Symmetries, Fields and Particles

Introduction
Lie groups
	Definitions
	Matrix Lie groups
	Properties of Lie groups
Lie algebras
	Lie algebras
	Differentiation
	Lie algebras from Lie groups
	The exponential map
Representations of Lie algebras
	Representations of Lie groups and algebras
	Complexification and correspondence of representations
	Representations of su(2)
	New representations from old
	Decomposition of tensor product of su(2) representations
Cartan classification
	The Killing form
	The Cartan basis
	Things are real
	A real subalgebra
	Simple roots
	The classification
	Reconstruction
Representation of Lie algebras
	Weights
	Root and weight lattices
	Classification of representations
	Decomposition of tensor products
Gauge theories
	Electromagnetism and U(1) gauge symmetry
	General case
Lie groups in nature
	Spacetime symmetry
	Possible extensions
	Internal symmetries and the eightfold way
Index