Math 261A, Fall 2009

Time: Tuesday & Thursday, 9:40 am - 11:00 am
Announcement: the classroom is being changed to:
Place: Room 31 Evans
Professor: Ian Agol
Office Hours: Tuesday, Wednesday 11am-12:30 pm
Office: 921 Evans
E-mail: ianagol at math dot berkeley dot edu
Course Control Number: 54812

Brian C. Hall, Lie Groups, Lie Algebras, and
Representations, Springer, 2003.

Supplementary Reading:
Notes from Haiman's class on Lie groups
Notes from Borcherds, Reshetikhin, Serganova's class on Lie Groups

Recommended Reading:
Fulton and Harris, Representation Theory: A First Course, Springer, 2008.

Samelson's Notes on Lie algebras

Homework will be due weekly on Thursdays

There will be an optional 1 hour oral final as practice for those
who would like to include Lie Groups as a topic
on their qualifying exam.

Homework 1, due Thursday September 3: Chapter 1, Exercises 4, 8, 11

The belt trick

Homework 2, due Thursday September 10: Chapter 1, Exercises 13, 21
Show that the fundamental group of a connected Lie group is commutative.

Computation of fundamental group of SO(n), U(n), Sp(n):
Example 4.55, pp. 383-384, Algebraic Topology, Hatcher

Homework 3, due Thursday September 17: Chapter 2, Exercises 8, 9, 14

Homework 4, due Thursday September 24: Chapter 2, Exercises 20, 22, 24, 28

Homework 5, due Thursday October 1: Chapter 3, Exercises 9, 13, 17

Notes on simple Lie algebras and Lie groups

Homework 6, due Thursday October 8:
Problem 1: Let G be a simply connected Lie group, with Lie algebra g. Let [G,G] be
the subgroup generated by commutators. Then [G,G] is a connected, closed
Lie subgroup of G, with Lie algebra Dg.
Problem 2: Which irreducible representations of SU(2) are representations of SO(3)= SU(2)/{+-I}?
Problem 3: Is the Lie algebra so(4) simple? If not, determine its structure.

Homework 7, due Thursday October 15:
Problem 1: For a 1-dimensional representation of a Lie algebra or Lie group, prove that
the tensor product with the dual representation is trivial.
Problem 2: If g is a direct sum of simple Lie algebras, then the only ideals of g are sums of the factors.
Problem 3: If g is semisimple, then the map ad: g -> gl(g) is an isomorphism of g to Der(g), the
derivations of g.

Homework 8, due Thursday October 22:
Problem 1: Let g be a Lie algebra, and  G < GL(g) be
the Lie group with Lie algebra ad(g). Then G is a subgroup
of O(B), where B is the Killing form on g.

Homework 9, due Thursday October 29:
1. Let g be a semisimple complex Lie algebra.
A subalgebra of g is a Cartan subalgebra iff it consists entirely
of semisimple elements and is maximal with respect to
this property.

2. Let g be a semisimple complex Lie algebra.
Show b = h ⊕α >0 gα
is a maximal solvable subalgebra (called the Borel subalgebra).

Homework 10, due Thursday November 5:
1. Show that all Borel subalgebras are equivalent.
2. prove that a semisimple Lie algebra is simple if and only if its
root system is irreducible.

Homework 11, due Thursday November 12:
1. Compute the Cartan matrices and determinants of the simple Lie algebras B4, C4, D4
, E6, and F4.
2. How many orbits of roots does each root system have under the action of the Weyl group?

Homework 12, due Thursday November 19:
1. Verify that if one starts with a semisimple Lie algebra with a given Dynkin diagram, the Serre
relations must hold.
2. A Coxeter element in the Weyl group W is the product of all simple reflections, one each, in
any order. Prove that any two Coxeter elements are conjugate in W.

Homework 13, due Thursday December 3:
1. Show using the 1/2-spin representations S+, S- that Spin2 ℂ = GL1ℂ = ℂ*, Spin4 ℂ = SL(S+) x SL(S-) = (SL2ℂ)2 , Spin6 ℂ = SL(S+)= SL 4

2. Compute the fundamental weights and ΛWR for the exceptional Lie algebra f4.
Read Lectures 27 & 28 of Borcherds et. al. about the construction of e8