Seminar

Representation Theory of Compact Lie Groups

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Organization Abstract Literature (Books) Literature (Articles)

Organization

This seminar is addressed to Bachelor's (\(\geq\)3rd semster) and Master's students. It is a regular elective whether it has, as of yet, appeared in the catalog or not.
The seminar will take place on Tusedays 16:15-17:45 in HS 6 (MI), broadcasted via BigBlueButton.

Abstract

A group \(G\) having the structure of an analytic manifold such that the mapping \((x,y) \mapsto xy^{-1}\) is analytic is called a Lie group. A compact Lie group is a finite dimensional real Lie group. Important examples are \(SO(n)\), \(SU(n)\), and \(Sp(n)\). Not only are they important in their own right with many applications in physics and related fields but their study is also a stepping stone to the theory of general Lie groups.
In practice, groups do not just arise as abstract algebraic object but usually through their action on other objects as, for instance, solutions to polynomial or differential equations or manifolds. Often there is, in some natural way, a vector space attached to such data which gives rise to a linear action of the group on it. Representation theory is then the study of this action as a way to understand the group better.
In this seminar, we want to study the representation theory of compact Lie groups following mainly the text MR0781344. Kirillov's book MR2069175 also has a chapter on computing the representations of compact Lie groups with the so-called orbit method. This would also be an appropriate talk for one or two students.
Depending on the number of participants and their interests, we may continue the seminar with the appropriate sections in MR3065085 or MR3469687, where pseudo-differential calculi on compact Lie groups are discussed.

Literature - Books

To get to the fulltext of the books via the OPAC of the University Library Göttingen, you may use vpn.gwdg.de to connect to the campus net.

Literature - Articles