Soit K un corps algébriquement clos de caractéristique nulle. Il est bien connu, d'après un résultat de Duflo, Khalgui et Torasso, qu'une algèbre de Lie algébrique quasi-réductive (définie sur K) est stable. La réciproque est fausse en général. Se pose la question de savoir, si pour certaines classes particulières d'algèbres de Lie non réductives, il y a équivalence entre ces deux notions. Plus généralement, les sous-algèbres biparaboliques forment une classe très intéressante (incluant la classe des sous-algèbres paraboliques et de Levi) d'algèbres de Lie qui ne sont pas toutes réductives. Panyushev conjecture que si une sous-algèbre biparabolique est stable, alors son stabilisateur générique est un tore. Cette conjecture peut être reformulée ainsi : une sous-algèbre de Lie biparabolique est stable si et seulement si elle est quasi-réductive. Compte tenu des résultats obtenus par ce dernier pour le cas des sous-algèbres paraboliques d'une algèbre de Lie simple de type A et C, on donne dans cette thèse une réponse positive à cette conjecture pour la classe des sous-algèbres paraboliques d'une algèbre de Lie simple. Au passage, nous montrons également qu'une sous-algèbre de Lie de gl(n, K) qui stabilise une forme bilinéaire alternée de rang maximal et un drapeau en position générique est stable si et seulement si elle est quasi-réductive. / Let K be an algebraically closed field of characteristic 0. It is well known by work of Duflo, Khalgui and Torasso that any quasi-reductive algebraic Lie algebra (defined over K) is stable. However, there are stable Lie algebras which are not quasi-reductive. This raises the question, if for some particular class of non-reductive Lie algebras, there is equivalence between stability and quasi-reductivity. More generally, biparabolic subalgebras form a very interesting class (including the class of parabolic subalgebras and of Levi subalgebras) of non-reductive Lie algebras. It was conjectured by Panyushev that these two notions are equivalent for biparabolic subalgebras of a reductive Lie algebra. In this thesis, we give by considering the results of Panyushev for parabolic subalgerbras of simple Lie algebra of type A and C a positive answer to this conjecture in the case of parabolic subalgebras. In passing, we prove that these two notions are equivalent for certain subalgebras of gl(n,K) which stabilize an alternating bilinear form of maximal rank and a flag in generic position.

Orientador: Geraldo Nunes Silva / Banca: Antonio Carlos Gardel Leitão / Banca: Fernando Manuel Ferreira Lobo Pereira / Resumo: Neste trabalho é feito um estudo aprofundado da estabilidade de sistemas alternantes, principalmente via teoria de Lie. Inicialmente são apresentados os principais conceitos básicos da álgebra de Lie, necessários para o estudo dos critérios de estabilidade dos sistemas alternantes. Depois são discutidos critérios de estabilidade para sistemas alternantes. É feita a exposição da demonstração de que para todo sistema linear da forma ? x = Apx p = 1, 2, ...,N, com as matrizes Ap assintóticamente estáveis e comutativas duas a duas, existe uma função de Lyapunov quadrática comum. Uma condição suficiente para estabilidade assintótica de um sistema linear alternante é apresentada em termos da álgebra de Lie gerada por uma família infinita de matrizes. A saber, se esta álgebra de Lie é solúvel, então o sistema alternante é estável para uma mudança arbitrária de sinal. Em seguida são estudadas condições mais fracas. Supondo que a álgebra de Lie não é solúvel, mas é decomponível na soma de um ideal solúvel e uma subálgebra com grupo de Lie compacto, então o sistema alternante é globalmente exponencialmente uniformemente estável. Entretanto, se o grupo de Lie não for compacto, verifica-se que é possível gerar uma família finita de matrizes estáveis tais que o correspondente sistema linear alternante não é estável. Finalmente, os resultados correspondentes de estabilidade local para sistemas alternantes não lineares são apresentados. / Abstract: In this work it is undertaken a deep study of stability for switched systems, mainly via Lie algebraic Theory. At first, the basic concepts and results from Lie algebra necessary for the study of stability of switched systems are presented. Criteria for stability are discussed. It is also done an exposition of the proof that all linear systems ? x = Apx, p = 1, 2, ...,N, with stable and pairwisely commutative matrices Ap, have common quadratic Lyapounov functions. A sufficient condition for asymptotic stability of switched linear systems is presented in term of the Lie algebra generated by a family infinite matrices. That is, if this Lie algebra is solvable, then the switched systems are stable for an arbitrary change of sinal. Next weaker conditions are studied. If the Lie algebra is decomposable into two subalgebras in which one is a solvable ideal and the other has a compact Lie group, then the switched systems are globally exponentially uniformly stable. However, if the Lie group is not compact, it is also possible to generate a finite family of stable matrices such that the corresponding switched linear systems are not stable. Finally, corresponding local stability results are presented for nonlinear systems. / Mestre

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

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Recent work has established that a group theoretical viewpoint of completely integrable dynamical systems with N degrees of freedom yields an algorithm that provides new information concerning the symmetry transformation group structure of this class of dynamical systems. The work presented here rests heavily on the results presented in reference and it is recommended that the reader consult this reference for a more rigorous discussion of the results given in this thesis.

In 1980 Feigin and Fuchs classified the length 2 bounded representations of Vec(R), the Lie algebra of polynomial vector fields on the line, as a result of their work on the cohomology of Vec(R). This dissertation is concerned mainly with the uniserial (completely indecomposable) representations of Vec(R) with a single Casimir eigenvalue and weights bounded below. Such representations are composed of irreducible representations with semisimple Euler operator action, bounded weight space dimensions, and weights bounded below. These are known to be the tensor density modules with lowest weight λ, for any non-zero complex number λ, and the trivial module C, with Vec(R) actions π_λ and π_C, respectively. Our proofs are cohomology arguments involving the first cohomology groups of Vec(R) with values in the space of homomorphisms between two irreducible representations. These results classify the finite length uniserial extensions, with a single Casimir eigenvalue, of admissible irreducible Vec(R) representations with weights bounded below. In almost every case there is at most one uniserial representation with a given composition series. However, in the case of an odd length extension with composition series {π_1,π_C,π_1,…,π_C,π_1}, there is a one-parameter family of extensions. We also give preliminary results on uniserial representations of the Virasoro Lie algebra.

The differential geometry software package in Maple has the necessary tools and commands to automate the classification process for complex simple Lie algebras. The purpose of this thesis is to write the programs to complete the classification for real simple Lie algebras. This classification is difficult because the Cartan subalgebras are not all conjugate as they are in the complex case. For the process of the real classification, one must first identify a maximally noncompact Cartan subalgebra. The process of the Cayley transform is used to find this specific Cartan subalgebra. This Cartan subalgebra is used to find the simple roots for the given real simple Lie algebra. With this information, we can then create a Satake diagram. Then we match our given algebra's Satake diagram to a Satake diagram of a known algebra. The programs explained in this thesis complete this process of classification.

This thesis was concerned with classifying the real indecomposable solvable Lie algebras with codimension one nilradicals of dimensions two through seven. This thesis was organized into three chapters. In the first, we described the necessary concepts and definitions about Lie algebras as well as a few helpful theorems that are necessary to understand the project. We also reviewed many concepts from linear algebra that are essential to the research. The second chapter was occupied with a description of how we went about classifying the Lie algebras. In particular, it outlined the basic premise of the classification: that we can use the automorphisms of the nilradical of the Lie algebra to find a basis with the simplest structure equations possible. In addition, it outlined a few other methods that also helped find this basis. Finally, this chapter included a discussion of the canonical forms of certain types of matrices that arose in the project. The third chapter presented a sample of the classification of the seven-dimensional Lie algebras. In it, we proceeded step-by-step through the classification of the Lie algebras whose nilradical was one of four specifically chosen because they were representative of the different types that arose during the project. In the appendices, we presented our results in a list of the multiplication tables of the isomorphism classes found.

We prove the positivity of Lusztig's q-analogue of weight multiplicity in a purely combinatorial way for rank 2 Lie algebras. Each summand in the polynomial can be interpreted as a linear combination of positive roots. We prove that all negative coefficients are cancelled in the polynomial. Further, the analysis of the root systems allows us to state formulae for every coefficient in Lusztig's q-analogue for rank 2 Lie algebras. / Ph. D.

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