Representation Theory IV Jack Shotton (April 27, 2026) Contents 1 Linear Lie groups and their Lie algebras 1.1 Linear Lie groups 1.2 The exponential map 1.3 One-parameter subgroups 1.4 Lie algebras 1.5 Lie group and Lie algebra homomorphisms 1.6 Complex Lie groups and holomorphic homomorphisms 1.7 Topological properties 1.8 The example of SU(2) and SO(3) 2 Representations of Lie groups and Lie algebras - generalities 2.1 Basics 2.2 Standard constructions for representations 2.2.1 Functional constructions 2.3 The adjoint representation 2.4 The homomorphism SU(2)→SO(3) 2.5 Representations of U(1) and Maschke’s theorem 3 SL2 3.1 Weights 3.1.1 Highest weights 3.2 Classification of representations of sl2,C. 3.3 Decomposing representations 3.4 Real forms and complete reducibility 3.5 SO(3) 3.5.1 Classification of irreducible representations 3.5.2 Harmonic functions 4 SL3 4.1 The Lie algebra sl3,C 4.2 Weights 4.3 Visualising weights 4.4 Representations and weights 4.5 Tensor constructions 4.6 Highest weights 4.7 Dominant weights and Weyl symmetry 4.8 Irreducible representations of sl3,C. 4.9 Proof of theorem 4.28 — nonexaminable 5 Problems for Epiphany 5.1 Problems for section 1 5.2 Problems for section 2 5.3 Problems for section 3 5.4 Problems for section 4 6 Solutions