Representation Theory of Quivers and BGG Category O for semi simple Lie Algebras (2024)

Convener(s)

 
Name: Amit Kuber Sachin Sharma
Mailing Address: Associate Professor
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
Uttar Pradesh 208016		 Associate Professor
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
Uttar Pradesh 208016
Email: askuber at iitk.ac.in sachinsh at iitk.ac.in

Representation theory of quivers(=finite-dimensional associative algebras) and Lie algebras are classical yet active topics of study and research, both having diverse applications in various branches of mathematics as well as theoretical physics. The two areas are interconnected but the former is underrepresented in India. The purpose of this school is to introduce the former area to the students and researchers from the Indian community while also building up the Lie theory side up to BGG Category O where quivers appear naturally. A more fundamental connection between the two can be seen via Dynkin diagrams, which appear in the classification of hereditary finite dimensional algebras (Gabriel’s theorem) as well as semisimple Lie algebras.

In the former theme, we will cover Gabriel’s theorem and Auslander-Reiten theory in the first two weeks while a guest speaker, Thomas Brustle, will introduce more contemporary topics to the audience. The latter theme will comprise of the basics of highest weight modules of semisimple Lie algebras in the first two weeks leading up to the Bernstein-Gelfand-Gelfand  (BGG) category O. This workshop will provide a unique opportunity as well as a common platform for students and researchers interested in either area.

 

Dates: 

Monday, December 2, 2024 - 09:00 to Saturday, December 21, 2024 - 21:00

Venue: 

Venue Address: 

Amit Kuber
Associate Professor
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
Uttar Pradesh 208016

 

Venue State: 

Venue City: 

PIN: 

208016

Syllabus: 

Syllabus:

 

Name of the Speaker with affiliation

 

No. of Lectures

+
Tutorials

 

Detailed Syllabus

Shraddha Srivastava

Dept. of Maths.

IIT Dharwad

     

   4.5+3

Lie algebras (definition and notation etc), classical Lie algebras, Semisimple Lie algebras, Weyl’s theorem, Representations of sl(2, C).

 

Sachin Sharma

Dept. of Maths and Stats

IIT Kanpur

 

   4.5+3

  

Root systems, Weyl groups, Dynkin diagrams: classification of irreducible root systems.

 

Upendra Kulkarni

Dept. of Maths.

CMI, Chennai

 

   9+6

Representations of quivers, Projective and injective representations, Dynkin diagrams and Gabriel’s theorem

 

R. Venkatesh

Dept. of Maths.

IISc, Bangalore

 

   4.5+3

Universal enveloping algebras, PBW theorem, Serre’s theorem.

 

B. Ravinder

Dept. of Maths and Stats

IIT Tirupati

 

    4.5+3

Representation theory of semisimple Lie algebras, highest weight modules, Verma modules, classification of finite dimensional modules. Characters, Harish-Chandra theorem

 

 

Amit Kuber

Dept. of Maths and Stats.

IIT Kanpur

 

   9+6

Path algebras and bound quiver algebras, Auslander-Reiten Theory: Almost split sequences, Auslander-Reiten quiver, Auslannder-Reiten formulas

 

Apoorva Khare

Dept. of Maths.

IISc, Bangalore

   

   9+6

Basic properties of Category O, projective covers of simple highest weight modules, splitting into blocks, Verma/standard flags of projective modules, endomorphism algebra of projective generator, BGG reciprocity.

 

Thomas Brüstle

Dept of Maths

Sherbrook University

Canada

 

   9+6

Recent trends in quiver representations

Part 1: Cluster algebras and quiver representations

Part 2: Gentle algebras and geometric models

 

References:

  1. Schiffler, Ralf. Quiver representations. (2014) CMS Books in Mathematics, Springer.
  2. Auslander, M., Reiten, I., & Smalo, S. (1995). Representation Theory of Artin Algebras (Cambridge Studies in Advanced Mathematics). Cambridge: Cambridge University Press.
  3. Assem, I., Skowronski, A., & Simson, D. (2006). Elements of the Representation Theory of Associative Algebras: Techniques of Representation Theory (London Mathematical Society Student Texts). Cambridge: Cambridge University Press.
  4. Crawley-Boevey W., Lectures on representations of quivers. Available at https://www.math.uni-bielefeld.de/~wcrawley/quivlecs.pdf
  5. Bernhard Keller, Cluster algebras, quiver representations and triangulated categories, arXiv:0807.1960 (sections 1-6)
  6. Bernhard Keller, Cluster algebras and derived categories, arXiv:1202.4161 (sections 1-5).
  7. Sebastian Opper, Pierre-Guy Plamondon, and Sibylle Schroll, A geometric model for the derived category of gentle algebras, arXiv:1801.09659.
  8. Humphreys, J. E. (2012). Introduction to Lie algebras and representation theory (Vol. 9). Springer Science \& Business Media.
  9. Humphreys, J. E. (2008). Representations of Semisimple Lie Algebras in the BGG Category O (Vol. 94). American Mathematical Soc.
  10. Bernstein, I. N. and Gel'fand, I. M. and Gel'fand, S. I. (1976). Category of g-modules. Functional analysis and its applications, 10(2), 87—92.

List of Tutors

Tutor
Affiliation
Dr. G.V. Krishna Teja 
ISI Bengaluru
Ritesh Pandey 
IIT Kanpur
Esha Gupta
Université Paris-Saclay
Subham Jaiswal
ISI Kolkata
Riju Bindua
IIT Delhi
 

 

Time Table: 

 


Day

Date

Lecture 1

Tutorial 1

Lecture 2

Tutorial 2

Week 1

 Inauguration at 9:20 by Prof. K.N. Raghavan

(9.30–11.00)

(11.30-12.30)

(2.00-3:30)

(4.00-5.00)

Mon

02/12/24

SSr

SSr+RP

UK

UK+SJ

Tues

03/12/24

SSr

SSr+RP

UK

UK+SJ

Wed

04/12/24

SSr

SSr+RP

UK

UK+SJ

Thu

05/12/24

SSh

SSh+RP

UK

UK+SJ

Fri

06/12/24

SSh

SSh+RP

UK

UK+SJ

Sat

07/12/24

SSh

SSh+RP

UK

UK+SJ

Week 2

 

(9.30–11.00)

(11.30-12.30)

(2.00-3:30)

(4.00-5.00)

Mon

09/12/24

RV

RV

AKu

AKu+EG

Tues

10/12/24

RV

RV

AKu

AKu+EG

Wed

11/12/24

RV

RV

AKu

AKu+EG

Thu

12/12/24

BR

BR

AKu

AKu+EG

Fri

13/12/24

BR

BR

AKu

AKu+EG

Sat

14/12/24

BR

BR

AKu

AKu+EG

Week 3

 

(11:00-12:30)

(12:45-1:45)

(2:45-4:15)

(4:30-5:30)

Mon

16/12/24

AKh

AKh+GKT

TB

TB+EG+RB

Tues

17/12/24

AKh

AKh+GKT

TB

TB+EG+RB

Wed

18/12/24

AKh

AKh+GKT

TB

TB+EG+RB

Thu

19/12/24

AKh

AKh+GKT

TB

TB+EG+RB

Fri

20/12/24

AKh

AKh+GKT

TB

TB+EG+RB

Sat

21/12/24

AKh

 AKh+GKT TB TB+EG+RB

UK: Upendra Kulkarni, SSr: Shraddha Srivastava, SSh: Sachin Sharma, AKu: Amit Kuber, RV: R. Venkatesh, BR: B. Ravinder, TB: Thomas Brüstle, AKh: Apoorva Khare, RP: Ritesh Pandey, SJ: Subham Jaiswal, EG: Esha Gupta, GKT: G. Krishna Teja, RB: Riju Bindua

 

 

 

 

Selected Applicants: 

 

 

Serial 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 58745 Mr. Sathish Kumar V Male IISc Bangalore Postdoc University of Madras 2018 12/06/2024
2 58746 Mr. Soumyadip Sarkar Male IMSc Chennai PhD Student The Institute of Mathematical Sciences HBNI 2023  
3 58756 Dr. Niranjan Nehra Male M.K Govt. Polytechnic Jattal Lecturer IIT KANPUR 2017 05/06/2024
4 58757 Mr. Annoy Sengupta Male IIT Kanpur PhD IIT Kanpur 2020  
5 58769 Ms Divya Setia Female Indian Institute of Science Education and Research Mohali Int.PhD Indian Institute of Science Education and Research Mohali 2020  
6 58780 Ms. Shushma Rani Female Harish Chandra Research Institute Postdoc Punjabi University Patiala 2015 27/06/2023
7 58781 Mr. Rishabh Goswami Male North-Eastern Hill University PhD Tezpur University 2021  
8 58798 Ms Shilpa Rani Female SRM UNIVERSITY- AP PhD St. Berchmans, Changanasserry 2020  
9 58814 Ms. Amita Amita Female Indira Ghandhi National Tribal University, Amarkantak(M.P.) - 484887 Ph.D. Scholar DBRAU, AGRA 2011  
10 58830 Ms Anju Kumari Yadav Female National Institute of Technology Rourkela PhD Ravenshaw University 2018  
11 58859 Dr. Sarjick Bakshi Male UM-DAE Centre for Excellence in Basic Sciences Postdoc Chennai Mathematical Institute   01/08/2021
12 58870 Mr Dhamothiran J Male National Institute of Technology PhD Periyar University, Salem. 2017  
13 58892 Mr. Aditya Singh Rana Male INDIAN INSTITUTE OF TECHNOLOGY DHARWAD Ph.D. Central University of Punjab, Bathinda/ M.Sc. Mathematics 2021  
14 58911 Dr Swati Agarwal Female Mangalayatan University Assistant Professor St. John's College (Dr. Bhimrao Ambedkar University), Agra 2011 11/01/2019
15 58918 Mrs. Suman Rani Female Harish Chandra Research Institute (HRI) PhD Chaudhary Devi Lal University 2020  
16 58960 Dr. Soumitra Das Male IISER Berhampur PDRF North-Eastern Hill University 2012 09/02/2021
17 59009 Mr. Tomasz Ciborski Male Nicolaus Copernicus University PhD Student Nicolaus Copernicus University; MSc in Mathematics 2022  
18 59063 Mr. Yogendra Singh Male IIT Kanpur PhD University of Delhi    
19 59073 Me Rupadarshi Ray Male IISER Mohali BSMS student BSMS, IISER Mohali Awaiting Result  
20 59085 Mr Dibyendu Das Male Indian Statistical Institute Bangalore Phd ISI Bangalore 2021  
21 59199 Mr. Theophilus Gera Male Sardar Vallabhbhai National Institute of Technology PhD Student Sardar Vallabhbhai National Institute of Technology 2024  
22 59549 Ms. Himanshi Chanana Female IIT Kanpur PhD IIT Ropar 2019  
23 59579 Ms Lakshmi Suresh Kumar Female National Institute of Technology Calicut JRF Cochin University of Science and Technology 2022  
24 59616 Dr. Katarzyna Zajac Female Nicolaus Copernicus University Assistant professor M.Sc.   20/12/2018
25 59830 Mr. Dipesh Meena Male Central University Of Rajasthan M.Sc. B.ED. MATHEMATICS Student Central University of Rajasthan Awaiting Result  
26 59844 Mr. Lovepreet Singh Male Indian Institute of Technology Patna PhD Indian Institute of Technology Patna 2023  
27 60000 Dr. Ashish Mishra Male The LNMIIT Jaipur Assistant Professor University of Delhi 2008 28/10/2015
28 60040 Mr. Sayantan Ghorai Male IIT Madras MSc Student IIT Madras Awaiting Result  
29 60073 Mr. Snehashis Mukherjee Male IIT Kanpur Post Doc University of calcutta 2018 19/02/2024
30 60419 Ms. Riddhi Vinay Patil Female Indian Institute of Technology Gandhinagar Ph.D. student University Of Mumbai 2021  
31 60449 Dr. Surendra Kumar Mishra Male W R S Govt. College Dehri Assistant Professor University of Allahabad, Prayagraj 2014 23/08/2021
32 60454 Mr. Sumanta Das Male University of Calcutta PhD student The University of Burdwan 2015  
33 60463 Mr. Subhradeep Ghosh Male Indian Institute of Science Education and Research Thiruvananthapuram MSc Student Indian Institute of Science Education and Research, Thiruvananthapuram    
34 60523 Ms. Tanisha Mandar Talekar Female Indian Institute of Science Education and Research - Thiruvananthapuram MSc Student Indian Institute of Science, Education and Research - Thiruvananthapuram    
35 60534 Mr Daksh Dheer Male IISER Thiruvananthapuram MS (Research) Student Currently pursuing    
36 60535 Mr. Mohammad Shane Alam Male ALIGARH MUSLIM UNIVERSITY PhD M.Sc. 2019  
37 60611 Mr, Aditya Bhan Ojha Male SCVB GOVT COLLEGE PALAMPUR Assistant Professor MDSU AJMER 2009  
38 60631 Dr. Saurabh Kumar Singh Male IIT Kanpur Associate professor IIT Kanpur 2010 09/09/2015
39 60640 Dr. Sanu Bera Male Gandhi Institute of Techonology and Management Asst. Prof. University of Calcutta 2018 10/02/2024
40 60720 Ms. Shivani Tushar Parab Female BITS Pilani K K Birla Goa Campus PhD St Joseph's College (Autonomous) Bangalore 2023  
41 60737 Mr. Sashadhar Dutta Male Ramakrishna Mission Vivekananda Centenary College, Rahara Msc student Ramakrishna Mission Vivekananda Centenary College    
42 60741 Mr. Samiran Nayek Male Indian institute of technology Tirupati PhD IIT TIRUPATI 2022  
43 60745 Ms Aisha Negi Female Chennai Mathematical institute PhD JNU 2022  
44 60777 Mr. Deepak Pal Male Indian Institute of Information Technology, Allahabad PhD UNIVERSITY OF ALLAHABAD 2018  
45 60788 Mr. Pavan Hasmukhbhai Adroja Male IIT Gandhinagar PhD Student Saurashtra University 2019  
46 60796 Mr. Anupam Sahoo Male IIT Kharagpur PhD University of Calcutta 2020  
47 60803 - Raghuram Sundararajan Male University of Chicago Ph.D. Student Not applicable    
48 60806 Mr. Prognadipto Majumder Male ISI Kolkata M Math 1st year student ISI Kolkata Awaiting Result  
49 60814 Mr. Pradeep Bisht Male Harish-Chandra Research Institute PhD University of Delhi 2019  
50 60820 Ms. Sneha Pushpakar Female IISER TVM MSc Student IISER TVM Awaiting Result  
51 60848 Mr Soumyadeep Chakraborty Male Indian Institute of Science Education and Research Kolkata MSc Completed Indian Institute of Science Education and Research Kolkata 2024  
52 60855 Mr. Tirtharaj Basu Male The Institute of Mathematical Sciences (IMSc) SRF The Insititute of Mathematical Sciences (IMSc) 2023  
53 60891 Mr Adhavan S V Male Indian Institute of Science BS Student      
54 60899 Mr. Subham Jaiswal Male Indian Statistical Institute Msc Student Indian Statistical Institute 2024  
55 60924 Mr Sanket Ramchandra Shete Male Bhaskaracharya prathisthana - IISER Bhopal 2023  
56 60929 Mr. Upamanyu Yaddanapudi Male IISER Kolkata PhD IISc Bengaluru, MS 2024  
57 60955 Ms. Ananya Gaur Female IISER Mohali PhD IISER Mohali 2023  
58 60980 Dr Sathya Seelan Male Sri Krishna Adithya College of Arts and Science Assistant Professor M.Sc 2009 30/12/2018
59 61027 Mr. Bappa Ghosh Male INDIAN STATISTICAL INSTITUTE PHD STUDENT Indian Statistical Institute 2020  
60 61035 Mr. Sadhanandh Vishwanath Male Chennai Mathematical Institute PhD Ramanujan Institute for Advanced studies in Mathematics, University of Madras 2019  
61 61038 Ms. Riju Bindua Female Indian Institute of Technology Delhi Undergraduate      
62 61039 Mr. Yerrapati Venkata Subbaiah Male Sardar Vallabhbhai National Institute of Technology MSc Student      
63 61042 Mr Sushruta Ghosh Male Shiv Nadar University PhD Student University of Hyderabad 2020  
64 61062 Mr. Ritesh Kumar Pandey Male IIT Kanpur PhD Student Banaras Hindu University 2019  
65 61064 Mr. Sugata Mandal Male Ramakrishna Mission Vivekananda Educational and Research Institute PhD M.Sc 2019  
66 61079 Ms. Dinny Ann Daniel Female Department of Mathematics, Cochin University of Science and Technology, Kerala, India Project Associate I of SERB-SURE Project Cochin University of Science and Technology 2023  
67 61092 Dr. Ambily A A Female Cochin University of Science and Technology Assistant Professor Cochin University of Science andTechnology 2007 18/07/2014
68 61094 Ms. Gayathry Pradeep Female Cochin University of Science And Technology Research Scholar Cochin University of Science And Technology 2020  
69 61099 Mr Krishna Teja Gv Male ISI Bangalore Postdoc CUTN 2016 20/09/2022

 

 

How to Reach: 

The school is online.

School Short Name: 

rtqbcssla

Last Date Application: 

Sunday, May 14, 2023

School Type: 

AIS

Separate faculty form: 

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