Maximal Subgroups of Compact Lie Groups

2011

In this PhD thesis, we consider two problems that are related to finite simple groups of Lie type. First of them is a problem mentioned in the Kourovka notebook: describe the finite simple groups in which every element is a product of two involutions. We consider the simple orthogonal groups in even characteristic, and solve the problem for them. Since other groups have been dealt with elsewhere, the problem is then solved completely. The second part of the thesis is related to Lie algebras. Every complex simple Lie algebra has a compact real form that is associated with a compact Lie group. In this thesis, we consider the Lie algebra of type E8, and give a new construction of its compact real form. The Lie product is defined using the irreducible subgroup of shape 25+10 ·GL5(2) of the automorphism group.

Proceedings of the American Mathematical Society

In this paper we determine the structure of the minimal ideal in the enveloping semigroup for the natural action of a connected semisimple Lie group on its maximal compact subgroup. In particular, if G = KAN is an Iwasawa decomposition of the group G, then the group in the minimal left ideal is isomorphic both algebraically and topologically with the normalizer M of AN in K. Complete descriptions are given for the enveloping semigroups in the cases G = SL(2, C) and G = SL(2, R).

arXiv: Group Theory, 2021

Chapter 1. Introduction Chapter 2. Notation and preliminaries Chapter 3. Subgroup structure of exceptional algebraic groups Chapter 4. Techniques for proving the results 4.1. The strategy 4.2. An example Chapter 5. Modules for groups of Lie type Chapter 6. Rank 4 groups for E 8 6.

In the end of 19 century, W. Killing and E. Cartan classified the complex simple Lie algebras, called A n , B n , C n , D n (classical type) and G 2 , F 4 , E 6 , E 7 , E 8 (exceptional type). These simple Lie algebras and the corresponding compact simple Lie groups have offered many subjects in mathematicians. Especially, exceptional Lie groups are very wonderful and interesting miracle in Lie group theory. Now, in the present book, we describe simply connected compact exceptional simple Lie groups G 2 , F 4 , E 6 , E 7 , E 8 , in very elementary way. The contents are given as follows. We first construct all simply connected compact exceptional Lie groups G concretely. Next, we find all involutive automorphisms σ of G, and determine the group structures of the fixed points subgroup G σ by σ. Note that they correspond to classification of all irreducible compact symmetric spaces G/G σ of exceptional type, and that they also correspond to classification of all non-compact exceptional simple Lie groups. Finally, we determined the group structures of the maximal subgroups of maximal rank. At any rate, we would like this book to be used in mathematics and physics.

Proceedings of the Edinburgh Mathematical Society, 1987

The major problem with which this paper is concerned is determining criteria that allow one to decide whether the subsemigroup generated by a subset B of a group G is all of G. Motivations for considering this problem arise from at least two sources.

International Journal of Algebra, 2017

Let G be a group. A set C of proper subgroups of G is called a cover for G if its set-theoretic union is equal to G. If the size of C is n, we call C an n-cover for the group G. A cover C for a group G is called irredundant if no proper subset of C is a cover for G. A cover C for a group G is called core-free if the intersection D = M ∈C M of C is core-free in G. A cover C for a group G is called maximal if all the members of C are maximal subgroups of G. A cover C for a group G is called a C n-cover whenever C is an irredundant maximal core-free n-cover for G and in this case we say that G is a C n-group. In this paper we give somer results on classification of C 8-groups having special maximal subgroups.

Group Theory

In this reprt, the compact Lie algebras are classified via the classification of complex simple Lie algebras. The classification of semisimple Lie algebras using the knowledge about simple Lie algebras follows. An application of this classification is given using representation theory of simple Lie algebras. Finally, a way to build infinite dimensional Lie algebras is shown in the example of the Kac-Moody algebras. The sections about simple and semisimple Lie algebras are largely based on J.E.Humphreys, [1]. Proofs can be generally found in the said reference. W.Fulton/J.Harris,[2] is mainly used as a second reference. The part covering Kac-Moody algebras can be found in the paper P.Goddard/D.Olive [3]

Taiwanese Journal of Mathematics, 2013

The 2-rank of a compact Lie group G is the maximal possible rank of the elementary 2-subgroup Z2 × • • • Z2 of G. The study of 2-ranks (and prank for any prime p) of compact Lie groups was initiated in 1953 by A. Borel and J.-P. Serre [9]. Since then the 2-ranks of compact Lie groups have been investigated by many mathematician. The 2-ranks of compact Lie groups relate closely with several important areas in mathematics. In this article, we survey important results concerning 2-ranks of compact Lie groups. In particular, we present the complete determination of 2-ranks of compact connected simple Lie groups G via the maximal antipodal sets A2G of G introduced in [16, 17].

Communications in Algebra, 2004

The maximality of certain symplectic subgroups of unitary groups P SU n (K), n ≥ 4, (K any field admitting a non-trivial involutory automorphism) belonging to the class C 5 of Aschbacher is proved. Furthermore some related geometry in the case n = 4 and K finite is investigated.

Journal of Pure and Applied Algebra

This paper is a continued investigation of the structure of Lie algebras in relation to their chief factors, using concepts that are analogous to corresponding ones in group theory. The first section investigates the structure of Lie algebras with a core-free maximal subalgebra. The results obtained are then used in section two to consider the relationship of two chief factors of L being L-connected, a weaker equivalence relation on the set of chief factors than that of being isomorphic as L-modules. A strengthened form of the Jordan-Hölder Theorem in which Frattini chief factors correspond is also established for every Lie algebra. The final section introduces the concept of a crown, a notion introduced in group theory by Gaschütz, and shows that it gives much information about the chief factors