Advanced Instructional School (sponsored by NBHM) on Lie Algebras
during 04--23 July 2011 at Chennai Mathematical Institute (CMI) and Institute of Mathematical Sciences (IMSc)

This is only one of many ATM Schools in 2011 funded by the National Board for Higher Mathematics (NBHM).

Tutorial sheets

General description: The theory of structure and representations of complex semisimple Lie algebras was developed towards the end of the nineteenth century, and so is over a hundred years old. It is an important basic subject in mathematics, a sound knowledge of it being a must for research in many diverse areas. This AIS aims to develop such basics of the theory as the classification of the algebras by means of root systems, the structure of an algebra in terms of a Cartan subalgebra and root spaces, complete reducibility of representations, parametrization of the irreducible representations by means of highest weights, and well known character formulas for representations. In addition, we will introduce two important twentieth century off-shoots of the theory:
(1) Chevalley groups and their basic properties;
(2) affine Kac-Moody Lie algebras and their representations, up to the Kac-Weyl character formula and the proof of the Macdonald identities.

Reference material: There are many texts on the subject, several of which may be termed classics (e.g., those by Bourbaki and Serre). While we would ideally not like to be tied to any one particular source, keeping in mind that the school is likely to be the first exposure to the subject for many participants, we will for the most part be following the standard text Introduction to Lie algebras and Representation Theory by Humphreys, which fortunately is now available in an affordable Indian paperback edition.

Conveners: Upendra Kulkarni (CMI), K. N. Raghavan (IMSc), S. Viswanath (IMSc).

Resource persons include: Punita Batra (HRI), Anuradha Garge (CEBS, Mumbai), Shripad Garge (IIT, Bombay), Senthamarai Kannan (CMI), Upendra Kulkarni (CMI), Brajesh Mishra (Allahabad University), K. N. Raghavan (IMSc), Ravindra P. Shukla (Allahabad University), Anupam Kumar Singh (IISER, Pune), K. V. Subrahmanyam (CMI), S. Viswanath (IMSc).

Pre-requisites: A basic knowledge of mathematics as taught in the foundational schools (particularly of algebra) will be assumed but pretty much nothing more. However, in order that substantial material is covered in the school, we will only state and not prove many basic statements (particularly relating to the structure and classification of complex semisimple Lie algebras; see the syllabus for details). Participants would do well therefore to come a little prepared. They are in particular advised to
(a) acquire their own personal copies of Humphreys's textbook mentioned above;
(b) become familiar with the contents of the book (especailly with those statements whose proofs are going to be skipped); and
(c) carry their personal copies to the school.

Syllabus (pdf file).

Unity of Mathematics lectures: In addition to the lectures on the syllabus proper, there will be two one hour lectures on relations to physics ("The uses of Lie groups and Lie algebras in physics") by N. Mukunda, on Tuesday 12th July and Wednesday 13th July, and one ninety minute lecture on relations to computational complexity ("Positivity and plethysms in geometric complexity theory") by Ketan Mulmuley, on Friday 22nd July. Click here for a pdf file of scanned hand-written notes for Mukunda's lectures, and here for the abstract of Mulmuley's lecture. Further details of these lectures will appear here as they become available.

Timetable (pdf file) (indicative but obsolete). Timetable for the first week (pdf file) (more precise than the indicative).

List of participants.

Advice for participants.