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|Sep. 29, 2019. 19:15||Preliminary meeting. Selection of a seminar topic, administrative details, etc.|
|Oct. 10, 2019. 20:30||Devkota, Prabhat|
Rudiments of Lie Groups. Lie algebra of a lie group, exponential mapping and adjoint representation. Discussion of su(2) and sl(2).
References: You can find notes here.
Oct. 18, 2019. 19:30
Representations and Irreducibility. Representations of lie groups and algebras, sub-representations, irreducible representations, common operations (direct sum, tensor product, etc), intertwining operators, and proof of Schur's lemma.
References: See Sections 4.1 to 4.4 in Alexander Kirillov's notes. For a review of common linear-algebraic constructions refer to Appendix B in Fulton and Harris' "Representation Theory: A First Course"
|Oct. 25, 2019.||No meeting.|
|Nov. 1, 2019.|
Young diagrams. Young diagrams and their connections with the representation theory of the symmetric groups and the simpler Lie algebras.
|Nov. 8, 2019|
First look at the classification of connected Abelian Lie groups and connected complex Abelian Lie groups.
References: The notes can be found here.
|Nov 15, 2019||Aryal, Deepak|
Killing form, solvable, nilpotent and semi-simple lie algebras, existence of maximal solvable and nilpotent radical, equivalent definitions of semisimple lie algebras (in terms of killing form and direct sum of simple lie algebra).
References: Page 88-90 and 95-98 of the book "Lie Groups: An approach through invariants and representations" by Claudio Procesi.
|Nov 22, 2019|
Introduction to Root Systems and Dynkin diagrams: First steps of classifying Lie Algebras geometrically.
|Nov 29, 2019||Oprea, Maria Antonia||Representations of sl(2, C)|
|Dec 28, 2019||Devkota, Prabhat||The Levi-Malcev theorem|
|Jan. 11, 2020||Pal, Abhik|
Classification theorem of semisimple Lie Algebras.
References: Humphreys "Introduction to Lie Algebra and Representation Theory" Part 3 sections 9-11.
|19:00. Jan. 19, 2020|
The Lie-Kolchin theorem and Borel's Fixed Point Theorem
Complexification and Real Forms
References: For background on algebraic groups: Malle-Testermann's Linear Algebraic Groups and Finite Groups of Lie Type (Chapters 1 and 6), or Milne's Algebraic Groups, available here. For real forms I used Onishchik-Vinberg's Lie Groups and Lie Algebras III.