published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Thesis (Ph. D.)--Ohio State University, 2004. / Title from first page of PDF file. Document formatted into pages; contains v, 88 p. Includes bibliographical references (p. 87-88). Available online via OhioLINK's ETD Center

Let g be a Lie algebra. The subalgebra classification problem is to create a list of all subalgebras of g up to equivalence. The purpose of this thesis is to provide a software toolkit within the Differential Geometry package of Maple for classifying subalgebras of In particular the thesis will focus on classifying those subalgebras which are isomorphic to the Lie algebra sl(2) and those subalgebras of which have a basis aligned with the root space decomposition (regular subalgebras).

This thesis studies collinearity graphs and commuting involution graphs associated with sporadic group geometries, and the conjugacy classes of semisimple elements in the exceptional Lie-type group E8(2).First we construct plane-line collinearity graphs for the sporadic simple groups and their associated minimal and maximal 2-local parabolic geometries. For such a group and geometry, the plane-line collinearity graph takes all planes of the geometry as its vertices and joins two vertices with an edge if their planes are collinear in the geometry. We construct these graphs for the groups M23, J4, Fi22, Fi23, He, Co3 and Co2. Additionally we construct a variety of collinearity graphs associated with the minimal 2-local geometries of McL.A second short study looks at the commuting involution graphs associated with the Baby Monster sporadic group. These are graphs which take a conjugacy class of involutions as its vertex set and joins two vertices with an edge if they commute. We detail information relating to two such graphs. Finally, we study the conjugacy classes of semisimple elements in the exceptional group E8(2). This study is a joint work with Ali Aubad, John Ballantyne, Alexander McGaw, Peter Neuhaus, Peter Rowley and David Ward in which we determine the structure of centralisers for all such elements including information such as fixed-space dimensions and powering up maps. The ultimate aim is to determine all maximal subgroups of E8(2). This is a lengthy ongoing project and this study forms part of that effort.

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Thesis (Ph. D.)--University of Washington, 1999. / Vita. Includes bibliographical references (p. 79-80).

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.

Soit G un groupe de Lie semisimple (de rang supérieur) et Γ un sous-groupe discret Zariski dense de G (de covolume infini). Dans cette thèse, on traite de deux questions reliées au cône limite de Benoist de Γ : l’une de marche aléatoire et l’autre de mélange topologique du flot directionnel des chambres de Weyl. Dans l’introduction, on énonce les résultats principaux de cette thèse dans leur contexte. Le second chapitre comporte des rappels sur les groupes de Lie et les éléments loxodromiques. Dans le troisième chapitre, on réalise tous les points de l’intérieur du cône limite par des vecteurs de Lyapunov. Dans le quatrième chapitre, on construit des coordonnées locales de G ainsi que des outils cruciaux pour la suite. Dans le cinquième chapitre, on introduit les ensembles invariants naturels de G. Dans le dernier chapitre de cette thèse, on prouve le critère de mélange topologique des flots directionnels réguliers des chambres de Weyl obtenu avec O. Glorieux et on généralise partiellement ce critère de mélange à Γ\G pour une classe de groupes de Lie incluant SL(n, R), SL(n, C), SO (p, p + 2). / Let G be a semisimple Lie group (of higher rank) and Γ a Zariski dense subgroup of G (of infinite covolume). In this thesis, we discuss two questions related to the Benoist limit cone of Γ : one concerns random walks, the other topological mixing of the directional Weyl chamber flow. In the introduction, we state the main results of this thesis in their context. In the second chapter, we recall some general facts about Lie groups and loxodromic elements. In the third chapter, we prove that every point of the interior of the limit cone is a Lyapunov vector. In the fourth chapter, we construct local coordinates of G and give key tools for the remaining parts. In the fifth chapter, we introduce the invariant subsets of G. In the last chapter of this thesis, we prove the topological mixing criterion of regular directional Weyl chamber flow obtained with O. Glorieux and we generalize this criterion to Γ\G for a class of Lie groups including SL(n, R), SL(n, C), SO(p, p + 2).

This research aims to refresh and reinterpret the radical theory of associative rings using the base radical and base semisimple class constructions. It also endeavours to generalise some results about ideals of rings in terms of accessible subrings. A characterisation of accessible subrings is included. By applying the base radical and base semisimple class constructions to many of the known results in established radical theory a number of gaps are uncovered and closed, with the goal of making the theory more accessible to advanced undergraduate and graduate students and mathematicians in related fields, and to open up new areas of investigation. After a literature review and brief reminder of algebra rudiments, the useful properties of accessible subrings and the U and S operators independent from radical class connections are described. The section on accessible subrings illustrates that replacing ideals with accessible subrings is indeed possible for a number of results and demonstrates its usefulness. The traditional radical and semisimple class definitions are included and it is shown that the base radical and base semisimple class constructions are equivalent. Diagrams illustrating the constructions support the definitions. From then on, all radical and semisimple classes mentioned are understood to have the base radical and base semisimple class form. Subject to the constraints of this work, many known results of traditional radical theory are reinterpreted with new proofs, illustrating the potential to simplify the understanding of radical theory using the base radical and base semisimple class constructions. Along with reinterpreting known results, new results emerge giving further insight to radical theory and its intricacies. Accessible subrings and the U and S operators are integrated into the development. The duality between the base radical and base semisimple class constructions is demonstrated in earnest. With a measure of the theory presented, the new constructions are applied to examples and concrete radicals. Context is supported by establishing the relationship between some well-known rings and the radical and related classes of interest. The title of the thesis, Towards a Reinterpretation of the Radical Theory of Associative Rings Using Base Radical and Base Semisimple Class Constructions, reflects the understanding that reinterpreting the entirety of radical theory is beyond the scope of this work. The conclusion includes an outlook listing further research that time did not allow.

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