Lie groups and Lie algebras

Resources

• General course information
• Mark scheme for fortnightly question sheets
• Mark scheme for workshop and project presentations
• Question sheet 1
• Question sheet 2
• Question sheet 3
• Question sheet 4
• Projects
• For pre-class exercises, see the pages for each week (e.g. Week 1).

Week 1, accessible version

Session 1

• Introductory session, accessible version 6.35

Session 2 58.38

• Matrix exponential, accessible version 8.25
• Properties of exp, accessible version 19.56
• Local logarithm, accessible version 19.16
• Baker-Campbell-Hausdorff formula, accessible version 11.01

Optional extras

• Matrix norms, accessible version 21.05
• Convergence of exp, accessible version 30.40

Week 2, accessible version

Session 1 40.59

• The Lie algebra of a matrix group, accessible version 15.45
• Example: O(n), accessible version 11.57
• Example: SL(2,C), accessible version 13.17

Session 2 35.47

• Lie algebras, accessible version 8.36
• Lie algebra as a tangent space, accessible version 14.36
• Proof that little g is a Lie algebra, accessible version 12.35

Optional Extras

• Local coordinates, accessible version 26.34

Week 3, accessible version

Session 1 37.16

• Smooth homomorphisms, accessible version 14.13
• One-parameter subgroups, accessible version 7.52
• Linearity of homomorphisms in exponential charts, accessible version 15.11

Session 2 34.38

• Lie's theorem, accessible version 12.42
• Example: SU(2) to SO(3), accessible version 21.56

Optional extras

• Topology of Lie groups, accessible version 25.49

Week 4, accessible version

Session 1 25.36

• Representations, accessible version 11.32
• Complete reducibility, accessible version 14.04

Session 2 23.20

• Representations of U(1), part 1, accessible version 7.02
• Representations of U(1), part 2, accessible version 16.14

Week 5, accessible version

Session 1 42.42

• SU(2): Overview, accessible version 17.51
• New representations from old, accessible version 24.51

Session 2 29.26

• Weight space decomposition, accessible version 13.12
• X and Y, accessible version 16.14

Week 6, accessible version

Session 1 37.17

• X and Y: example, accessible version 16.33
• Classification of SU(2) representations, accessible version 20.44

Session 2 16.57

• Decomposing into irreducibles, accessible version 16.57

Optional extras

• Invariant theory, accessible version 19.00

Week 7, accessible version

Session 1 39.36

• SU(3): Overview, accessible version 21.05
• The adjoint representation, accessible version 18.31

Session 2 31.32

• Root vectors acting on weight spaces, accessible version 20.28
• Weyl symmetry, accessible version 11.04

Week 8, accessible version

Session 1 30.39

• Classification of SU(3) representations, accessible version 17.57
• Decomposing SU(3) representations, accessible version 12.42

Session 2 28.07

• The quark model, accessible version 14.41
• Duals, accessible version 13.26

Optional extras

• Proof of classification theorem, accessible version 19.15
• Uniqueness of irreps, accessible version 7.11
• Strangeness minus 3 - a BBC Horizon documentary about the prediction and discovery of the Omega minus particle. Featuring interviews and discussion with Gell-Mann, Ne'eman and Feynman.

Week 9, accessible version

Session 1 26.53

• Maximal tori: Overview, accessible version 13.20
• Weights, accessible version 9.16

Session 2 35.12

• The Killing form, accessible version 13.33
• The dual Killing form, accessible version 21.39

Optional extras

• Finding compact abelian subgroups, accessible version 13.46
• Compact connected abelian groups are tori, accessible version 18.02
• Any torus is contained in a maximal torus, accessible version 8.54

Week 10, accessible version

Session 1 43.26

• sl(2,C) subalgebras, 1, accessible version 19.37
• sl(2,C) subalgebras, 2, accessible version 11.09
• Root spaces are 1-dimensional, accessible version 12.40

Session 2 42.00

• Root systems, accessible version 16.05
• Classification of root systems, 1, accessible version 9.40
• Classification of root systems, 2, accessible version 16.15

Optional extras

• Dihedral angles, accessible version 6.43
• Simple roots, accessible version 19.55
• End of proof, accessible version 15.21
• Further reading (7.07)