AIC - Compact Lie groups and their representations (2023)

Speakers and Syllabus

Abstract:
  • 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 to 5:30 pm

  • 9th Jan to 14th Feb

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