Introduction to finite and infinite dimensional lie (super)algebras

Description

Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. Introduction to Finite and Infinite Dimensional Lie Algebras and Superalgebras introduces the theory of Lie superalgebras, their algebras, and their representations. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semi-simple Lie algebras. While discussing all classes of finite and infinite dimensional Lie algebras and Lie superalgebras in terms of their different classes of root systems, the book focuses on Kac-Moody algebras. With numerous exercises and worked examples, it is ideal for graduate courses on Lie groups and Lie algebras. Discusses the fundamental structure and all root relationships of Lie algebras and Lie superalgebras and their finite and infinite dimensional representation theoryClosely describes BKM Lie superalgebras, their different classes of imaginary root systems, their complete classifications, root-supermultiplicities, and related combinatorial identitiesIncludes numerous tables of the properties of individual Lie algebras and Lie superalgebrasFocuses on Kac-Moody algebras Content: Front Cover Introduction to Finite and Infinite Dimensional Lie (Super) algebras Copyright Dedication Contents About the author Acknowledgement Preface Author Acknowledgements Chapter 1: Finite-dimensional Lie algebras 1.1 Basic definition of Lie algebras with examples and structure constants A Lie algebra can also be defined starting from the definition of an algebra Lie algebras of one, two, and three dimensions and their structure constants 1.2 Subalgebras of Lie algebras and different classes of subalgebras of gl(n, C) 1.2.1 Different subalgebras of gl(n, C). Four families of classical Lie algebras, namely, An, Bn, Cn, and Dn and their bases1.3 Ideals, quotient Lie algebras, derived sub Lie algebras, and direct sum 1.4 Simple Lie algebras, semisimple Lie algebras, solvable and nilpotent Lie algebras 1.5 Isomorphism theorems, Killing form, and some basic theorems Examples for the matrix of the Killing form 1.6 Derivation of Lie algebras 1.7 Representations of Lie algebras and representations of sl(2,C) Representation of sl(2,C) in an (n + 1)-dimensional vector space. General theory of the representation of sl(2,C). Throughout this section G denotes sl(2,C)1.8 Rootspace decomposition of semisimple Lie algebras Basic properties of root systems Root space decomposition and properties of Killing form 1.9 Root system in Euclidean spaces and root diagrams 1.10 Coxeter graphs and Dynkin diagrams 1.11 Cartan matrices, ranks, and dimensions of simple Lie algebras Cartan matrices of classical simple Lie algebras 1.12 Weyl groups and structure of Weyl groups of simple Lie algebras. 1.13 Root systems of classical simple Lie algebras and highest long and short roots1.14 Universal enveloping algebras of Lie algebras The above definition can also be written as follows The universal mapping property 1.15 Representation theory of semisimple Lie algebras 1.16 Construction of semisimple Lie algebras by generators and relations 1.17 Cartan-Weyl bas