Theory of groups and symmetries: Finite groups, Lie groups and Lie algebras

Abstract

This book provides a complete and concise review of the main methods used in the study of algebraic structures of symmetries for use in models of theoretical and mathematical physics, namely groups, Lie groups, Lie algebras, their representations and deformations. Theory of Groups and Symmetries is systematic, challenging and well suited for advanced undergraduate and graduate students as well as early career researchers in theoretical and mathematical physics. The textbook is based on a number of lectures in Dubna and Moscow. The theory of groups and conformal symmetries is a significant part of today’s theoretical physics, especially quantumfield theory and quantum gravity, representing the basis of the AdS/CFT correspondence. The authors give a clear and comprehensive introduction to the representation theory of groups focussing on Lie groups and moving later to advanced group theory and diverse notions from topology, analysis, and linear algebra, with numerous suggestions useful to the modern theoretical physicist. The textbook also contains a selected bibliography section that provides the reader with additional references to the topics covered. A large number of problems and their solutions completes this well-organized textbook. At the end of each chapter, the authors introduce several remarks and concrete examples, to help better understanding of the presented concepts and further gain working knowledge of the theory. Most suggested examples have direct applications in modern theoretical and mathematical physics, used in several models of quantum field theory, gravity and statistical physics. The first chapter presents Groups and Transformations, from Matrix Groups. Linear, Unitary, Orthogonal and Symplectic Groups to Homomorphisms, Groups of Transformations and the Conformal Group. The next two chapters focus on Lie Groups (Manifolds, Tangent Spaces and Haar Measure) and Lie Algebras (Matrix Lie Algebras, Lie Algebras of Classical Series, Lie Algebra of Conformal Group). Chapter four contains Representations of Groups and Lie Algebras (Linear Representations of Lie Groups, Representations of Lie Algebras, Direct Product and Direct Sum of Representations, Reducible and Irreducible Representations, Representations of Finite Groups and Compact Lie Groups. Group Algebra and Regular Representations, Character Theory for Finite Groups and Compact Lie Groups, Universal Enveloping Algebra. Casimir Operators and Yangians). The following chapters provide elements of Compact Lie Algebras, Root Systems and Classification of Simple Lie Algebras (Cartan Subalgebra. Rank of Lie Algebra and Cartan–Weyl Basis, Root Systems of Simple Lie Algebras). The last chapter discusses Homogeneous Spaces and their Geometry (Homogeneous Spaces and their examples, Induced Representations, Invariant metrics, Models of Lobachevskian Geometry and Geometry of Spaces AdS and dS, Metrics, Spherical functions and Laplace Operators in Homogeneous Spaces). The book is organised, well presented and establishes a strong conceptual foundation of the topics discussed, with insightful but accessible content aiming the physicist’s inquiring mind rather than a mathematically oriented reader. Eric Howard Department of Physics and Astronomy, Macquarie University, Sydney, Australia Centre for Quantum Dynamics, Griffith University, Brisbane, Australia [email protected] http://orcid.org/0000-0002-8133-8323