Compact Lie groups and their representations (2022)

In this course I will first talk about the structure theory of compact Lie groups, beginning with the fact that a compact connected Lie group is an almost direct product of the identity connected component of its centre and its commutator subgroup (which is closed subgroup) conjugacy of maximal tori and the fact that every element is contained in a maximal torus. In the course of proving these results, some results on the topology of compact Lie groups will also be proved. I will then establish Weyl's theorem which asserts that if $G$ is a compact connected Lie group and $[G,G]=G$, $\pi_1(G,e)$ is finite (and hence the universal covering of a compact group whose abelianization is trivial is a compact. Then I will introduce roots and weights and the Dynkin diagram of the compact group and sketch proof of the fact that the Dynkin diagram determines the group locally. The remaining lectures will be devoted to representation theory. I will establish the bijective correspondence between 'Dominant Weights' and irreducible representations. The course will end with the Weyl Character Formula for the character of an irreducible representation corresponding to a 'dominant' weight. The entire theory is essentially the same as the representation theory of reductive algebraic groups. I will off and on indicate how the two are related.
I will be assuming some familiarity with the basic theory of Lie groups such as the correspondence between Lie sub-algebras of the Lie group and Lie subgroups of the Lie groups; also with some basic results from algebraic topology.

Time Table:

Speaker

Prof. M. S. Raghunathan,
Centre for Basic Sciences, Mumbai

Date and time

Every Monday at 4 pm
17 October-5 December
- Except on 24th October. The lecture on 24th October will be delivered on 26 October at 4 pm.

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
1	43643	Mrs Revathi Devi	Female	SRM INSTITUTE OF SCIENCE AND TECHNOLOGY	RESEARCH SCHOLAR	MKU EVENING COLLEGE DINDIGUL	2012
2	43657	Mr. Supriyo Jana	Male	IIT Madras	PhD Scholar	IIT Madras	2021
3	43670	Mr. Suvam Mukherjee	Male	Muralidhar Girls' College	Lecturer	WBSU	2019
4	43676	Ms. Manika Gupta	Female	Institute of Mathematical Sciences	PhD Student	Kerala School of Mathematics	2022
5	43677	Mr. Bidhan Paul	Male	IIT MADRAS	PhD student	IIT MADRAS	2019
6	43681	Ms. Chanchan Oinam	Female	Manipur University	Pre-PhD	Manipur University	2019
7	43690	Mr. Pinaki Ranjan Ghosh	Male	The University of Burdwan	PhD	Presidency University, Kolkata	2016
8	43710	Mr. Soham Mondal	Male	IIT MADRAS	Ph.D	Visva-Bharati	2017
9	43711	Mr Arnab Kundu	Male	IIIT Guwahati	PhD Student	Ramakrishna Mission Vivekananda Educational and Research Institute	2021
10	43721	Ms Anju Kumari Yadav	Female	NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA	PhD	RAVENSHAW UNIVERSITY	2018
11	43733	Mr. Abinash Pradhan	Male	NIT,Rourkela	PhD	Tezpur University	2019
12	43734	Mr. Raja Kundu	Male	Indian Institute of Technology Madras	PhD Student	Indian Institute of Technology Madras	2020
13	43745	Mr. Jitender Sharma	Male	Indian Institute of Science Education and Research Kolkata	I-PhD	Indian Institute of Science Education and Research Kolkata, Int-PhD (Mathematics)	Appeared / Awaiting Result
14	43754	Dr Loganathan Seenivasan	Male	SETHUPATHY GOVERNMENT ARTS COLLEGE	Assistant Professor	Gandhigram Rural University	2006
15	43770	Mr. Shayon Bhadra	Male	Kalna College	Project Student under Dr. Abhishek Mukherjee	Tezpur University	2019
16	43771	Mr. Manish Chaurasia	Male	IIT(BHU)	Ph.D.	IIT(Bhubaneswar)	2018
17	43784	Mr. Promod Sharma	Male	IIT(BHU), VARANASI	PhD Scholar	Assam University Silchar	2017
18	43791	Ms. Mansi Mishra	Female	IIT(BHU)	PhD	IIT Kanpur	2019
19	43834	Mr. Sashadhar Dutta	Male	Ramakrishna Mission Vivekananda Centenary College	Msc student	Ramakrishna Mission Vivekananda Centenary College	Awaiting Result
20	43847	Mr. Dev Karan Singh	Male	Motilal Nehru National Institute of Technology Allahabad,	Ph.D. (SRF)	IIT Guwahati	2018
21	43880	Ms. Marisha Singh	Female	National Institute of Technology	Msc. Student
22	44032	Mr. Ashid Aboobacker	Male	Birla Institute of Technology and Science Pilani, Hyderabad Campus	PhD	University of Hyderabad	2020

Compact Lie groups and their representations (2022)

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