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Contents

• V Full version
• 1 Introduction
• 2 Lie groups
• 2.1 Definitions
• 2.2 Matrix Lie groups
• 2.3 Properties of Lie groups
• 3 Lie algebras
• 3.1 Lie algebras
• 3.2 Differentiation
• 3.3 Lie algebras from Lie groups
• 3.4 The exponential map
• 4 Representations of Lie algebras
• 4.1 Representations of Lie groups and algebras
• 4.2 Complexification and correspondence of representations
• 4.3 Representations of su(2)
• 4.4 New representations from old
• 4.5 Decomposition of tensor product of su(2) representations
• 5 Cartan classification
• 5.1 The Killing form
• 5.2 The Cartan basis
• 5.3 Things are real
• 5.4 A real subalgebra
• 5.5 Simple roots
• 5.6 The classification
• 5.7 Reconstruction
• 6 Representation of Lie algebras
• 6.1 Weights
• 6.2 Root and weight lattices
• 6.3 Classification of representations
• 6.4 Decomposition of tensor products
• 7 Gauge theories
• 7.1 Electromagnetism and U(1) gauge symmetry
• 7.2 General case
• 8 Lie groups in nature
• 8.1 Spacetime symmetry
• 8.2 Possible extensions
• 8.3 Internal symmetries and the eightfold way