Introduction to Lie algebras 7X0E7E1DD2

URI	`m2://res/7X0E7E1DD2`
Kind	course
Creators	R Venkatesh (instructor)
Date	2024-01-17
Scope	in
Ids	course 04dese585:111108414

Expressions

e1: edition

Language	en
Date	2024-01-17

m1: page, IISc Bangalore (record). nptel 111108414
m2: page, 2024-01-17 (record). nptel 111108414

Contents

part 1: Week 1 Basic concepts of Lie algebras
session 1: Lec 01 -An introduction to Lie algebras.
session 2: Lec 02 - Lie subalgebra and Homomorphism
session 3: Lec 03 - Some Problems.
session 4: Lec 04 - Ideals and Quotient algebras.
part 2: Week 2 Isomorphism Theorems and classification of low dimensional Lie algebras
session 5: Lec 05 - Isomorphism theorems
session 6: Lec 06 - Correspondence between ideals
session 7: Lec 07 - Low dimensional Lie algebra-1
session 8: Lec 08 - Low dimensional Lie algebra-2.
part 3: Week 3 Solvable and Nilpotent Lie algebras
session 9: Lec 09 - Some more definitions.
session 10: Lec 10 - Solvable and nilpotent Lie algebra.
session 11: Lec 11 Nilpotent Lie algebra
session 12: Lec 12 The invariance Lemma
part 4: Week 4 Engel’s Theorem, Lie’s Theorem and some Representatation Theory
session 13: Lec 13 Engel’s and Lie’s theorem
session 14: Lec 14 Engel’s and Lie’s theorem (continued)
session 15: Lec 15 Lie’s theorem
session 16: Lec 16 Basics of Representation theory.
session 17: Lec 17 Basics of Representation theory (continued).
part 5: Week 5 sl2(C)-Representation Theory.
session 18: Lec 18 Schur’s lemma
session 19: Lec 19 Finite dimensional representations of of sl2(C).
session 20: Lec 20 Classification of finite dimensional representations of of sl2(C).
session 21: Lec 21 Complete reducibility of finite dimensional representation of sl2(C), part 1.
part 6: Week 6 Complete reducibility for sl2(C) and some applications of Lie’s and Engel’s Theorem.
session 22: Lec 22 Complete reducibility of finite dimensional representation of sl2(C), part 2.
session 23: Lec 23 Applications of Lie’s and Engel’s theorem.
session 24: Lec 24 Applications of Weyl’s Theorem for sl2(C).
part 7: Week 7 Combinatorics of representations of sl2(C), Jordan-Chevalley decomposition and Cartan’s criteion.
session 25: Lec 25 New representations form given representations.
session 26: Lec 26 Primary decomposition Theorem and Jordan-Chevalley decomposition.
session 27: Lec 27 Cartan’s criteria for solvability.
part 8: Week 8 Cartan’s criteria, abstract Jordan decomposition and the complete reducibility of representations.
session 28: Lec 28 Cartan’s criteria for semisimplicity and its consequences
session 29: Lec 29 Abstract Jordan decomposition in semisimple Lie algebras.
session 30: Lec 30 Casimir element of a representation of a semisimple Lie algebra.
session 31: Lec 31 Weyl’s Theorem of complete reducibility for semisimple Lie algebras.
part 9: Week 9 Semisimple Lie algebras and root space decomposition.
session 32: Lec 32 Root space decomposition of semisimple Lie algebras.
session 33: Lec 33 Centralizer of a maximal toral subalgebra.
session 34: Lec 34 Properties of roots.
session 35: Lec 35 More properties of roots.
part 10: Week 10 Abstract root systems and the classification of abstract root systems of low rank.
session 36: Lec 36 Rationality of roots.
session 37: Lec 37 Abstract root system and Weyl groups
session 38: Lec 38 Isomorphism of Root systems and dual root systems
session 39: Lec 39 Root systems of Ranks 1 and 2
session 40: Lec 40 Classification of rank 2 root systems
part 11: Week 11 Weyl groups, length function, Cartan matrices and Dynkin diagrams.
session 41: Lec 41 Base of a root system
session 42: Lec 42 Classification of bases
session 43: Lec 43 Basic properties of simple roots
session 44: Lec 44 Characterization of length function
session 45: Lec 45 Decomposition of root systems
session 46: Lec 46 Root lengths, Cartan Matrices
session 47: Lec 47 Cartan matrices and Dynkin diagrams
part 12: Week 12 Isomorphism Theorems and Serre presentation.
session 48: Lec 48 Classification of Root systems
session 49: Lec 49 Classification of Root systems - Continued
session 50: Lec 50 Concrete description of root systems
session 51: Lec 51 Uniqueness of root systems
session 52: Lec 52 Isomorphism theorem
session 53: Lec 53 Isomorphism theorem - continued
session 54: Lec 54 Generators and relations
session 55: Lec 55 Serre presentation
part 13: Live Session
session 56: Live Session 14-03-2024
session 57: Live Session 17-03-2025
session 58: Live Session 15-04-2025

Catalogue records

nptel: 111108414 (manifestation)
nptel_video: 111108414 (manifestation)