Lie groups and algebraic groups. (Math 535, Term II 2009/2010) Tuesday, Thursday 11am-12:30pm, at Math 102.
My office: Math 217.
  • Course description
  • References:

    Related links: (under construction) Scheduling changes: No class on April 6 and 15. Presentations on April 20, 22 instead.

    Tentative list of Presentations (April 8, 13, 20, 22):

    (please e-mail me if you'd like to change topic, etc.)
  • Tannaka's theorem (see Springer, section 2.5). (Shane?)
  • Bruhat decomposition (Andrew?)
  • Flag varieties; cohomology of Grassmannians (this can generate any munber of presentations). Sources to look at, to start: Bump "Lie groups", Chapter 50; Springer section 8.5. (Robert?)
  • Symmetric spaces (consider Chapter 31 of Bump, "Lie groups") Fulton-Harris 23.3 (Kael and Jerome?)
  • Connections with Riemannian geometry (eg., maximal tori and geodesics, etc., see Bump, for example) (Maxim?)
  • Haar measure on real groups; Weyl integration formula (Athena)

    Approximate detailed syllabus:

  • A growing list of problems. We will discuss some of these (depending on your interest) on Tuesday January 26 and Thursday the 28th; the list of problems will grow, and they will be discussed as necessary. The ones discussed already will be marked with a check mark. "The textbook" mentioned in the problems is Onischik-Vinberg, unless otherwise specified.

  • January 5-12 : Definition of an algebraic group; examples (the classical groups); the connected component at the identity; subgroups and homomorphisms; connectedness of a subgroup generated by a family of closed connected subgroups. References: OV 3.1.1, 3.1.4, or Humphreys 7.1 -- 7.5.
  • January 19-21 : Quasiprojective varieties; algebraic actions: existence of a closed orbit. Quasiprojective structure on a quotient. Linearization of an affine algebraic group. References: OV 2.1.8; 2.2.1 -- 2.2.6; 3.1.5, 3.1.6. Humphreys Chapter 8, mostly.
  • January 26 : Discussed Problems 1-8 on the list; the most important piece was the exact sequence of homotopy groups, that can be used to compute essentailly all fundamental groups of Lie groups. Discussed algebraic tori and quasitori (see OV Chapter 3, section 2.3)
  • January 28 : The Jordan decomposition in algebraic groups (OV Chapter 3, section 2.4), Commutative and solvable algebraic groups (most notably, Borel's Theorem) (OV, Chapter 3, sections 2.5--2.7), or the corresponding sections in either Borel, Humphreys or Springer.
  • February 2 : no class;
  • February 4 : The tangent algebra (OV, 3.3, and a some of Springer, Chapter 4).
  • February 9 : Semisimple and reductive Lie algebras; Killing form; characterization of the reductive linear Lie algebras in terms of the form Tr(XY); Borel subgroups. References: OV, 4.1.1-4.1.3, and also 3.1.7, 3.2.9; Springer Section 6.2.7.
  • February 11 : Conjugacy of all Borel subgroups over an alg. closed field; review of the root systems <---> semisimple Lie algebras correspondence; Reductive algebraic groups (definition). The adjoint action of the maximal torus of on the Lie algebra, and the resulting orthogonal decompostion of the Lie algebra.
    References: OV Chapter 4 Sections 1 and 4; Springer 6.3.5 (the conjugacy of maximal tori reference)
    Recommended review reading: OV Chapter 4, Section 5 (about the classical Lie algebars)
    A clumsy and extremely detailed calculation of the root system for sl(3) .
  • Monday March 1, 12-1 in Math 125 (the new seminar room in the lounge): make-up class. The correspondence between Weyl chambers and Borel subgroups. Reference: OV Chapter 4, sections 2.3, 2.4.
  • Tuesday March 2: Discussion of exercises!
  • Thursday March 4: The root and weight lattices; coroots, coweights, the character lattice. Reference: OV, Chapter 4, Section 2.8
  • Tuesday March 9: The centre and the fundamental group: characters, weights, and simple connectedness. Reference: OV, Chapter 4, Section 3.4
  • March 11: Classification theorems (OV, chapter 4, and section 3.4) Bruhat decomposition; some odds and ends (algebraicity of complex semisimple groups, etc.), Main reference: Springer, Chapter 8.
  • March 16-18 : "How to feel at home in E_8" -- guest lectures by Bill Casselman
  • Tuesday March 23: The notion of real form os an algebraic variety and algebraic group. Involutions. Also, example: not all real Lie groups are algebraic. Reference: OV Chapter 2, sections 3.4 -- 3.7, and Chapter 5, section 1.1.
  • Thursday March 25: Examples: real forms of the classical groups and Lie algebras. Toward classification of real forms of Lie algebras: the automorphism group of a Lie algebra. References: OV Chapter 5, section 1.2, and Chapter 4, sections 4.1--4.2.
  • Next week: Finishing the classification of the autmorphisms (Kac diagrams); the compact real form and related magic; an atlas demo.
  • In April: presentations (see the list of possible topics above, and feel free to suggest your favourite relevant topics instead).