AIS - Lie Groups and Lie Algebras (2023)
Speakers and Syllabus
|
Name of the Speakers with their affiliation. |
No. of Lectures |
Detailed Syllabus |
|
Ved Datar |
6 |
Review of multivariable calculus including the inverse function theorem, smooth manifolds with examples, smooth functions, partition of unity, tangent spaces and the tangent bundle, vector fields and flows. |
|
Vamsi Pritham Pingali |
6 |
Lie groups with examples (and motivation), left invariant vector fields and the Lie bracket, one parameter subgroups and the exponential map, Baker-Campbell-Hausdorff formula, Closed Subgroup Theorem. |
|
Subhojoy Gupta |
6 |
Frobenius theorem and its proof, with commuting flows as a special case; correspondence between Lie subgroups and Lie subalgebras, correspondence between simply-connected Lie groups and finite-dimensional Lie algebras. |
|
Apoorva Khare |
6 |
Structure of semisimple Lie algebras: theorems of Engel and Lie (for nilpotent and solvable Lie algebras), Cartan’s criterion and the Killing form, Weyl’s theorem of complete reducibility, sl(2)-theory, toral subalgebras, root spaces. |
|
R. Venkatesh |
6 |
Root systems, simple bases, classification via Dynkin diagrams, root and weight lattices, isomorphism and existence theorems for semisimple Lie algebras. Theorems of Ado and Levi. |
|
Muna Naik |
6 | Compact Lie groups, Lie algebras, representations, weights of representations, Weyl group, highest weight theorem for compact semisimple Lie groups. |
References:
- V. S. Varadarajan, Lie groups, Lie algebras and their representations, Sringer 1984.
- Brian Hall, Lie groups, Lie algebras and representations, Springer 2003.
- Barry Simon, Representations of finite and compact groups, AMS 1996.
- A. W. Knapp, Representation theory of semismiple Lie groups. An overview based on examples, Princeton university press 2002.
- S. Kumaresan S, A course in differential geometry and Lie groups, Texts and Readings in Mathematics, 22. Hindustan Book Agency, New Delhi, 2002.
- F. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983.
- J. E. Humphreys, Introduction to Lie algebras and Representation theory, Springer-Verlag, 1972.
- J.-P. Serre, Complex Semisimple Lie Algebras, Springer, 2001.
- W. Fulton, and J. Harris, Representation theory, Springer-Verlag. 1991.
- Roger Carter, Lie Algebras of Finite and Affine Type, Cambridge University Press, 2010.
Time Table
Time-Table
(with initials of speakers and course associates for entire days of the programme)
|
Day |
Date |
Lec 1 9.30 |
Tea 11.00 |
Lec 2 11.30 |
Lunch 1.00 |
Tut 1 2.30 |
Tea 3.30 |
Tut 2 4.00 |
Snacks 5.00 |
|
|
|
|
(The name of the speaker) |
|
(The name of the speaker) |
|
(The name of the speaker/tutor) |
|
(The name of the speaker/tutor) |
|
|
|
Mon |
19/06/ 2023 |
VD |
Tea |
AK |
Lunch |
VD+IH+SK |
Tea |
RV+IH+SK |
Snacks |
|
|
Tues |
20/06/ 2023 |
VD |
Tea |
AK |
Lunch |
VD+IH+SK |
Tea |
RV+IH+SK |
Snacks |
|
|
Wed |
21/06/ 2023 |
VD |
Tea |
AK |
Lunch |
VD+IH+SK |
Tea |
RV+IH+SK |
Snacks |
|
|
Thu |
22/06/ 2023 |
VD |
Tea |
AK |
Lunch |
VD+IH+SK |
Tea |
RV+IH+SK |
Snacks |
|
|
Fri |
23/06/ 2023 |
VD |
Tea |
AK |
Lunch |
VD+IH+SK |
Tea |
RV+IH+SK |
Snacks |
|
|
Sat |
24/06/ 2023 |
VD |
Tea |
AK |
Lunch |
VD+IH+SK |
Tea |
RV+IH+SK |
Snacks |
|
|
SUNDAY : OFF |
||||||||||
|
Mon |
26/06/ 2023 |
VP |
Tea |
RV |
Lunch |
VP+IH+SK |
Tea |
RV+IH+SK |
Snacks |
|
|
Tues |
27/06/ 2023 |
VP |
Tea |
RV |
Lunch |
VP+IH+SK |
Tea |
RV+IH+SK |
Snacks |
|
|
Wed |
28/06/ 2023 |
VP |
Tea |
RV |
Lunch |
VP+IH+SK |
Tea |
RV+IH+SK |
Snacks |
|
|
Thu |
29/06/ 2023 |
VP |
Tea |
RV |
Lunch |
VP+IH+SK |
Tea |
RV+IH+SK |
Snacks |
|
|
Fri |
30/06/ 2023 |
VP |
Tea |
RV |
Lunch |
VP+IH+SK |
Tea |
RV+IH+SK |
Snacks |
|
|
Sat |
01/07/ 2023 |
VP |
Tea |
RV |
Lunch |
VP+IH+SK |
Tea |
RV+IH+SK |
Snacks |
|
|
SUNDAY : OFF |
||||||||||
|
Mon |
03/07/ 2023 |
SG |
Tea |
MN |
Lunch |
SG+IH+SK |
Tea |
MN+IH+SK |
Snacks |
|
|
Tues |
04/07/ 2023 |
SG |
Tea |
MN |
Lunch |
SG+IH+SK |
Tea |
MN+IH+SK |
Snacks |
|
|
Wed |
05/07/ 2023 |
SG |
Tea |
MN |
Lunch |
SG+IH+SK |
Tea |
MN+IH+SK |
Snacks |
|
|
Thu |
06/07/ 2023 |
SG |
Tea |
MN |
Lunch |
SG+IH+SK |
Tea |
MN+IH+SK |
Snacks |
|
|
Fri |
07/07/ 2023 |
SG |
Tea |
MN |
Lunch |
SG+IH+SK |
Tea |
MN+IH+SK |
Snacks |
|
|
Sat |
08/07/ 2023 |
SG |
Tea |
MN |
Lunch |
SG+IH+SK |
Tea |
MN+IH+SK |
Snacks |
|
SG: Subhojoy Gupta, VP: Vamsi Pritham Pingali, VD: Ved Datar, MN: Muna Naik, RV: R. Venkatesh, AK: Apoorva Khare, IH: Irfan MD Habib, SK: Sathish Kumar
List of Speaker and Tutors:
|
S. No. |
Name |
Affiliation |
|
1 |
Subhojoy Gupta (SG) |
Mathematics, IISc Bangalore |
|
2 |
Vamsi Pritham Pingali (VP) |
Mathematics, IISc Bangalore |
|
3 |
Ved Datar (VD) |
Mathematics, IISc Bangalore |
|
4 |
Muna Naik (MN) |
Mathematics, IHRI Prayagraj |
|
5 |
R. Venkatesh (RV) |
Mathematics, IISc Bangalore |
|
6 |
Apoorva Khare (AK) |
Mathematics, IISc Bangalore |
|
7 |
Irfan Md. Habib (IH) |
Mathematics, IISc Bangalore |
|
8 |
V. Sathish Kumar (SK) |
Mathematics, IMSc Chennai |