AIS Lie Algebras (2011)

Venue: Chennai Mathematical Institute (CMI) and Institute of Mathematical Sciences (IMSc)
Dates: 4 Jul - 23 Jul, 2011


Convener(s) Speakers, Syllabus and Time table Applicants/Participants


School Convener(s)

Name U. Kulkarni K. N. Raghavan S. Viswanath
Mailing Address Chennai Mathematical Institute
Plot H1, SIPCOT IT Park
Siruseri, Padur PO
Chennai 603 103
Institute of Mathematical Sciences
CIT Campus
Taramani, Chennai 600 113
Institute of Mathematical Sciences
CIT Campus
Taramani, Chennai 600 113


Speakers and Syllabus 

Resource persons

  1. Punita Batra (HRI, Allahabad)
  2. Shripad Garge (IIT, Bombay)
  3. Senthamarai Kannan (CMI, Chennai)
  4. Upendra Kulkarni (CMI, Chennai)
  5. Amit Kulshrestha (IISER, Mohali)*
  6. K. N. Raghavan (IMSc, Chennai)
  7. Ravindra P. Shukla (Allahabad University)
  8. Anupam Kumar Singh (IISER, Pune)
  9. K. V. Subrahmanyam (CMI, Chennai)
  10. S. Viswanath (IMSc, Chennai)

* means participation not yet confirmed


  • I (2 lectures; KNR) Lie algebras and homomorphisms, ideals; representations; sl2C and its representations.
  • II (3 lectures but only 2 tutorials; UK) linear Lie algebras; derivations; automorphisms. Nilpotent and solvable Lie algebras; Engel's theorem; Lie's theorem. Cartan's criterion for solvability; semisimplicity; Killing form; Cartan's criterion for semisimplicity. (Proofs to be omitted; statements to be emphasized.)
  • III (no lecture but 1 tutorial; SV) Basic multilinear algebra.
  • IV (1 lecture; KNR) Complete reducibility of representations; Casimir element; Weyl's theorem; preservation of Jordan-Chevalley decomposition. (Proofs to be omitted; statements to be emphasized.)
  • V (4 lectures; SV) Root systems: axiomatics, simple roots, Weyl group, bases, irreducibility; Classification: Coxeter graphs, Dynkin diagrams; Construction of root systems.
  • VI (2 lectures; UK) Root space decomposition; maximal toral subalgebras; orthogonality, integrality, and rationality properties.
  • VII (1 lecture; AKS) Universal enveloping algebras and the PBW theorem; free Lie algebras.
  • VIII (2 lectures; AKS) Weights and maximal vectors; standard cyclic modules; finite dimensional modules: weight strings, weight diagrams.
  • IX (3 lectures; PB) Central extension of loop algebras; affine algebras; root system; Weyl group; normalized invariant form.
  • X (3 lectures; SK) Character and multiplicity formulas: Freudenthal's formula, formulas of Weyl, Kostant, and Steinberg. Harish-Chandra homomorphism.
  • XI (1 lecture: KNR) Isomorphism theorem: root system determines the Lie algebra; Cartan subalgebras and their conjugacy; Borel subalgebras and their conjugacy; Automorphism groups. (Proofs to be omitted; statements to be emphasized.)
  • XII-A (2 lectures; PB) Highest weight modules over affine Kac-Moody algebras.
  • XII-B (2 lectures; KNR) Highest weight modules over affine Kac-Moody algebras (continued); Kac-Weyl character formula.
  • XIII (5 lectures but only 4 tutorials; SV) Weight system of integrable highest weight modules; Macdonald's eta function identities.
  • XIV (4 lectures but only 3 tutorials; RPS) Chevalley groups: Chevalley basis, construction of the groups; Kostant's theorem; Admissible lattices.
  • XV (1 lecture; UK) Generators and relations for complex semisimple Lie algebras; Serre's theorem.
  • UK: Upendra Kulkarni
  • KNR: K. N. Raghavan
  • SV: S. Viswanath
  • SK: Senthamarai Kannan
  • AKS: Anupam Kumar Singh
  • PB: Punita Batra
  • RPS: Ravindra Shukla

Selected Applicants

List of selected candidates

  1. Abhishek Dang
  2. Akash Sengupta
  3. Amita Malik
  4. Amiya Mondal
  5. Anand Deshpande
  6. Ananth Shankar
  7. Anil Kumar Pranesachar
  8. Anup Dixit
  9. Anuradha Ahuja
  10. Apurv Nakade
  11. Arpan Dutta
  12. Balchand Prajapati
  13. Biswajit Ransingh
  14. Debika Dey
  15. Dinesh Valluri
  16. Divyum Sharma
  17. Gayathri V
  18. Geetha Thangavelu
  19. Kali Pada Samanta
  20. Kamalakshya Mahatab
  21. Krishnaveni Veni
  22. Mahendra Kumar Verma
  23. Mamta Balodi
  24. Muniasamy G
  25. Narasimhachary B
  26. Neha Hooda
  27. Neha Makhijani
  28. Padmavathy Srinivasan
  29. Pampa Paul
  30. Rahul Gupta
  31. Rahul Kitture
  32. Rahul Singh (CMI, Chennai)
  33. Ravinder B
  34. Rohini Ramadas
  35. Sachin Sharma
  36. Sampath Lonka
  37. Senthil Raani
  38. Siddharth Mathur
  39. Sudipta Kolay
  40. Sunil Prajapati
  41. Suprajo Das
  42. Venkata Raghu Tej Pantnagi
  43. Venkatesh R
  44. Vipul Kakkar
  45. Yajnaseni Dutta


  1. Students of IMSc and Ph.D students of CMI who have been selected
    will not be provided accommodation as part of the AIS. They must
    retain their current rooms.
  2. All other participants will be provided shared housing on the CMI campus.
  3. As per ATM school policy, all participants are eligible for travel
    reimbursement, upto a maximum of two-way AC 3 tier train fare.
    A copy of the tickets must be submitted. State transport
    AC bus fare can be claimed in lieu of train fare. Auto/taxi fare is
    not reimbursable whether in Chennai or in the city of origin/destination.
    (It's OK to be claiming reimbursement for travel from city A to Chennai
    and from Chennai to another city B, provided there is sound justification.)
  4. Selected applicants must confirm their participation by sending
    email to by May 25, 2011 with "confirmation" in the subject
  5. All online applications received by May 15, 2011 have been considered,
    even if the hardcopy of the application has been delayed.
  6. We received a large number of enthusiastic applications for participation
    in the school. Unfortunately, due to various logistical constraints such as
    availability of suitable accommodation, we could not accept all of them.
    It seems to us that, for many B.Sc., M.Sc, and early Ph.D. applicants,
    the Annual Foundational Schools (AFS) would be more beneficial than a school such as this, at this stage of their
    careers. We strongly recommend that they apply to these. Other workshops and
    training programmes such as the MT&TS

    might also be of interest.

Link for how to reach the venue