The Lie Algebras Su(n): An Introduction

ISBN-13: 9783764324186 / Angielski / Miękka / 2003 / 432 str.

Walter Pfeifer; W. Pfeifer

Opis

Lie algebras are efficient tools for analyzing the properties of physical systems. Concrete applications comprise the formulation of symmetries of Hamiltonian systems, the description of atomic, molecular and nuclear spectra, the physics of elementary particles and many others. This work gives an introduction to the properties and the structure of the Lie algebras su(n). First, characteristic quantities such as structure constants, the Killing form and functions of Lie algebras are introduced. The properties of the algebras su(2), su(3) and su(4) are investigated in detail. Geometric models of the representations are developed. A lot of care is taken over the use of the term "multiplet of an algebra."
The book features an elementary (matrix) access to su(N)-algebras, and gives a first insight into Lie algebras. Student readers should be enabled to begin studies on physical su(N)-applications, instructors will profit from the detailed calculations and examples.

Spis treści

1 Lie algebras.- 1.1 Definition and basic properties.- 1.1.1 What is a Lie algebra?.- 1.1.2 The structure constants.- 1.1.3 The adjoint matrices.- 1.1.4 The Killing form.- 1.1.5 Simplicity.- 1.1.6 Example.- 1.2 Isomorphic Lie algebras.- 1.3 Operators and functions.- 1.3.1 The general set-up.- 1.3.2 Further properties.- 1.4 Representation of a Lie algebra.- 1.5 Reducible and irreducible representations.- 2 The Lie algebras su(N).- 2.1 Hermitian matrices.- 2.2 Definition.- 2.3 Structure constants of su(N).- 3 The Lie algebra su(2).- 3.1 The generators of the su(2)-algebra.- 3.2 Operators constituting the algebra su(2).- 3.3 Multiplets of su(2).- 3.4 Irreducible representations of su(2).- 3.5 Direct products of irreducible representations.- 3.6 Reduction of direct products of su(2).- 3.7 Graphical reduction of direct products.- 4 The Lie algebra su(3).- 4.1 The generators of the su(3)-algebra.- 4.2 Subalgebras of the su(3)-algebra.- 4.3 Step operators and states in su(3).- 4.4 Multiplets of su(3).- 4.5 Individual states of the su(3)-multiplet.- 4.6 Dimension of the su(3)-multiplet.- 4.7 The smallest su(3)-multiplets.- 4.8 The fundamental multiplet of su(3).- 4.9 The hypercharge Y.- 4.10 Irreducible representations of the su(3) algebra.- 4.11 Casimir operators.- 4.12 The eigenvalue of the Casimir operator C1 in su(3).- 4.13 Direct products of su(3)-multiplets.- 4.14 Decomposition of direct products of multiplets.- 5 The Lie algebra su(4).- 5.1 The generators of the su(4)-algebra, subalgebras.- 5.2 Step operators and states in su(4).- 5.3 Multiplets of su(4).- 5.4 The charm C.- 5.5 Direct products of su(4)-multiplets.- 5.6 The Cartan—Weyl basis of su(4).- 6 General properties of the su(N)-algebras.- 6.1 Elements of the su(N)-algebra.- 6.2 Multiplets of su(N).- References.