MAT 552: Introduction to Lie Groups and Lie Algebras

Spring 2017
Department of Mathematics
SUNY at Stony Brook




We will follow the books

Anthony W. Knapp Lie Groups Beyond an Introduction Birkhäuser; 2nd edition (August 21, 2002)

and by Fulton W., Harris J. Representation theory. A first course (Springer, 1991)


Time/Location: TuTh 10:00am- 11:20am Physics P123

Instructor: Mikhail Movshev
Math Tower 4-109. Phone: 632-8271. email: mmovshev at math dot sunysb dot edu
Office Hours: TuTh 12:00 - 1: 30, or drop-in, or by appointment.


Prerequisites: Students are expected to be familiar with most of the material of Math 530-531 (Geometry/Topology I-II) and Math 534-535 (Algebra I-II). Manifolds and group/algebra/module theory will be frequently used.

There will be homework assignments and a take home midterm (more assignments and its solutions to be posted). Though not mandatory, homework make a good supplement to classes.






Jan Tu 24

pp.2-15 Definitions and Examples. Ideals. Field Extensions and the Killing Form

Jan Th 26

pp.15-22 Semidirect Products of Lie Algebras. Solvable Lie Algebras and Lie's Theorem

Jan Tu 31

pp.22-37 Nilpotent Lie Algebras and Engel's Theorem. Cartan's Criterion for Semisimplicity Examples of Semisimpie Lie Algebras

Homework 1

Feb Th 2

pp37-55 Representations of sI(2, C) Elementary Theory of Lie Groups

Homework 2

Feb Tu 7

pp.55-62 Automorphisms and Derivations. Semidirect Products of Lie Groups

Homework 3

Feb Th 9

pp.62-73 Nilpotent Lie Groups. Classical Semisimpie Lie Groups

Homework 4

Feb Tu 14

pp79-92 Classical Root Space Decompositions Existence of Cartan Subalgebras

Homework 5

Feb Th 16

pp.92-116 Uniqueness of Cartan Subalgebras.Roots Abstract Root Systems

Homework 6

Feb Tu 21

Uniqueness of Cartan Subalgebras Roots Abstract Root Systems(cont)

Homework 7

Feb Th 23

pp.116-138 Weyl Group Classification of Abstract Cartan Matrices

Homework 8

Feb Tu 28

Weyl Group Classification of Abstract Cartan Matrices(cont)

Homework 9

Mar Th 2

pp.138-149 Classification of Nonreduced Abstract Root Systems Serre Relations

Homework 10

Mar Tu 7

pp149-156.Isomorphism Theorem Existence Theorem

Homework 11

Mar Th 9

pp164-172 Universal Mapping Property Poincare-Birkhoff-WittTheorem

Mar Tu 14

Spring Recess

Mar Th 16

Spring Recess

Mar Tu 21

Pp.172-179 Associated Graded Algebra Free Lie Algebras

Mar Th 23

pp.181-191 Examples of Representations Abstract Representation Theory

Mar Tu 28

pp.191-206  Peter-Weyl Theorem. Compact Lie Algebras. Centralizers of Tori

Mar Th 30

Peter-Weyl Theorem. Compact Lie Algebras. Centralizers of Tori(cont)

Apr Tu 4

pp.206-215  Analytic Weyl Group Integral Forms Weyl's Theorem

Apr Th 6

pp.220-229  Weights Theorem of the Highest Weight

Apr Tu 11

pp.229-246  Verma Modules. Complete Reducibility

Apr Th 13

Verma Modules. Complete Reducibility(cont.)

Apr Tu 18

pp.246-259  Harish-Chandra Isomorphism

Apr Th 20

pp.259-269  Weyl Character Formula

Apr Tu 25

pp. 269-283  Parabolic Subalgebras Application to Compact Lie Groups

Apr Th 27

pp. 291-304  Existence of a Compact Real Form Cartan Decomposition on the Lie Algebra Level

May Tu 2

pp. 304-320  Cartan Decomposition on the Lie Group Level Iwasawa Decomposition

May Th 4

Uniqueness Properties of the Iwasawa Decomposition 320 Cartan Subalgebras 326


Disabilities: If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Disability Support Services, ECC (Educational Communications Center) Building, room 128, (631) 632-6748. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students requiring emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information, go to the following web site: