Related papers: Elements of Group Theory
In physics, Lie groups represent the algebraic structure that describes symmetry transformations of a given system. Then, the descending Lie algebra of those groups are necessarily real. In most cases, the complexification of those Lie…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. In this second part we introduce the fundamental concepts of topological spaces, convergence, and continuity, as…
General properties of ternary semigroups and groups are considered. The bi-element representation theory in which every representation matrix corresponds to a pair of elements is built, connection with the standard theory is considered and…
This is a survey of recent developments in combinatorics. The goal is to give a big picture of its many interactions with other areas of mathematics, such as: group theory, representation theory, commutative algebra, geometry (including…
We functorially characterize groupoids as special dagger Frobenius algebras in the category of sets and relations. This is then generalized to a non-unital setting, by establishing an adjunction between H*-algebras in the category of sets…
In this paper we define a new algebraic object: the disguised-groups. We show the main properties of the disguised-groups and, as a consequence, we will see that disguised-groups coincide with regular semigroups. We prove many of the…
This is a survey on appearances of reflection groups, real and complex, in algebraic geometry. We also include a brief introduction into the theory of reflection groups.
The aim of this paper (Part III) is formulating GR as a scalar field theory. The basic structural elements of it are a generating function, a generalized density and a generalized temperature. One of the axioms of this theory is a…
The paper has a form of a talk on the given topic. It consists of three parts. The first part of the paper contains main notions, the second one is devoted to logical geometry, the third part describes types and isotypeness. The problems…
We consider the problem of existence of representations of topological groupoids on a principal bundle and the classification of such representations up to gauge transformation. Such representations naturally occur in various contexts such…
We present here definitions and constructions basic for the theory of monoidal and tensor categories. We provide references to the original sources, whenever possible. Group-theoretical categories are used as examples
This is a short, self-contained expository survey, focused on algebraic and analytic aspects of quantum groups. Topics covered include the definition of ``quantum group,'' the Yang-Baxter equation, quantized universal enveloping algebras,…
We provide a brief overview of tensor models and group field theories, focusing on their main common features. Both frameworks arose in the context of quantum gravity research, and can be understood as higher-dimensional generalizations of…
It is described the group of arrowy permutations (that is extension of symmetric group) and the consequent process of generation of GL(n) and some its subgroups by this combinatoric group and its subgroups.
Group Theory has become an invaluable tool in the physics community. Despite numerous introductory books, the subject remains challenging for beginners. Mathematica has emerged as a popular tool for research and education, offering various…
The purpose of this paper is to prove the First and Second Fundamental Theorems of invariant theory for the complex special linear supergroup and discuss the superalgebra of invariants, via the super Plucker relations.
A deformation of the canonical algebra for kinematical observables of the quantum field theory in Minkowski space-time has been considered under the condition of Lorentz invariance. A relativistic invariant algebra obtained depends on…
This paper is the third in a series dedicated to the fundamentals of sub-Riemannian geometry and its implications in Lie groups theory: "Sub-Riemannian geometry and Lie groups. Part I", math.MG/0210189, available at…
We first recall two equivalent definitions of Lie $2$-algebras, categorification of Lie algebras and $2$-term $L_\infty$-algebras. Then we present four different kinds of Lie $2$-algebras from $2$-plectic manifolds, Courant algebroids,…
This is a preliminary version of the first chapter of a book project on the character theory of finite groups of Lie type. It provides the foundations from the general theory of reductive algebraic groups over a finite field.