The structure of Cartan subgroups in Lie groups

Ukrainian Mathematical Journal, 1990

Archiv der Mathematik, 1991

Symmetry, Integrability and Geometry: Methods and Applications, 2009

The discrete cocompact subgroups of the five-dimensional connected, simply connected nilpotent Lie groups are determined up to isomorphism. Moreover, we prove if G = N × A is a connected, simply connected, nilpotent Lie group with an Abelian factor A, then every uniform subgroup of G is the direct product of a uniform subgroup of N and Z r where r = dim(A).

Journal of the London Mathematical Society, 2019

Let G be a group definable in an o-minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G. When M is an expansion of a real closed field we give a characterization of Cartan subgroups of G via their Lie algebras which allow us to prove firstly, that every Cartan subalgebra of the Lie algebra of G is the Lie algebra of a definable subgroup -a Cartan subgroup of G -, and secondly, that the set of regular points of G -a dense subset of G -is formed by points which belong to a unique Cartan subgroup of G.

Mathematical Notes, 1991

Journal of Lie theory

We give explicit, practical conditions that determine whether or not a closed, connected subgroup H of G = SU(2, n) has the property that there exists a compact subset C of G with CHC = G. To do this, we fix a Cartan decomposition G = KA + K of G, and then carry out an approximate calculation of (KHK) ∩ A + for each closed, connected subgroup H of G. This generalizes the work of H. Oh and D. Witte for G = SO(2, n).

This report aims at giving a general overview on the classification of the maximal subgroups of compact Lie groups (not necessarily connected). In the first part, it is shown that these fall naturally into three types: (1) those of trivial type, which are simply defined as inverse images of maximal subgroups of the corresponding component group under the canonical projection and whose classification constitutes a problem in finite group theory, (2) those of normal type, whose connected one-component is a normal subgroup, and (3) those of normalizer type, which are the normalizers of their own connected one-component. It is also shown how to reduce the classification of maximal subgroups of the last two types to: (2) the classification of the finite maximal Σ-invariant subgroups of centerfree connected compact simple Lie groups and (3) the classification of the Σ-primitive subalgebras of compact simple Lie algebras, where Σ is a subgroup of the corresponding outer automorphism group. In the second part, we explicitly compute the normalizers of the primitive subalgebras of the compact classical Lie algebras (in the corresponding classical groups), thus arriving at the complete classification of all (non-discrete) maximal subgroups of the compact classical Lie groups. Mathematics Subject Classification 2000: 22E15.

International Journal of Algebra and Computation, 2013

We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for Lie algebras. Some open problems are discussed as well.

Archiv der Mathematik, 2017

Let G be a connected Lie group. In this paper, we study the density of the images of individual power maps P k : G → G : g → g k. We give criteria for the density of P k (G) in terms of regular elements, as well as Cartan subgroups. In fact, we prove that if Reg(G) is the set of regular elements of G, then P k (G) ∩ Reg(G) is closed in Reg(G). On the other hand, the weak exponentiality of G turns out to be equivalent to the density of all the power maps P k. In linear Lie groups, weak exponentiality reduces to the density of P2(G). We also prove that the density of the image of P k for G implies the same for any connected full rank subgroup.

Advances in Linear Algebra & Matrix Theory

This paper is made out of necessity as a doctoral student taking the exam from Lie groups. Using the literature suggested to me by the professor, I felt the need to, in addition to that literature, and since there was more superficial in that book with some remarks about the examples given in relation to the left group. I decided to try a little harder and collect as much literature as possible, both for the needs of me and the others who will take after me. Searching for literature in my mother tongue I could not find anything, in English as someone who comes from a small country like Montenegro, all I could find was through the internet. I decided to gather what I could find from the literature in my own way and to my observation and make this kind of work. The main content of this paper is to present the Lie group in the simplest way. Before and before I started writing or collecting about Lie groups, it was necessary to say something about groups and subgroups that are taught in basic studies in algebra. In them I cited several deficits and an example. The following content of the paper is related to Lie groups primarily concerning the definition of examples such as The General Linear Group GL(n, R), The Complex General Linear Group GL(n, C), The Special Linear Group