# 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

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

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

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

Samelson's Notes on Lie algebras

Homework:
Homework will be due weekly on Thursdays

Exams:
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