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.
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Date |
Topic |
Homework |
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Jan Tu 24 |
pp.2-15 Definitions and Examples. Ideals. Field Extensions and the Killing Form |
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Jan Th 26 |
pp.15-22 Semidirect Products of Lie Algebras. Solvable Lie Algebras and Lie's Theorem |
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Jan Tu 31 |
pp.22-37 Nilpotent Lie Algebras and Engel's Theorem. Cartan's Criterion for Semisimplicity Examples of Semisimpie Lie Algebras |
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Feb Th 2 |
pp37-55 Representations of sI(2, C) Elementary Theory of Lie Groups |
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Feb Tu 7 |
pp.55-62 Automorphisms and Derivations. Semidirect Products of Lie Groups |
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Feb Th 9 |
pp.62-73 Nilpotent Lie Groups. Classical Semisimpie Lie Groups |
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Feb Tu 14 |
pp79-92 Classical Root Space Decompositions Existence of Cartan Subalgebras |
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Feb Th 16 |
pp.92-116 Uniqueness of Cartan Subalgebras.Roots Abstract Root Systems |
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Feb Tu 21 |
Uniqueness of Cartan Subalgebras Roots Abstract Root Systems(cont) |
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Feb Th 23 |
pp.116-138 Weyl Group Classification of Abstract Cartan Matrices |
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Feb Tu 28 |
Weyl Group Classification of Abstract Cartan Matrices(cont) |
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Mar Th 2 |
pp.138-149 Classification of Nonreduced Abstract Root Systems Serre Relations |
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Mar Tu 7 |
pp149-156.Isomorphism Theorem Existence Theorem |
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Mar Th 9 |
pp164-172 Universal Mapping Property Poincare-Birkhoff-WittTheorem |
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Mar Tu 14 |
Spring Recess |
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Mar Th 16 |
Spring Recess |
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Mar Tu 21 |
Pp.172-179 Associated Graded Algebra Free Lie Algebras |
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Mar Th 23 |
pp.181-191 Examples of Representations Abstract Representation Theory |
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Mar Tu 28 |
pp.191-206 Peter-Weyl Theorem. Compact Lie Algebras. Centralizers of Tori |
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Mar Th 30 |
Peter-Weyl Theorem. Compact Lie Algebras. Centralizers of Tori(cont) |
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Apr Tu 4 |
pp.206-215 Analytic Weyl Group Integral Forms Weyl's Theorem |
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Apr Th 6 |
pp.220-229 Weights Theorem of the Highest Weight |
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Apr Tu 11 |
pp.229-246 Verma Modules. Complete Reducibility |
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Apr Th 13 |
Verma Modules. Complete Reducibility(cont.) |
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Apr Tu 18 |
pp.246-259 Harish-Chandra Isomorphism |
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Apr Th 20 |
pp.259-269 Weyl Character Formula |
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Apr Tu 25 |
pp. 269-283 Parabolic Subalgebras Application to Compact Lie Groups |
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Apr Th 27 |
pp. 291-304 Existence of a Compact Real Form Cartan Decomposition on the Lie Algebra Level |
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May Tu 2 |
pp. 304-320 Cartan Decomposition on the Lie Group Level Iwasawa Decomposition |
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May Th 4 |
Uniqueness Properties of the Iwasawa Decomposition 320 Cartan Subalgebras 326 |
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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: http://www.www.ehs.stonybrook.edu/fire/disabilities.shtml