Learn · Introduction to Lie Algebras
Lecture 38 of 58Week 10 Abstract root systems and the classification of abstract root systems of low rank.
Lec 38 Isomorphism of Root systems and dual root systems
Prof. R Venkatesh · IISc Bangalore
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- Introduction to Lie Algebras, IISc Bangalore (at NPTEL)
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All the lectures
- Lec 01 -An introduction to Lie algebras.
- Lec 02 - Lie subalgebra and Homomorphism
- Lec 03 - Some Problems.
- Lec 04 - Ideals and Quotient algebras.
- Lec 05 - Isomorphism theorems
- Lec 06 - Correspondence between ideals
- Lec 07 - Low dimensional Lie algebra-1
- Lec 08 - Low dimensional Lie algebra-2.
- Lec 09 - Some more definitions.
- Lec 10 - Solvable and nilpotent Lie algebra.
- Lec 11 Nilpotent Lie algebra
- Lec 12 The invariance Lemma
- 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).
- 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.
- 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).
- Lec 25 New representations form given representations.
- Lec 26 Primary decomposition Theorem and Jordan-Chevalley decomposition.
- Lec 27 Cartan’s criteria for solvability.
- 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.
- 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.
- 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
- 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
- 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
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