Theory discussed in lectures Lie groups
(Mastermath, WISL 519, 2019)

See also: exercises


In the list below, references are given to the 2010 version of the lecture notes.
Fall semester

1. Week 37 (Sep 12)
  • Prepare yourself for this lecture beforehand, by reading the basic material of Section 1: Groups.
  • Section 2: Lie groups, basic definitions
  • Theorem 2.9 (proof follows later)
  • Lemma 2.13, including the application to SL(V), Example 2.14 and Lemma 2.15.
  • Thm. 2.16 about closed subgroups. This theorem will be proved later. Read the proof of implication (b) ==> (a).
  • application to the examples in the rest of section 2.
  • 3: invariant vector fields: bijection with tangent space
    • 2. Week 38 (Sep 19)
  • Section 3: invariant vector fields, (maximal) integral curve, its domain
  • definition of exp, one parameter subgroups, up to Lemma 3.9.
  • Section 4: Ad, example GL(V), up to Lemma 4.6.
    • 3. Week 39 (Sep 26)
  • Section 4: Lie algebra Cor. 4.14
  • functiorial property of taking Lie algebra: Lemma 4.16.
  • Lemma 5.2: product of exponentials for commuting elements.
  • Lemma 5.3: a useful tool for differentiation.
  • Def 54. up to Lemma 5.8: component of the identity.
  • Theorem 5.11: Commutativity of group and algebra.
  • Def 7.1 of Lie subgroup, Lemma 7.6, Cor 7.6'

    • 4. Week 40 (Oct 3)
  • Lemma 7.7: char of Lie subalgebra, Examples 7.8, 7.8, mention of Thm 7.6.
  • One parameter subgroups: Lemma 7.2, read on your own: example of dense subgroup, Lemma 7.3.
  • 7.8 skipped.
  • 9.1: closed subgroup theorem and its proof.
    • 5. Week 41 (Oct 10)
  • continuous homomorphism is smooth Cor. 9.3 and Lemma 9.4.
  • SU(2) and SO(3): section 10 entirely.
  • Section 11: Group actions.
    • 6. Week 42 (Oct 17)
  • Section 12: smooth actions, principal fiber bundles
  • smoothness principle: Lemma 12.4.
  • Theorem 12.5, Def 12.7, Lemma 1.28
  • Sect. 13: Proper free actions, Theorem 13.5.
    • 7. Week 43 (Oct 24)
  • Infinitesimal action, Lemma 15.1,
  • Stabilizer and its lie algebra, Lemma 15.3
  • Orbit is immersed, Lemma 15.4.
  • Smooth transitive action: Prop. 15.5. Application of Baire.
  • mention of the Baire category theorem, no proof.
  • Examples 15.6, 15.7
  • Prop 17.1 already done in 6. Isomorphism thm: 17.4.
  • Application to adjoint group.
  • Ideal of a Lie algebra and normal subgroup. Intro to HW2.
  • 18: skipped
  • 19: densities and integration, up to Prop 19.3.

    • 8. Week 44 (Oct 31)
  • Invariant integration on the group: Proposition 19.8
  • normalized Haar measure, Lemma 19.10
  • right and left invariant measure, Lemma 19.12, Cor. 19.14, Lemma 19.14.
  • the notion of continuous representation, Def 20.1, Examples 20.3
  • Lemma 20.4: smoothness, rep of Lie algebra, derived representation.
  • Def 20.11 invariant subspace,
  • unitary representation, unitarizability of fin dim rep of compact group , Prop 20.17.
  • complement of invariant subspace for unitary represention, deco in irreducibles, Cor 20.19.
  • Def 20.21 of matrix coefficient.
  • Equivariant or intertwining map. Equivalence of representation.
  • mention of Irreducible representations for SU(2).
  • Schur's lemma, Lemma 20.27.

    • 9. Week 45 (Nov 7)
  • Def 20.21: matrix coefficient, 20.23 Equivalence of representations
  • Irreducibility of the representations pi_n of SU(2), Prop. 2.28.
  • Lemma 20.30 no intertwining operators; Lemma 20.31: uniqueness inner product
  • Par 21: span of matrix coefficients (22); relation with rep: Lemma 21.1, Cor 21.2
  • Schur orthogonality Theorem 21.3; proof; Cor 21.5.
  • Formulation of the Peter-Weyl theorem, Thm 23.5.
  • 27: Abelian groups and Fourier series, sketch, read on your own.
  • character of a representation,
  • orthogonality of characters: Lemma 22.10, multiplicity of irreducible subrepresentation: Lemma 22.31.
    • 10. Week 46 (Nov 14) different starting time: 13:30
  • Par 22: properties of characters, character of contragredient, direct sum, tensor product
  • Equivalence classes of representations are determined by their character.
  • skip Par 24, 25
  • 26: Peter-Weyl for class functions, formula for the projection operator L^2(G) --> C(G)_\pi.
  • 28: Prop 28.1 - Cor. 28.6: completeness of the set of characters of SU(2) and classification of the irreducible representations.
  • Lie algebra representations, Lemma 29.1, 29.2, representation of the complexified algebra.
  • Representations of sl(2,C) up to Theorem 30.3, Remark 30.4.

    • 11. Week 47 (Nov 21): Start at 14:00 in BBG 023.
  • Def of weight space and Lemma 30.5
  • def of primitive vector and Lemma 30.8 and proof
  • structure of rep: Lemma 30.7 and proof by induction
  • Corollary 30.8 and completion of the proof of Theorem 30.3
  • Section 31: roots and weights
  • weights for the action of a torus, Lemma 31.1; simultaneous diagonalisation of the action: 31.2
  • definition of maximal torus in g, Lemma 31.4.
  • deco in weight spaces of a finite dimensional rep of a compact group.
  • definition of root of t in g, root space decomposition, Cor 31.7
  • Example 31.8, Lemma 31.9,
  • roots and weights, Lemma 31.10, Cor. 31.12.
    • 12. Week 48 (Nov 28)
  • completion of Section 31:
  • bracket of root spaces: Cor 31.11, minus a root is root: Lemma 31.14.
  • Weyl chambers and system of positive roots, Lemma 3.15, sum of positive root spaces: Lemma 31.16,
  • highest weight vector and Lemma 31.8, definition cyclic vector.
  • structure fin dim rep with cyclic weight vector: Prop. 31.20.
  • application: irrep determined by highest weight: Thm 31.22.
  • Section 32: Lemma 32.2, study proof yourself.
  • Lemma 32.3 and Cor 32.4
  • 33: Aut(g) has Lie algebra Der(g)
  • Killing form and characterization of compact algebra
  • I made changes to Section 31 of the lecture notes. Here is the new text.
    • 13. Week 49 (Dec 5): no lecture, nor exercise class, no exercises.
    14. Week 50 (Dec 12)
  • 34: Killing form,
  • 35: Lemma 35.1, characterization of compact Lie algebra
  • 36: root systems for compact algebras, Lemma 36.1,
  • Cor 36.2: orthogonality of root space deco
  • Lemma 36.2: existence of sl 2 triple
  • Prop. 36.6,
  • Weyl group reflection, definition
  • Weyl group by interior automomorphisms: Lemma 36.7,
  • Lemma 36.9: finiteness Weyl group
  • read yourself: Definition 36.11: defi of abstract root system.
  • Def 38.6 of basis (fundamental system, system of simple roots).
    • 15. Week 51 (Dec 19)
  • 38: Description of the classification of root systems.


    • Written Exam: TBA

    Retake: TBA
  • Material: theory and exercises dealt with in this course.
  • Open book:You are allowed to bring lecture notes and personal notes, but no exercises, worked exercises or notes on exercises.
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    Last update: 22/11-2019