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Schedule

DateSpeakerDetails
Sep. 29, 2019. 19:15
Preliminary meeting. Selection of a seminar topic, administrative details, etc.
Oct. 10, 2019. 20:30Devkota, 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.

References: See Appendix A of Lando's Graphs on Surfaces and their Applications for a quick overview (also here). Chapters 1-2 of Serre's Linear Representations of Finite Groups are also useful.

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, 2019Aryal, 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.

  • Cartan subalgebras, root sets, dual of Cartan subalgebras, root systems, Dynkin diagrams.

References: Fulton and Harris, Part IV section 21.1Frederic Schuller - Classification of Lie algebras and Dynkin diagrams

Nov 29, 2019Oprea, Maria AntoniaRepresentations of sl(2, C)
Dec 28, 2019Devkota, PrabhatThe Levi-Malcev theorem
Jan. 11, 2020Pal, 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.


General Information

Meeting time, location

19:30. Room 120 (Seminar Room), Research


Participants

Pal, AbhikAryal, Deepak Irungu, Martin Waiharo Falkenburg, Pia Cosma Devkota, Prabhat Blloshmi, Denida Mele, Crystal Oprea, Maria Antonia