Theory discussed in lectures Lie groups and Lie algebras
(Mastermath, WISL 535, 2018)

See also: exercises


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

1. Week 6 (Feb 9)
  • 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: informally: integral curves, up to the definition of exp
    • 2. Week 7 (Feb 16)
  • 3. (maximal) integral curve, definition of exp
  • exp for GL(n)
  • properties of exp: Lemma 3.6.
  • definition and classification of one parameter subgroups, Lemma 3.9.
  • Definition 4.1: the adjoint representation; Lemmas 4.3, 4.4
  • Definition of ad and properties of ad: Lemma 4.6, Example 4.7.
  • Definition Lie bracket: 4.8
  • mention of Lemma 4.16.
    • 3. Week 8 (Feb 23)
  • anti-symmetry of Lie bracket, Lemma 4.9
  • Lie homomorphism and preservation of bracket: Lemma 4.10.
  • The Jacobi-identity, Cor 4.12. Conclusion Lemma 4.16.
  • Commuting elements and exp: Lemma 5.2.
  • Component of the identity, Def 5.4
  • Lie group with commutative algebra: Theorem 5.11.
  • Overview of Section 6 and read the section on your own.

    • 4. Week 9 (Mar 2)
  • Definition 7.1 of Lie subgroup, Lemma 7.2, Example 7.5, read on your own: Lemma 7.3, Cor 7.4.
  • Lemma 7.6, Cor. 7.6': Lie algebra of a Lie subgroup; application to examples: 7.8, 7.9. Theorem 7.11.
  • Read the proof of Thm 7.11, in Section 8, on your own.
  • The closed subgroup theorem: Section 9, Theorem 9.1.

    • 5. Week 10 (Mar 9)
  • Application of closed subgroup theorem: Cor. 9.3, Lemma 9.4.
  • Section 10, the groups SU(2) and SO(3).
  • Section 11 Actions: left and right, continuous and smooth
  • quotient topology on quotient, continuous functions on quotient
  • Hausdorfness: Lemma 11.8
  • Smooth action of PFB type: defi 12.2.
  • Smoothness principle: Lemma 12.4
  • Theorem 12.5: manifold structure on quotient for action of PFB type; Lemma 12.8 justifies terminology.

    • 6. Week 11 (Mar 16)
  • Proof of Thm 12.5, construction of atlas; Def 12.7: PFB
  • Section 13: proper free actions, and Theorem 13.5; Lemma 13.6: read on your own; Slice Lemma (3.17)
  • Section 14: coset spaces; smoothness of the action: Cor 14.2. Cor 14.3: tangent space at origin
  • Section 15: orbits: defi infinitesimal action $X \mapsto \ga_X.$
  • Lemmma 15.3: Lie algebra of stabilizer; Lemma 15.4, Example of the sphere.
  • 16: the Baire category theorem: this tool from topology is used in the proof of Cor 16.6, which is applied in Prop. 15.5.
  • Section 17: Theorem 17.4. The rest of the section has been partially discussed in the lecture. Read the rest on your own.
  • Section 18 is skipped.
    • 7. Week 12 (Mar 23)
  • Section 19: the notion of density on a manifold, integration of such a density
  • Left invariant density on a group, Haar measure.
  • Properties of the invariant integral.
  • Unimodular group, Cor 19.13, Lemma 19.14.
  • Read on your own: left invariant density on a homogeneous space
  • In the lecture we will do left invariant density on the quotient of two unimodular groups, on $S^n.$
  • Section 20: the notion of finite dimensional continuous representation, Def 20.1. Lemma 20.4, Def 20.5.
  • Natural appearance with actions, Representations of SU(2)
  • Def 20.11: invariant subspace of a representation, notion of irreducibility.
  • Unitary and unitarizable representation.
  • Application of invariant integration, unitarizability of finite dim rep of compact group: Prop. 20.17.
    • xx. Week 13 (Mar 30) Goede vrijdag, no lecture.

    8. Week 14 (Apr 6)
  • Irreducibility of reps of SU(2) : Prop. 20.28
  • Definition of matrix coefficient, and space spanned by them
  • Theorem 21.3 Schur orthogonality and proof
  • Defi 21.6 of character, Rem 21.7
  • Lemma 21.8
  • Section 22, characters Lemma 22.1, 22.2, 22.3
  • read yourself: Defi 22.4 Lemma 22.5 and Lemma 22.5
  • Lemma 22.7: character of direct sum
  • orthonormality of characters: Lemma 22.10
  • Cor 20.19: decomposition of unitarizable finite dimensional representation.
  • Lemma 22.11: multiplicity of irreducibles
  • Corollary 22.13: character as complete invariant.
  • read on your own: cor 22.14

    • 9. Week 15 (Apr 13)
  • constructions of reps: contragredient and tensor product, their characters: Lemmas 22.6, 22.8.
  • Equivariance of matrix coefficients, Lemma 21.1.
  • Schur orthogonality in the form of Cor. 21.5.
  • Section 23: The Peter-Weyl theorem, formulation without proof
  • We skip the proof of the PW theorem, which is contained in 24 and 25. Exercise 49 suggests the proof for finite groups. Those interested in functional analysis are strongly urged to read sections 24 and 25.
  • Class functions and characters: Corollary 26.2
  • Fourier series for compact abelian groups: Cor. 27.3, Cor. 27.4.
  • Section 28: irreducible reps for SU(2).
  • Section 29: Lie algebra representations
  • Section 30: representations of sl(2,C), up to Theorem 30.3.
    • 10. Week 16 (Apr 20)
  • Section 30, Thm: 30.3: irreducible finite dimensional modules for sl(2,C)
  • Consequence for SU(2) mentioned in Remark 30.4.
  • weight spaces for H, Lemma 30.5, primitive vector, Lemma 30.6
  • Lemma 30.7 and its proof in detail
  • Corollary 30.8: irreducible module determined by weight of primitive vector.
  • the structure of weight spaces of a module over an abelian Lie algebra: Lemma 31.1, Lemma 31.2.
  • Def 31.4: torus and maximal torus in a compact Lie algebra.
  • the weight space decomposition of a finite dimensional representation, Lemma 31.5.
  • root system of a compact Lie algebra
  • Roots and weights. Lemma 31.10, Cor. 31.11.
    • xx. Week 17 (Apr 27) King's day, no lecture.

    11. Week 18 (May 4)
  • The weight space decomposition of a finite dimensioanal irreducible module Lemma 31.12
  • System of positive roots: Lemma 31.14, Lemma 31.15
  • the sum of the positive root spaces. Lemma 31.16
  • Definition of highest weight vector. Def 31.17
  • Definition of cyclic vector, 31.19
  • Proposition 31.20 about cyclic highest weight vector with proof.
  • Uniqueness of highest weight, and Theorem 31.23.
    • xx. Week 19 (May 11) Hemelvaart, no lecture
    12. Week 20 (May 18)
  • Automorphisms and derivations: section 33
  • Killing form: section 34
  • compact Lie algebras: Lemma 35.1, Lemmas 35.2, 35.3, 35.4
  • Thm 35.5, equivalence of (a) and (c).
  • Lemma 35.6, Proposition 35.8
  • Root system for compact Lie algebras: Lemma 36.1, Lemma 36.3, Ex. 36.4, Lemma 36.5,
  • Prop 36.6.
    • 13. Week 21 (May 25)


    Written Exam: June 22, 10:00 - 13:00, SP H0.08

    Retake: July 13, 10:00 - 13:00, SP A1.04
  • 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: 25/5-2016