Course11110841458 lectures
Introduction to Lie Algebras
Prof. R Venkatesh · IISc Bangalore
Lectures
Week 1 Basic concepts of Lie algebras
- Lec 01 -An introduction to Lie algebras.
- Lec 02 - Lie subalgebra and Homomorphism
- Lec 03 - Some Problems.
- Lec 04 - Ideals and Quotient algebras.
Week 2 Isomorphism Theorems and classification of low dimensional Lie algebras
- Lec 05 - Isomorphism theorems
- Lec 06 - Correspondence between ideals
- Lec 07 - Low dimensional Lie algebra-1
- Lec 08 - Low dimensional Lie algebra-2.
Week 3 Solvable and Nilpotent Lie algebras
- Lec 09 - Some more definitions.
- Lec 10 - Solvable and nilpotent Lie algebra.
- Lec 11 Nilpotent Lie algebra
- Lec 12 The invariance Lemma
Week 4 Engel’s Theorem, Lie’s Theorem and some Representatation Theory
- Lec 13 Engel’s and Lie’s theorem
- Lec 14 Engel’s and Lie’s theorem (continued)
- Lec 15 Lie’s theorem
- Lec 16 Basics of Representation theory.
- Lec 17 Basics of Representation theory (continued).
Week 5 sl2(C)-Representation Theory.
- Lec 18 Schur’s lemma
- Lec 19 Finite dimensional representations of of sl2(C).
- Lec 20 Classification of finite dimensional representations of of sl2(C).
- Lec 21 Complete reducibility of finite dimensional representation of sl2(C), part 1.
Week 6 Complete reducibility for sl2(C) and some applications of Lie’s and Engel’s Theorem.
- Lec 22 Complete reducibility of finite dimensional representation of sl2(C), part 2.
- Lec 23 Applications of Lie’s and Engel’s theorem.
- Lec 24 Applications of Weyl’s Theorem for sl2(C).
Week 7 Combinatorics of representations of sl2(C), Jordan-Chevalley decomposition and Cartan’s criteion.
- Lec 25 New representations form given representations.
- Lec 26 Primary decomposition Theorem and Jordan-Chevalley decomposition.
- Lec 27 Cartan’s criteria for solvability.
Week 8 Cartan’s criteria, abstract Jordan decomposition and the complete reducibility of representations.
- Lec 28 Cartan’s criteria for semisimplicity and its consequences
- Lec 29 Abstract Jordan decomposition in semisimple Lie algebras.
- Lec 30 Casimir element of a representation of a semisimple Lie algebra.
- Lec 31 Weyl’s Theorem of complete reducibility for semisimple Lie algebras.
Week 9 Semisimple Lie algebras and root space decomposition.
- Lec 32 Root space decomposition of semisimple Lie algebras.
- Lec 33 Centralizer of a maximal toral subalgebra.
- Lec 34 Properties of roots.
- Lec 35 More properties of roots.
Week 10 Abstract root systems and the classification of abstract root systems of low rank.
- Lec 36 Rationality of roots.
- Lec 37 Abstract root system and Weyl groups
- Lec 38 Isomorphism of Root systems and dual root systems
- Lec 39 Root systems of Ranks 1 and 2
- Lec 40 Classification of rank 2 root systems
Week 11 Weyl groups, length function, Cartan matrices and Dynkin diagrams.
- Lec 41 Base of a root system
- Lec 42 Classification of bases
- Lec 43 Basic properties of simple roots
- Lec 44 Characterization of length function
- Lec 45 Decomposition of root systems
- Lec 46 Root lengths, Cartan Matrices
- Lec 47 Cartan matrices and Dynkin diagrams
Week 12 Isomorphism Theorems and Serre presentation.
- Lec 48 Classification of Root systems
- Lec 49 Classification of Root systems - Continued
- Lec 50 Concrete description of root systems
- Lec 51 Uniqueness of root systems
- Lec 52 Isomorphism theorem
- Lec 53 Isomorphism theorem - continued
- Lec 54 Generators and relations
- Lec 55 Serre presentation
Live Session
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IISc Bangalore
- Department
- Mathematics
- Teacher
- Prof. R Venkatesh
- Level
- Undergraduate
- Language
- English
- Subjects
- Mathematics
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
- NPTEL gives no rating. Invisible Network shows it to Tweens profiles and older, judged from what the source says about it. How courses are placed
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