Department of Mathematics Syllabus
This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.
MAT 261: Lie Groups & Lie Algebras
Approved: 2021-05-01, Gorskiy/Kapovich
Units/Lecture:
4units
Suggested Textbook: (actual textbook varies by instructor; check your
instructor)
Brian C. Hall, Lie Groups, Lie Algebras and Representations. Second Edition. Graduate Texts in Mathematics vol. 222. Springer International Publishing, 2015.
Prerequisites:
MAT 147, 150A or equivalent; recommended 250A, 215A and 239, the latter can be taken concurrently with 261.
Course Description:
Lie groups, examples and topological properties. Lie algebras and representation theory, and semisimple Lie algebras.
Suggested Schedule:
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| Lectures | Sections | Topics/Comments |
|---|---|---|
| 1 | Lie groups, examples. Topological properties. Homomorphisms. | |
| 2 | Exponential map, one-parameter subgroups. Closed subgroup theorem (w/o proof). | |
| 3 | Lie algebras: definition, basic properties, homomorphisms. Simple, solvable and nilpotent Lie algebras. Lie algebras for matrix Lie groups. | |
| 4.6 | Representation theory of sl(2,C) | |
| 7 | 5. Semisimple Lie algebras. Roots, Cartan subalgebras, Weyl group. Killing form | |
| 9 | (Time permitting) Representations of semisimple Lie algebras: weights, Verma modules, finite-dimensional representations | |
Additional Notes:
Additional reading: W. Ziller. Lie Groups, Representation Theory and
Symmetric spaces. https://www2.math.upenn.edu/~wziller/math650/LieGroupsReps.pdf
W. Fulton, J. Harris. Representation Theory, A First Course. Graduate Texts
in Mathematics vol.129, Springer.