Lie Groups and Lie Algebras (2023)

Convener(s)

 
Name: Apoorva Khare R. Venkatesh
Mailing Address: Associate Professor
Department of Mathematics
Indian Institute of Science
Bangalore 560012		 Assistant Professor
Department of Mathematics
Indian Institute of Science
Bangalore 560012
Email: khare at iisc.ac.in rvenkat at iisc.ac.in

Lie groups and Lie algebras are one of the cornerstones of mathematics. They are well-studied in geometry, (harmonic) analysis, representation theory, number theory, string theory, and particle physics, to name several different fields – in addition to featuring in probability theory, control systems, and other areas. This AIS will focus on discussing and developing the basics of topological groups, Lie groups, and Lie algebras, with both their structure and their representations in mind.

The lectures aim to cover the basics of the structure theory of (complex semisimple) Lie algebras, root systems and Dynkin diagrams, topological and Lie groups, the Peter-Weyl theorem, and the correspondence between Lie algebras and Lie groups (via exponentiation). The lectures are meant for an audience of students and postdocs interested in Lie theory. Each lecture (in the morning) is accompanied by a tutorial session (in the afternoon).

Dates: 

Monday, June 19, 2023 - 09:00 to Saturday, July 8, 2023 - 18:00

Venue: 

Venue Address: 

Lecture Hall - I
Department of Mathematics
Indian Institute of Science
Bengaluru 560012

Venue State: 

Venue City: 

PIN: 

560012

Syllabus: 

 

Name of the Speakers with their affiliation.

No. of Lectures

Detailed Syllabus

Ved Datar
Dept. of Maths.
IISc, Bangalore

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
Dept. of Maths.
IISc, Bangalore

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
Dept. of Maths.
IISc, Bangalore

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
Dept. of Maths.
IISc, Bangalore

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
Dept. of Maths.
IISc, Bangalore

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
Dept. of Maths.
HRI, Prayagraj

 6 Compact Lie groups, Lie algebras, representations, weights of representations, Weyl group, highest weight theorem for compact semisimple Lie groups.

References:

  1. V. S. Varadarajan, Lie groups, Lie algebras and their representations, Sringer 1984.
  2. Brian Hall, Lie groups, Lie algebras and representations, Springer 2003.
  3. Barry Simon, Representations of finite and compact groups, AMS 1996.
  4. A. W. Knapp, Representation theory of semismiple Lie groups. An overview based on examples, Princeton university press 2002.
  5. S. Kumaresan S, A course in differential geometry and Lie groups, Texts and Readings in Mathematics, 22. Hindustan Book Agency, New Delhi, 2002.
  6. F. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983.
  7. J. E. Humphreys, Introduction to Lie algebras and Representation theory, Springer-Verlag, 1972.
  8. J.-P. Serre, Complex Semisimple Lie Algebras, Springer, 2001.
  9. W. Fulton, and J. Harris, Representation theory, Springer-Verlag. 1991.
  10. 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
to
11.00

Tea

11.00
to
11.30

Lec 2

11.30
to
1.00

Lunch

1.00
to
2.30

Tut 1

2.30
to
3.30

Tea

3.30
to
4.00

Tut 2

4.00
to
5.00

Snacks

5.00
to
5.30

 

 

(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

Selected Applicants: 

 

Sr.n SID Full Name Gender Affiliation Position in College/ University University /Institute M.Sc./ M.A. Year of Passing M.Sc./ M.A Ph.D. Degree Date
1 47398 Ms. Manika Gupta Female Institute of Mathematical Sciences PhD Student Kerala School of Mathematics 2022  
2 47426 Mr. Aniruddha Vishwas Deshmukh Male Indian Institute of Technology Indore PhD Student Sardar Vallabhbhai National Institute of Technology Surat 2020  
3 47459 Mr. S Viswanathan Male International Centre for Theoretical Sciences PhD Kerala School of Mathematics 2022  
4 47478 Mr. Sougata Mandal Male IIT Jodhpur Phd Scholar Vidyasagar University 2018  
5 47502 Mr. Kaustabh Mondal Male INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH Integrated Ph.D Student IISER Pune 2022  
6 47758 Ms. Prachi Saini Female IISER Pune PhD IIT Hyderabad 2021  
7 47780 Ms Sahanawaj Sabnam Female NISER BHUBANESWAR PhD JADAVPUR UNIVERSITY 2019  
8 47860 Mr. Tejbir Lohan Male Indian Institute of Science Education and Research Mohali PhD Student IIT Kanpur 2018  
9 48188 Mr. Paritosh Ghosh Male Jadavpur University PhD Jadavpur University 2018  
10 48198 Dr. Kiran Meena Female Harish-Chandra Research Institute, Prayagraj PDF University of Rajasthan, Jaipur 2018 25/02/2023
11 48201 Mr. Sourav Ghosh Male Indian Institute of Science (IISc) Integrated PhD Student      
12 48242 Mr Dibyendu Das Male Indian Statistical Institute PhD Indian Statistical Institute Bangalore 2021  
13 48286 Mr. Sayan Pal Male Indian Statistical Institute PhD IIT Kharagpur 2021  
14 48323 Mr. Mohammad Aqib Male Harish-Chandra Research Institute SRF IITGN 2019  
15 48357 Mr Muthuraj T Male Indira Gandhi Center for Atomic Research Senior Research Fellow - PhD Bharathiar university 2018  
16 48210 Dr. Choiti Bandyopadhyay Female Harish-Chandra Research Institute Post-doctoral Fellow Indian Institute of Science Bangalore, India 2012 07/06/2018
17 48260 Mr. Praful Rahangdale Male Indian Institute of Science Education and Research (IISER) Kolkata 5-Year BS-MS Student Indian Institute of Science Education and Research Kolkata Appeared / Awaiting Result  
18 48265 Mr. Sumanta Das Male UNIVERSITY OF CALCUTTA PhD student The University of Burdwan 2015  
19 48268 Ms. Pooja Rani Female IIT BHU PhD M.Sc 2018  
20 48434 Mr. Arkadeepta Roy Male Harish-Chandra Research Institute PhD Student Ramakrishna Mission Vidyamandira 2019  
21 48444 Mr. Pradeep Bisht Male Harish-Chandra Research Institute PhD University of Delhi 2019  
22 48478 Ms Aswathy Byju Female Kerala School of Mathematics PhD student, First year Kerala School of Mathematics 2022  
23 48642 Mr Somnath Ghosh Male TIFR - Centre for Applicable Mathematics Postdoctoral fellow IIT Guwahati 2015 30/06/2021
24 48693 Mr. Himanshu Bimal Male ICTS-TIFR, Bengaluru Long term visiting student NISER 2022  
25 48698 Ms Manali Sajjan Female IIT Guwahati PhD University of Calcutta 2019  
26 48833 Ms. Shubhanshi Gupta Female Institute of Infrastructure, Technology, Research And Management PhD Student The Maharaja Sayajirao University of Baroda, Vadodara 2019  
27 48899 Ms. Harmandeep Kaur Female Central University of Punjab PhD (Research Scholar) Panjab University 2019  
28 48920 Ms. Gargi Biswas Female NISER, Bhubaneswar M.Sc. student NISER, Bhubaneswar Appeared / Awaiting Result  
29 49396 Mr Yogesh Kumar  Prajapaty Male Indian Institute of Science Education and Research Thiruvananthapuram PhD Indian Institute of Science Education and Research Thiruvananthapuram 2018  
30 49420 Mr. Supriyo Jana Male IIT Madras Research Scholar IIT Madras 2021  
31 49508 Mr Mohith Raju Male Indian Institute of Science BSc - MSc Indian Institute of Science    
32 49566 Mr. Sourav Das Male INDIAN INSTITUTE OF TECHNOLOGY KANPUR PhD INDIAN INSTITUTE OF TECHNOLOGY HYDERABAD 2019  
33 49586 Mr. Rahul Kaushik Male IISER Pune Post Doc MSc Mathematics, University of Delhi 2015  
34 49679 Mr Sundara Narasimhan Male Tata Institute of Fundamental Research PhD student Indian Institute of Science Education and Research Mohali 2022  
35 49797 Mr. Ritesh Kumar Pandey Male IIT Kanpur PhD Banaras Hindu University 2019  
36 49808 Ms. Archita Gupta Female Indian Institute of Technology, Kanpur PhD Student Indian Institute of Technology, Kharagpur 2020  
37 49897 Mr. Monideep Ghosh Male TIFR- Centre For applicable Mathematics Research Scholar TIFR    
38 49944 Mr. Anant Narayanan Male Indian Institute of Science Integrated PhD Student      
39 49976 Ms. Chaithra P Female Indian Institute of Science, Bangalore PhD Student St.Joseph's college (Autonomous) Devagiri, Calicut 2020  
40 50015 Mr Samiran Nayek Male IISER MOHALI PHD IIT TIRUPATI    
 41   Rupkatha Chand   IISER Mohali        

 

 

How to Reach: 

Click the following link for further details

https://iisc.ac.in/about/general-information/how-to-reach-iisc/

School Short Name: 

lgla

Last Date Application: 

Sunday, May 21, 2023

List of Conveners: 

1) Apoorva Khare, Associate Professor
2) R. Venkatesh, Assistant Professor

School Type: 

AIS

Separate faculty form: 

0