Lie Algebra - MTH430
August-November 2017
Instructor : Dr. Anupam Singh
Audience : UG, iPhD, and PhD
Schedule : Thursday 2-3 and Friday 10:50-11:50. Optional tutorial will be run by Dr. Uday Sharma and Dr. Dilpreet Kaur.
Evaluation : Test I - 20%
Mid-sem examination- 30%
Test II- 20%
End-sem examination- 30%
Prerequisite : Linear Algebra
Goal of the course : To understand Lie algebras and classification of simple Lie algebras---- This subject has wide application in Physics (for example, standard model SU(2)), Chemistry (for example, Crystallography), and forms backbone to the topics such as Number Theory, Algebraic Groups, Lie Groups, Differential Geometry, Dynamical Systems, Kac-Moody Lie algebras, Quantum Groups etc.
Proposed content : Lie algebra, solvable and nilpotent Lie algebras, simple and semisimple Lie algebras, Root systems and classification of simple Lie algebras.
Courses taught related to this topic in past: Linear Algebraic Groups (in 2015) and Classical Groups (in 2007)
References :
Humphreys : Introduction to Lie algebra and representation theory
Erdmann and Wildon : Introduction to Lie algebras
Carter : Lie algebras of finite and affine type
3 August 2017 Definition and example of gl(n)
4 August 2017 Examples gl(n), sl(n), t(n), n(n) etc
Recommended reading - Missed Opportunities by Freeman Dyson
10 August 2017 Story of Classification of Finite Simple Groups, example of classical Lie algebras, A_l, B_l, C_l, D_l
11 August 2017 Definitions: Ideals, homomorphism, Solvable Lie algebra, Simple Lie algebra etc.
17 August 2017 Solvable Lie algebra
18 August 2017 Tutorial
24 August 2017 Nilpotent Lie algebra, ad-nilpotent
25 August 2017 Holiday
31 August 2017 Engel's Theorem -Characterization of nilpotent Lie algebra
01 September 2017 Lie's theorem - Characterization of solvable Lie algebra
Assignment II - Test your linear algebra
02 September 2017 Test - I
07 September 2017 - Cartan's criteria, Killing form
08 September 2017 - classification of Semisimple Lie algebras vs. Killing form
14 September 2017 Semisimple Lie algebra as direct sum of simple Lie algebras
15 September 2017 Representations of sl(2)
20-27 September - Mid sem exam
5 October 2017 More on Representations of sl(2,C)
6 October 2017 Cartan decomposition of a semisimple Lie algebra
7 October 2017 (extra class - 2 hours) Associating root system to a semisimple Lie algebra
12 October 2017 Associating a root system to a semisimple Lie algebra
13 October 2017 Associating a root system to a semisimple Lie algebra
19 October 2017 Holiday
20 October 2017 No class
26 October 2017 Tutorial - root systems of classical Lie algebras
27 October 2017 Tutorial - roots systems of classical Lie algebras
28 October 2017 Test - II
2 November 2017 Root system, classification of 2-dimensional root systems
3 November 2017 Classical root system, demo of rank 2 and 3 root systems
see Brian Hall - Instructions for building roots systems in Zometool: kit.pdf
9 November 2017 Weyl group of a root system, Base and positive roots, Coxeter graphs
10 November 2017 Dynkin diagrams, Classification of root systems
16 November 2017 Tutorial - Write down all classical roots systems, simple root system and compute Dynkin diagram and Weyl groups
17 November 2017 Tutorial
30 November 2017 End Semester Examination
What next?
Chevalley groups (Simple groups of Lie type : Carter)
Representation theory of semisimple Lie algebras, highest weight theory, Verma modules etc (Continue with Humphreys book)
Lie groups, compact Lie groups (Warner or Adams : Lie groups)
Algebraic groups (Humphreys)