#
Columbia University G4343

Lie Groups and Representations

## Basic information

## Syllabus

Representation theory of finite groups.

McKay correspondence.

Representation theory of the symmetric group.

Basics of Lie groups and Lie algebras.

Solvable and nilpotent Lie algebras.

Universal enveloping algebras. PBW theorem.

sl(2) and its representations.

Classification of simple Lie algebras. Dynkin diagrams.

Representations of sl(n) and Schur-Weyl duality.

Representations of simple Lie algebras. Complete reducibility.

Haar measure. Representations of compact Lie groups.
## Online resources:

M. Alexandrino and R. Bettiol,
Introduction to Lie groups, adjoint action and its generalizations

J. Gallier,
Notes on Lie group actions: manifolds, Lie groups and Lie algebras

A. Kirillov, Jr.,
Introduction to Lie groups and Lie algebras

S. Sternberg,
Lie Algebras

D. Milicic, Lectures on Lie Groups

Notes for
Lie algebras class by Victor Kac.

Brian Hall,
An Elementary Introduction to Groups and Representations

Peter Woit, Lie groups and representations

Hans Samelson,
Notes on Lie algebras

Eckhard Meinrenken,
Clifford algebras and Lie groups

A brief summary
Root systems and Weyl groups, by Jeffrey Adams.

Online notes for MIT course
Introduction to Lie groups

Vera Serganova,
Representation theory: representations of finite groups, symmetric groups, GL(n),
quivers.

#### Representation theory overview:

Constantin Teleman,
Representation theory

P.Etingof et al., Introduction to
representation theory.

## Books:

The following books will be placed on reserve in the math library:

J.E.Humphreys, * Introduction to Lie algebras and representation
theory. *

W.Fulton and J.Harris, * Representation Theory: A First Course. *

R. Carter, G. Segal, and I. MacDonald,
* Lectures on Lie Groups and Lie Algebras. * A good supplementary reading
for our course is
Chapter II, by Segal.

Additional books on Lie groups and Lie algebras:

Daniel Bump, * Lie groups, * Graduate Texts in Mathematics, Vol. 225.

T.Brocker and T.Dieck, * Representations of Compact Lie Groups. *

Anthony Knapp, * Lie groups, Lie algebras, and cohomology. *

Anthony Knapp, * Lie groups Beyond an Introduction. *

J.-P. Serre, * Complex Semisimple Lie algebras. *

## Homework

** Homework 1 **
** Homework 2 **

** Homework 3 **

** Homework 4 **

** Homework 5 **

** Homework 6 **

** Homework 7 **

** Homework 8 **