MAT 552: Introduction to Lie Groups and Lie Algebras

Spring 2014
Department of Mathematics
SUNY at Stony Brook

In this course we adopt a global approach to studying compact connected Lie groups and their representations. It should appeals to topologists and geometers.

We will follow books by Adams J.F. “Lectures on Lie groups” (Benjamin, 1969) and by Fulton W., Harris J. Representation theory. A first course (Springer, 1991)

Time/Location: TuTh 1:00pm- 2:20pm Physics P127

Instructor: Mikhail Movshev
Math Tower 4-109. Phone: 632-8271. email:
 mmovshev at math dot sunysb dot edu
Office Hours: TuTh 2:30 - 3: 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.

Requirements: There will be regular homework assignments (to be posted) due each class (see a table below), midterm and a final exam (both take home)

Grading System: The relative significance of exams and problem sets in determining final grades is as follows.



Final Exam


Problem Sets/Class Participation





Jan Tu 28

No class

Jan Th 30

No class

Feb Tu 4

p.1-7 Basic definitions

Feb Th 6

p.7-12 One-parameter subgroups

Homework 1

Feb Tu 11

p.12-18 Properties of the exponential map

Homework 2

Feb Th 13

p.19-25 Elementary Representation Theory. Representations over R,C and H

Feb Tu 18

p.26-32 Restriction and extension of scalars. Tensor product of representations.

Homework 3

Feb Th 20

p.33-39 Dual representation. Theory of integration over compact groups. Complete reducibility of representations.

Homework 4

Feb Tu 25

p.40-46 Schur's lemma. K-functor.

Homework 5

Feb Th 27

p.47-53 Characters. Orthogonality relation.

Homework 6

Mar Tu 4

p.54-60 Peter-Weyl theorem

Homework 7

Mar Th 6

p.61-67 More on real and quaternionic structures on complex representations,self-conjugate representations.

Homework 8

Mar Tu 11

p.68-74 Adams operations. Irr. rep. for a products of groups. Representations of compact abelian groups

Homework 9

Mar Th 13

p.75-81 Maximal Tori In Lie Groups. Monogenic subgroups.


Homework 10

Mar Tu 18

Spring Recess

Homework 11

Mar Th 20

Spring Recess

Mar Tu 25

p.82-88 Roots. Examples:SU,Sp,SO(2n)

Mar Th 27

p.89-95 Examples cont.:SO(2n+1). Conjugacy of maximal tori. Weyl group W. Regular and singular elements. Action of W on roots

Apr Tu 1

p.96-102 Subgroups U_theta, labelled by root theta. Geometry Of The Stiefel Diagram Examples of diagrams for U(2),SO(4),Sp(2)

Apr Th 3

p.103-108 Examples cont.SU(3). Intersection of U_theta, Linear relations between roots

Apr Tu 8

p.109-115 Action of W on roots cont. Action of W on Weyl chambers. Generators for W. Examples: SU,Sp,SO

Apr Th 10

p.116-122 A formula for the generators. Weights. Angles comprised by roots. Series of roots.Examples SU(3),Sp(2). A fundamental Weyl chamber. Simple roots.

Apr Tu 15

p.123-129 The Dynkin diagram.The fundamental dual Weyl chamber. Half sum of positive roots. The extended Weyl group. Description of the fundamental group in terms of the root data.

Apr Th 17

p.130-136 The fundamental group cont.Examples of computations: SU,Sp, Groups Spin(n)

Apr Tu 22

p.137-143 Representation Theory Weyl Integration Formula. Symmetries of characters. Characterization of anti-symmetric characters. Elementary symmetric sums

Apr Th 24

p.144-150 The elementary alternating sum. Characters of Spin(n)

Apr Tu 29

p.151-157 Two partial orders on weight lattice.

May Th 1

p.158-164 The maximal weight of an irr.rep.

May Tu 5

p.165-171 K(G) is a polynomial algebra for simply-connected G. Description of the complex representation rings K(U(n)) and K(Sp(n)).

May Th 8

p.172-180 Computation of K(SO(n)).

May Mon 19

Final Exam

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: