Synopsis
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.
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