Spring 2017
Department of Mathematics
SUNY at Stony
Brook
We will follow the books
Anthony W. Knapp Lie Groups Beyond an Introduction BirkhĂ¤user; 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 4109. Phone:
6328271. email: mmovshev at math dot sunysb dot edu
Office
Hours: TuTh 12:00  1: 30, or dropin, or by appointment.
Prerequisites: Students are expected to be familiar with most of the material of Math 530531 (Geometry/Topology III) and Math 534535 (Algebra III). 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.
Date 
Topic 
Homework 
Jan Tu 24 
pp.215 Definitions and Examples. Ideals. Field Extensions and the Killing Form 

Jan Th 26 
pp.1522 Semidirect Products of Lie Algebras. Solvable Lie Algebras and Lie's Theorem 

Jan Tu 31 
pp.2237 Nilpotent Lie Algebras and Engel's Theorem. Cartan's Criterion for Semisimplicity Examples of Semisimpie Lie Algebras 

Feb Th 2 
pp3755 Representations of sI(2, C) Elementary Theory of Lie Groups 

Feb Tu 7 
pp.5562 Automorphisms and Derivations. Semidirect Products of Lie Groups 

Feb Th 9 
pp.6273 Nilpotent Lie Groups. Classical Semisimpie Lie Groups 

Feb Tu 14 
pp7992 Classical Root Space Decompositions Existence of Cartan Subalgebras 

Feb Th 16 
pp.92116 Uniqueness of Cartan Subalgebras.Roots Abstract Root Systems 

Feb Tu 21 
Uniqueness of Cartan Subalgebras Roots Abstract Root Systems(cont) 

Feb Th 23 
pp.116138 Weyl Group Classification of Abstract Cartan Matrices 

Feb Tu 28 
Weyl Group Classification of Abstract Cartan Matrices(cont) 

Mar Th 2 
pp.138149 Classification of Nonreduced Abstract Root Systems Serre Relations 

Mar Tu 7 
pp149156.Isomorphism Theorem Existence Theorem 

Mar Th 9 
pp164172 Universal Mapping Property PoincareBirkhoffWittTheorem 

Mar Tu 14 
Spring Recess 

Mar Th 16 
Spring Recess 

Mar Tu 21 
Pp.172179 Associated Graded Algebra Free Lie Algebras 

Mar Th 23 
pp.181191 Examples of Representations Abstract Representation Theory 

Mar Tu 28 
pp.191206 PeterWeyl Theorem. Compact Lie Algebras. Centralizers of Tori 

Mar Th 30 
PeterWeyl Theorem. Compact Lie Algebras. Centralizers of Tori(cont) 

Apr Tu 4 
pp.206215 Analytic Weyl Group Integral Forms Weyl's Theorem 

Apr Th 6 
pp.220229 Weights Theorem of the Highest Weight 

Apr Tu 11 
pp.229246 Verma Modules. Complete Reducibility 

Apr Th 13 
Verma Modules. Complete Reducibility(cont.) 

Apr Tu 18 
pp.246259 HarishChandra Isomorphism 

Apr Th 20 
pp.259269 Weyl Character Formula 

Apr Tu 25 
pp. 269283 Parabolic Subalgebras Application to Compact Lie Groups 

Apr Th 27 
pp. 291304 Existence of a Compact Real Form Cartan Decomposition on the Lie Algebra Level 

May Tu 2 
pp. 304320 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) 6326748. 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: http://www.www.ehs.stonybrook.edu/fire/disabilities.shtml