New Developments in Lie Theory and Their Applications

Contemporary Mathematics, 2011

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Abstract and Applied Analysis, 2013

This special issue was planned to focus on most the recent advances in the applications of Lie groups. It covered a wide area of topics in interdisciplinary studies in mathematics, mechanics, physics, and finance. We were particularly interested in receiving novel contributions devoted to Lie groups, in particular, applications to specific problems in applied sciences. We aimed to bring together contributions across a variety of applications of Lie groups and invite researchers to submit original research and/or domain reviews in various topics. Potential topics included, but were not limited to the following: Lie algebras and Lie pseudogroups, optimal control, topological groups, representation theory of Lie algebras, differential geometry, finance, dynamical systems, quantum mechanics, super-symmetry and superintegrability, information theory, Lie theory and symmetry methods in differential, fractal differential, integrodifferential, and difference equations, and further applications in physics and mechanics.

2020

Lie groups are important to describe symmetries, both in mathematics and in applications (physics, chemistry, engineering,. . . ). The classical Lie groups are for example the orthogonal groups O(n), the unitary groups U(n), but mathematicians and physicists are also fascinated by more exotic examples such as the symmetry group of the octonions which is discussed a lot in modern mathematical physics. Many of these Lie groups can be represented as subgroups of Gl(k,R) for some sufficiently large k, but there are also Lie groups which cannot. Lie groups are manifolds G together with a multiplication μ : G×G→ G which is a smooth map, such that (G,μ) is a group.1 Lie groups and their representation is a mighty theory which allows effect calculations both for problems inside mathematics and also for applications outside.

The main goal of this poster, which is written in the form of a survey and tries to show some aspects of the research of authors on Lie algebras, is to pay homage to the memory of Pilar Pisón Casares, who was firstly teacher of some of them and later colleague of all of them during different stages of her stay as a member of the Departamento deÁlgebra, Computación, Geometría y

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

2020

In this research article, Lie Groups and Lie Algebras are projected in a distinct direction and with innovative proofs. We develop the necessary and sufficient condition for a topological group to be Hausdorff. The criteria of topological group and Haussdorff to be connected is also derived. This research article mainly explores the concept of left in variant field and presents an important hypothesis namely ―any Lie group is parallelizable‖ with a very simple accurate logical and analytical reasoning. Moreover this paper covers the impervious of another property namely every one parameter subgroup is an integral curve of some left-invariant vector field.

Journal of Generalized Lie Theory and Applications, 2012

Hom-algebra structures are given on linear spaces by multiplications twisted by linear maps. Hom-Lie algebras and general quasi-Hom-Lie and quasi-Lie algebras were introduced by Hartwig, Larsson and Silvestrov as algebras embracing Lie algebras, super and color Lie algebras and their quasi-deformations by twisted derivations. In this paper we introduce and study Hom-associative, Hom-Leibniz and Hom-Lie admissible algebraic structures generalizing associative, Leibniz and Lie admissible algebras. Also, we characterize flexible Hom-algebras and explain some connections and differences between Hom-Lie algebras and Santilli's isotopies of associative and Lie algebras.

2012

The purpose of the book, as announced in the Preface, is to supply a collection of problems in group theory, Lie group theory and Lie algebras. Each Chapter contains 100 completely solved problems.

Lecture Notes in Mathematics, 1984

2014

In this paper we prove that matrix groups are manifolds and use them as a special case to introduce the concepts of Lie groups, Lie algebras, and the exponential map.