# Part III - Symmetries, Fields and Particles

## Lectured by N. Dorey, Michaelmas 2016

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# Contents

- V Full version
- 1 Introduction
- 2 Lie groups
- 3 Lie algebras
- 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
- 8 Lie groups in nature