Дисертації: "Lie groups and Lie algebras"

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

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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3

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).

4

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.

5

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.

6

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

7

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

8

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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9

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.

10

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

11

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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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).

13

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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14

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.

15

Martini, Alessio. "Algebras of differential operators on Lie groups and spectral multipliers." Doctoral thesis, Scuola Normale Superiore, 2010. http://hdl.handle.net/11384/85663.

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Let (X, μ) be a measure space, and let L1, . . . ,Ln be (possibly unbounded) selfadjoint operators on L2(X, μ), which commute strongly pairwise, i.e., which admit a joint spectral resolution E on Rn. A joint functional calculus is then defined via spectral integration: for every Borel function m : Rn → C, m(L) = m(L1, . . . ,Ln) = ∫ Rn m(λ) dE(λ) is a normal operator on L2(X, μ), which is bounded if and only if m - called the joint spectral multiplier associated to m(L) - is (E-essentially) bounded. However, the abstract theory of spectral integrals does not tackle the following problem: to find conditions on the multiplier m ensuring the boundedness of m(L) on Lp(X, μ) for some p ≠ 2. We are interested in this problem when the measure space is a connected Lie group G with a right Haar measure, and L1, . . . ,Ln are left-invariant differential operators on G. In fact, the question has been studied quite extensively in the case of a single operator, namely, a sublaplacian or a higher-order analogue. On the other hand, for multiple operators, only specific classes of groups and specific choices of operators have been considered in the literature. Suppose that L1, . . . ,Ln are formally self-adjoint, left-invariant differential operators on a connected Lie group G, which commute pairwise (as operators on smooth functions). Under the assumption that the algebra generated by L1, . . . ,Ln contains a weighted subcoercive operator --- a notion due to [ER98], including positive elliptic operators, sublaplacians and Rockland operators---we prove that L1, . . . ,Ln are (essentially) self-adjoint and strongly commuting on L2(G). Moreover, we perform an abstract study of such a system of operators, in connection with the algebraic structure and the representation theory of G, similarly as what is done in the literature for the algebras of differential operators associated with Gelfand pairs. Under the additional assumption that G has polynomial volume growth, weighted L1 estimates are obtained for the convolution kernel of the operator m(L) corresponding to a compactly supported multiplier m satisfying some smoothness condition. The order of smoothness which we require on m is related to the degree of polynomial growth of G. Some techniques are presented, which allow, for some specific groups and operators, to lower the smoothness requirement on the multiplier. In the case G is a homogeneous Lie group and L1, . . . ,Ln are homogeneous operators, a multiplier theorem of Mihlin-H\"ormander type is proved, extending the result for a single operator of [Chr91] and [MM90]. Further, a product theory is developed, by considering several homogeneous groups Gj , each of which with its own system of operators; a non-conventional use of transference techniques then yields a multiplier theorem of Marcinkiewicz type, not only on the direct product of the Gj , but also on other (possibly non-homogeneous) groups, containing homomorphic images of the Gj . Consequently, for certain non-nilpotent groups of polynomial growth and for some distinguished sublaplacians, we are able to improve the general result of [Ale94].

16

Lampetti, Enrico. "Nilpotent orbits in semisimple Lie algebras." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/23595/.

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This thesis is dedicated to the introductory study of the so-called nilpotent orbits in a semisimple complex Lie algebra g, i.e., the orbits of nilpotent elements under the adjoint action of the adjoint group Gad with Lie algebra g. These orbits have an extremely rich structure and lie at the interface of Lie theory, algebraic geometry, symplectic geometry, and geometric representation theory. The Jacobson and Morozov Theorem relates the orbit of a nilpotent element X in a semisimple complex Lie algebra g with a triple {H,X,Y} that generates a subalgebra of g isomorphic to sl(2,C). There is a parabolic subalgebra associated to this triple that permits to attach a weight to each node of the Dynkin diagram of g. The resulting diagram is called a weighted Dynkin diagram associated with the nilpotent orbit of X. This is a complete invariant of the orbit that one can use in order to show that there are only _nitely many nilpotent orbits in g. The thesis is organized as follows: the first three chapters contain some preliminary material on Lie algebras (Chapter 1), on Lie groups (Chapter 3) and on the representation theory of sl(2,C) (Chapter 2). Chapter 4 and 5 are the heart of the thesis. Namely, Jacobson-Morozov, Kostant and Mal'cev Theorems are proved in Chapter 4 and Chapter 5 is dedicated to the construction of weighted Dynkin diagrams. As an example the conjugacy classes of nilpotent elements in sl(n,C) are described in detail and a formula for their dimension is given. In this case, as well as in the case of all classical Lie algebras, the description of the orbits can be done in terms of partitions and tableaux.

17

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

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18

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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19

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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20

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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21

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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22

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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23

Gouthier, Daniele. "LCR-structures and LCR-algebras." Doctoral thesis, SISSA, 1996. http://hdl.handle.net/20.500.11767/4388.

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24

Okeke, Nnamdi, and University of Lethbridge Faculty of Arts and Science. "Character generators and graphs for simple lie algebras." Thesis, Lethbridge, Alta. : University of Lethbridge, Faculty of Arts and Science, 2006, 2006. http://hdl.handle.net/10133/532.

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We study character generating functions (character generators) of simple Lie algebras. The expression due to Patera and Sharp, derived from the Weyl character formula, is ¯rst re- viewed. A new general formula is then found. It makes clear the distinct roles of \outside" and \inside" elements of the integrity basis, and helps determine their quadratic incompati- bilities. We review, analyze and extend the results obtained by Gaskell using the Demazure character formulas. We ¯nd that the fundamental generalized-poset graphs underlying the character generators can be deduced from such calculations. These graphs, introduced by Baclawski and Towber, can be simpli¯ed for the purposes of constructing the character generator. The generating functions can be written easily using the simpli¯ed versions, and associated Demazure expressions. The rank-two algebras are treated in detail, but we believe our results are indicative of those for general simple Lie algebras.
vii, 92 leaves ; 29 cm.

25

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.

26

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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27

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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28

Lacerda, Conrado Damato de 1986. "Grupos de Lie compactos." [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/305808.

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Orientador: Luiz Antonio Barrera San Martin
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
Made available in DSpace on 2018-08-18T06:52:41Z (GMT). No. of bitstreams: 1 Lacerda_ConradoDamatode_M.pdf: 1208692 bytes, checksum: 167da419a80e3fe06963795a1b3fea2d (MD5) Previous issue date: 2011
Resumo: Neste trabalho apresentamos os principais resultados da teoria dos grupos de Lie compactos e provamos o Teorema de Weyl sobre os seus grupos fundamentais
Abstract: In this work we present the main results about compact Lie groups and prove Weyl's Theorem on their fundamental groups
Mestrado
Teoria de Lie
Mestre em Matemática

29

Graner, Nicholas. "Canonical Coordinates on Lie Groups and the Baker Campbell Hausdorff Formula." DigitalCommons@USU, 2018. https://digitalcommons.usu.edu/etd/7232.

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Lie Groups occur in math and physics as representations of continuous symmetries and are often described in terms of their Lie Algebra. This thesis is concerned with finding a concrete description of a Lie group given its associated Lie algebra. Several calculations toward this end are developed and then implemented in the Maple Differential Geometry package. Examples of the calculations are given.

30

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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31

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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32

Chopp, Mikaël. "Lie-admissible structures on Witt type algebras and automorphic algebras." Electronic Thesis or Diss., Metz, 2011. http://www.theses.fr/2011METZ020S.

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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- Novikov
The 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

33

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- Novikov
The 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

34

Rivezzi, Andrea. "Lie bialgebras and Etingof-Kazhdan quantization." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2020. http://amslaurea.unibo.it/21784/.

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In questa tesi viene presentata la soluzione data da Pavel Etingof e David Kazhdan al problema della quantizzazione delle bialgebre di Lie, formulato da Vladimir Drinfeld nel 1992. Il problema consiste nel trovare un funtore che, data una bialgebra di Lie, costruisca una algebra di Hopf che la quantizzi. Nel primo capitolo vengono presentati gli aspetti di teoria delle categorie necessarie per la lettura. Nel secondo capitolo, introduciamo le nozioni di algebra, coalgebra, bialgebra e algebra di Hopf, con particolare attenzione alla loro teoria delle rappresentazioni. Nel terzo capitolo, presentiamo le nozioni base della teoria delle algebre di Lie, per poi definire le nozioni di coalgebra di Lie e di bialgebra di Lie. Vengono quindi definite le triple di Manin e il doppio di Drinfeld di una bialgebra di Lie. Nel quarto capitolo definiamo la nozione di quantizzazione di una bialgebra di Lie, e presentiamo i quantum groups di Drinfeld e Jimbo, che ne sono un esempio nel caso delle algebre di Kac-Moody simmetrizzabili. Infine, nel quinto ed ultimo capitolo presentiamo la costruzione della quantizzazione di Etingof e Kazhdan. Tale tecnica di quantizzazione si suddivide in diversi passi, ed è basata sulla dualità di Tannaka-Krein. In un primo momento, analizziamo il caso in cui la bialgebra di Lie è di dimensione finita. In seguito, adattiamo la costruzione del caso finito dimensionale al caso infinito dimensionale.

35

Xanthopoulos, Stilianos. "On a question of Verma about indecomposable representations of algebraic groups and of their lie algebras." Thesis, Queen Mary, University of London, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.413244.

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36

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

37

Bernhardt, Karen 1977. "The generalized Harish-Chandra homomorphism for Hecke algebras of real reductive Lie groups." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/28922.

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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.
Includes bibliographical references (p. 73-74).
For complex reductive Lie algebras g, the classical Harish-Chandra homomorphism allows to link irreducible finite dimensional representations of g to those of certain subalgebras l. The Casselman-Osborne theorem establishes an extension of this link to infinite dimensional irreducible representations. In this paper we present a generalized Harish-Chandra homomorphism construction for Hecke algebras, and establish the corresponding generalized Casselman-Osborne theorem. This homomorphism can be used to link representations of (g, L n K)-pairs to those of (g, L n K)-pairs, where is a certain subalgebra of g as in the classical case. Since representations of such pairs are closely related to those of the underlying Lie group G, this construction is a good first approximation to lifting the Harish-Chandra homomorphism from the Lie algebra to the Lie group level.
by Karen Bernhardt.
S.M.

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

Ng, Ka-chun, and 吳嘉俊. "Total positivity in some classical groups." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2008. http://hub.hku.hk/bib/B40987838.

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40

Ka-chun, Ng. "Total positivity in some classical groups." Click to view the E-thesis via HKUTO, 2008. http://sunzi.lib.hku.hk/hkuto/record/B40987838.

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41

Lin, Qian Ph D. Massachusetts Institute of Technology. "Modules over affine lie algebras at critical level and quantum groups by Qian Lin." Thesis, Massachusetts Institute of Technology, 2010. http://hdl.handle.net/1721.1/60195.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 45-47).
There are two algebras associated to a reductive Lie algebra g: the De Concini- Kac quantum algebra and the Kac-Moody Lie algebra. Recent results show that the principle block of De Concini -Kac quantum algebra at an odd root of unity with (some) fixed central character is equivalent to the core of a certain t-structure on the derived category of coherent sheaves on certain Springer Fiber. Meanwhile, a certain category of representation of Kac-Moody Lie algebra at critical level with (some) fixed central character is also equivalent to a core of certain t-structure on the same triangulated category. Based on several geometric results developed by Bezurkvanikov et al. these two abelian categories turn out to be equivalent. i.e. the two t-structures coincide.
Ph.D.

42

Nishiyama, Kyo. "Representations of Weyl groups and their Hecke algebras on virtual character modules of a semisimple Lie group." 京都大学 (Kyoto University), 1986. http://hdl.handle.net/2433/86366.

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43

Faccin, Paolo. "Computational problems in algebra: units in group rings and subalgebras of real simple Lie algebras." Doctoral thesis, Università degli studi di Trento, 2014. https://hdl.handle.net/11572/368142.

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In the first part of the thesis I produce and implement an algorithm for obtaining generators of the unit group of the integral group ring ZG of finite abelian group G. We use our implementation in MAGMA of this algorithm to compute the unit group of ZG for G of order up to 110. In the second part of the thesis I show how to construct multiplication tables of the semisimple real Lie algebras. Next I give an algorithm, based on the work of Sugiura, to find all Cartan subalgebra of such a Lie algebra. Finally I show algorithms for finding semisimple subalgebras of a given semisimple real Lie algebra.

44

Faccin, Paolo. "Computational problems in algebra: units in group rings and subalgebras of real simple Lie algebras." Doctoral thesis, University of Trento, 2014. http://eprints-phd.biblio.unitn.it/1182/1/PhdThesisFaccinPaolo.pdf.

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In the first part of the thesis I produce and implement an algorithm for obtaining generators of the unit group of the integral group ring ZG of finite abelian group G. We use our implementation in MAGMA of this algorithm to compute the unit group of ZG for G of order up to 110. In the second part of the thesis I show how to construct multiplication tables of the semisimple real Lie algebras. Next I give an algorithm, based on the work of Sugiura, to find all Cartan subalgebra of such a Lie algebra. Finally I show algorithms for finding semisimple subalgebras of a given semisimple real Lie algebra.

45

Sigurdsson, Gunnar. "Canoniical involutions and bosonic representations of three-dimensional lie colour algebras." Licentiate thesis, KTH, Physics, 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-1750.

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46

Wickramasekara, Sujeewa, and sujeewa@physics utexas edu. "Symmetry Representations in the Rigged Hilbert Space Formulation of." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi993.ps.

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47

Zwicknagl, Sebastian. "Equivariant Poisson algebras and their deformations /." view abstract or download file of text, 2006. http://proquest.umi.com/pqdweb?did=1280144671&sid=2&Fmt=2&clientId=11238&RQT=309&VName=PQD.

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Thesis (Ph. D.)--University of Oregon, 2006.
Typescript. Includes vita and abstract. "In this dissertation I investigate Poisson structures on symmetric and exterior algebras of modules over complex reductive Lie algebras. I use the results to study the braided symmetric and exterior algebras"--P. 1. Includes bibliographical references (leaves 150-152). Also available for download via the World Wide Web; free to University of Oregon users.

48

Meyer, Philippe. "Représentations associées à des graduations d'algèbres de Lie et d'algèbres de Lie colorées." Thesis, Strasbourg, 2019. http://www.theses.fr/2019STRAD001/document.

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Soit k un corps de caractéristique différente de 2 et de 3. Les algèbres de Lie colorées généralisent à la fois les algèbres de Lie et les superalgèbres de Lie. Dans cette thèse on étudie des représentations V d'algèbres de Lie colorées g provenant de structures d'algèbres de Lie colorées sur l'espace vectoriel g⨁V. En premier lieu, on s'intéresse à la structure générale des algèbres de Lie simples de dimension 3 sur k. Puis, on classifie à isomorphisme près les superalgèbres de Lie de dimension finie dont la partie paire est une algèbre de Lie simple de dimension 3. Ensuite, pour un groupe abélien ᴦ et un facteur de commutation ɛ de ᴦ, on développe l'algèbre multilinéaire associée aux espaces vectoriels ᴦ-gradués. Dans ce contexte, les algèbres de Lie colorées jouent le rôle des algèbres de Lie. Ce langage nous permet d'énoncer et prouver un théorème de reconstruction d'une algèbre de Lie colorée ɛ-quadratique g⨁V à partir d'une représentation ɛ-orthogonale V d'une algèbre de Lie colorée ɛ-quadratique g. Ce théorème fait intervenir un invariant qui prend ses valeurs dans la ɛ-algèbre extérieure de V et généralise des résultats de Kostant et Chen-Kang. Puis, on introduit la notion de représentation ɛ-orthogonale spéciale V d'une algèbre de Lie colorée ɛ-quadratique g et on montre qu'elle permet de définir une structure d'algèbre de Lie colorée ɛ-quadratique sur l'espace vectoriel g⨁sl(2,k)⨁V⨂k². Enfin on donne des exemples de représentations ɛ-orthogonales spéciales, notamment des représentations orthogonales spéciales d'algèbres de Lie dont : une famille à un paramètre de représentations de sl(2,k)xsl(2,k) ; la représentation fondamentale de dimension 7 d'une algèbre de Lie de type G₂ ; la représentation spinorielle de dimension 8 d'une algèbre de Lie de type so(7)
Let k be a field of characteristic not 2 or 3. Colour Lie algebras generalise both Lie algebras and Lie superalgebras. In this thesis we study representations V of colour Lie algebras g arising from colour Lie algebras structures on the vector space g⨁V. Firstly, we study the general structure of simple three-dimensional Lie algebras over k. Then, we classify up to isomorphism finite-dimensional Lie superalgebras whose even part is a simple three-dimensional Lie algebra. Next, to an abelian group ᴦ and a commutation factor ɛ of ᴦ, we develop the multilinear algebra associated to ᴦ-graded vector spaces. In this context, colour Lie algebras play the rôle of Lie algebras. This language allows us to state and prove a theorem reconstructing an ɛ-quadratic colour Lie algebra g⨁V from an ɛ-orthogonal representation V of an ɛ-quadratic colour Lie algebra g. This theorem involves an invariant taking its values in the ɛ-exterior algebra of V and generalises results of Kostant and Chen-Kang. We then introduce the notion of a special ɛ-orthogonal representation V of an ɛ-quadratic colour Lie algebra g and show that it allows us to define an ɛ-quadratic colour Lie algebra structure on the vector space g⨁sl(2,k)⨁V⨂k². Finally we give examples of special ɛ-orthogonal representations and in particular examples of special orthogonal representations of Lie algebras amongst which are: a one-parameter family of representations of sl(2,k)xsl(2,k) ; the 7-dimensional fundamental representation of a Lie algebra of type G₂ ; the 8-dimensional spinor representation of a Lie algebra of type so(7)

49

Severi, Claudio. "Clifford algebras and spin groups, with physical applications." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/18387/.

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In questo lavoro viene esposta la teoria delle algebre di Clifford e dei gruppi di Spin, con attenzione alle applicazioni fisiche, in particolare l'equazione di Dirac per particelle quantistiche con spin 1/2. I primi due capitoli sono dedicati ad una descrizione generale delle algebre di Clifford reali e complesse, che vengono costruite e classificate. Il terzo capitolo è dedicato ai gruppi di Spin ed alle loro algebre di Lie. Gli ultimi due capitoli illustrano un'applicazione fisica: viene esposta la teoria quantistica dello spin e del momento angolare, e si deriva l'equazione di Dirac con un principio variazionale. Dopo una discussione delle proprietà generiche di questa equazione, si dimostra che descrive accuratamente la struttura fine dello spettro dell'atomo di idrogeno.

50

Athapattu, Mudiyanselage Chathurika Umayangani Manike Athapattu. "Chevalley Groups." OpenSIUC, 2016. https://opensiuc.lib.siu.edu/theses/1986.

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In this thesis, we construct Chevalley groups over arbitrary fields. The construction is based on the properties of semi-simple complex Lie algebras, the existence of Chevalley bases and notion of universal enveloping algebras. Using integral lattices in universal enveloping algebras and integral properties of Chevalley bases, we present a method which produces, for any complex simple Lie group, an analogous group over an arbitrary field.

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