirect product reduction to the nonholonomic case. It also sets a convenient stage for the study of Hamiltonization of certain nonholonomic systems.Our examples include the Suslov and the Veselova problems of constrained motion of a rigid body, and the Chaplygin sleigh.In addition we study the almost Poisson reduction of the Chaplygin sphere. We show that the bracket given byBorisov and Mamaev is obtained by reducing a nonstandard almost Poisson bracket that is obtained by projecting a non-canonical bivector onto the constraint submanifold using the Lagrange-D'Alembert principle.The examples that we treat show that it is possible to cast the reduced equations of motion of certain nonholonomic systems in Hamiltonian form (in the Poisson formulation) either by multiplication by a conformal factor, by the use of nonstandard brackets or simply by reduction methods.

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Dissertations / Theses on the topic 'Lie groups'

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