Lie groups and algebraic groups.
(Math 535, Term II 2009/2010)
Tuesday, Thursday 11am-12:30pm, at Math 102.
My office: Math 217.
No class on April 6 and 15. Presentations on April 20, 22 instead.
Fulton and Harris.
Onischik and Vinberg "Lie groups and Algebraic groups".
- J. E. Humphreys, "Linear Algebraic Groups"
- A. Borel, "Linear Algebraic Groups"
- Essays on the structure of reductive groups
Notes by W. Casselman.
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).
Bruhat decomposition (Andrew?)
Flag varieties; cohomology of Grassmannians (this can generate any
munber of presentations). Sources to look at, to start: Bump "Lie groups",
Springer section 8.5.
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
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 :
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
References: OV, 4.1.1-4.1.3, and also 3.1.7, 3.2.9; Springer Section
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
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
Reference: OV, Chapter 4, Section 3.4
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).