Instructor: S. S. Kannan, kannan@cmi.ac.in
TA: Somnath Dake, somnath@cmi.ac.in, somnathcmi@gmail.com. Since we have limit on cmi inbox size, for submission of quizes, assignments or any other document, use gmail id.
Book: "Introduction to Lie Algebras and Representation Theory" by James E. Humphreys
Prerequisite: Any course in Linear Algebra or equivalent.
Grading:
Mid-term exam: 20 23 marks.
Scheduled on Thu, Mar 17 10:00 to 13:00 including scanning and submission time. Syllabus: Lectures 1 to 14.
Question Paper
End-term exam: 40 marks. Exam will be offline in CMI and is scheduled on Thu, May 12 09:30 to 12:30. Syllabus: All lectures including presentations. Lectures 8 onwards are main focus. Some practice questions: link
One assignments: 15 12 marks
Three quizzes: 3x5 = 15 marks. Quiz 1
One presetation: 10 marks
Class link: https://cmi-ac-in.zoom.us/j/7918100714?pwd=L0ZpWU52dWNQRnJsZ00wbzRyUUR6QT09
Lecture 1, Tue, Jan 24, 2022: Definition of Lie algebra, homomorphism, isomorphism, subalgebras, gl(n,F) and gl(V). link
Lecture 2, Thu, Jan 26, 2022: Trace of a matrix and linear transformation and properties of trace, Lie algebras sl(n,F), so(n,F), sp(2n,F), t(n,F), etc. link
Lecture 3, Tue, Feb 1, 2022: Abelian Lie algebra, derivations, ad(x) is derivation, ideals, center of L, simple Lie Algebras. link
Lecture 4, Thu, Feb 3, 2022: sl(2,C) is simple, quotient of Lie algebras, normalizer of L, centralizer, more about homomorphisms, representations, adjoint representation. link
Lecture 5, Tue, Feb 8, 2022: more about t(n,F), derived subalgebras, derived series, solvable Lie algebra and properties, radical of L, semisimple Lie algebra, central series, nilpotent Lie algebras. link
Lecture 6, Thu, Feb 10, 2022: properties of nilpotent Lie algebras, ad-nilpotent element, Engel's theorem: proof of two required results. link
Lecture 7, Tue, Feb 15, 2022: proof of Engel's theorem, few corollaries of proof of Engels' theorem. link
Lecture 8, Thu, Feb 17, 2022: Lie's theorem. Quiz 1. link
Lecture 9, Tue, Feb 22, 2022: corollaries of Lie's theorem, Jordan-Chevalley decomposition: semisimple element in End(V), Chinese remainder theorem. link
Lecture 10, Thu, Feb 24, 2022 : Jordan-Chevalley decomposition: theorem statement and proof, Killing form. link
Lecture 11, Tue, Mar 1, 2022 : Some excercises. (By Sadha Vishwa) link
Lecture 12, Thu, Mar 3, 2022 : Lemma required for Cartan's criteria (By Arkadev) link
Lecture 13, Tue, Mar 8, 2022 : Cartan's criteria, criterian for semisimplicity using Killing form, L semisimple is direct sum of simple ideals. link
Lecture 14, Thu, Mar 10, 2022 : L semisimple is direct sum of simple ideals, for L semisimple every derivation is inner derivation, Jordan-Chevalley decomposition in Der(A), abstract Jordan-Chevalley decomposition for semisimple lie algebras. link
Lecture 15, Tue, Mar 22, 2022 : L-module definition and representation corresponding to it, L-submodule, irreducible L-module, homomorphism of L-modules, Schur's Lemma, dual representation, tensor product of L-modules, induced L-module structure on Hom(V,W), Casimir element. link
Lecture 16, Thu, Mar 24, 2022 : Properties of Casimir element, complete reducibility, Weyl's theorem. link
Lecture 17-18, Tue-Thu, Mar 29-31, 2022 : for semisimple Lie subalgebras of gl(V) abstract Jordan-Chevalley decomposition coincide with usual Jordan decomposition, preservation of Jordan-Chevalley decomposition. representation theory of sl(2,F): basis of sl(2,F) {h,x,y} and Lie-bracket, weights, weight spaces, maximal vector, basis of arbitrary sl(2,F)-module, action of sl(2,F) basis elements on defined basis of sl(2,F)-module. link. Quiz questions and presentation assignment link
Lecture 19, Tue, Apr 5, 2022 : classification of sl(2,F)-modules, toral Lie subalgebra of semisimple Lie algebra, toral Lie subalgebras are abelian, maximal toral subalgebra, Cartan's decomposition, sl(n,F) link-1. roots of L, lemmas about roots, restriction of Killing form to centralizer is nondegerate link-2.
Lecture 20, Thu, Apr 7, 2022 : H = C(H), H* identified with H, properties of roots and root spaces. link
Lecture 21, Tue, Apr 12, 2022 : more properties of roots and root spaces link-1 link-2
Lecture 22, Sat, Apr 16, 2022 : link
Lecture 23, Tue, Apr 19, 2022 : Presentation by Abhinav link
Lecture 24, Thu, Apr 21, 2022 : Presentation by Russell link
Lecture 25, Tue, Apr 26, 2022 : Presentation by Dhivya link
Lecture 26, Thu, Apr 28, 2022 : Presentation by Sumantha link
Lecture 27, Mon, May 02, 2022 : Presentation by Archi link
Lecture 28, Fri, May 07, 2022 : Presentation by Rohan link
Students: Abhinav Kumar, Rohan Bajaj, Aritra Kundu(noncredit), Russell Dcruz, Archi Kaushik, Dhivya Prakash R V