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.

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:

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 ⊕

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 B

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

2. Compute the fundamental weights and Λ

Read Lectures 27 & 28 of Borcherds et. al. about the construction of e