Sommersemester 2018: Seminar on Lie groups, Lie algebras and their representations

Prof. Dr. Katrin Wendland
Dr. Santosh Kandel

Wann und wo: Di 14 - 16, SR 403, Eckerstr. 1

Vorbesprechung: Di 06.02.2018, 12:30, SR 318, Eckerstr. 1
Um teilzunehmen, kommen Sie bitte in die Vorbesprechung des Seminares; eine Teilnehmerliste wird nicht vorab ausliegen.

Topic:
The concept of a Lie group arises naturally by putting together the algebraic notion of a group with the geometric notion of a smooth manifold. A Lie group is a smooth manifold with a group structure such that the group operations are smooth. Lie groups arise in a natural way as symmetries of a geometric object. The general linear group GL_n(R) is our guiding example of a Lie group. A representation of a Lie group G on a vector space V is a group homomorphism ρ: G→GL(V). Similarly, the notion of Lie algebra appears naturally. For example, the tangent space 𝔤 at the identity element of a Lie group G has this structure. For G=GL_n(R), this yields 𝔤 = Mat_R(n×n) with the composition rule given by the commutator [X,Y]=XY-YX. A vector space with such a composition rule is called a Lie algebra.

In this seminar, we will study (matrix) Lie groups, Lie algebras and their representations. We will introduce the notion of Lie groups and Lie algebras and discuss the correspondence between them. Since finite dimensional "semisimple" Lie algebras can be viewed as elementary building blocks of more complicated Lie algebras, we will study them with an emphasis on the structure theory and their representations. Finally, we will discuss representations of Lie groups and prove a version of the Peter-Weyl Theorem, which is a statement about using irreducible representations of a compact Lie group G to study the Hilbert space of square integrable functions on G with respect to the so-called Haar measure. As a corollary of the Peter-Weyl theorem, it follows that every compact Lie group can be realized as a matrix Lie group.

Literatur:
Die Links führen auf Webseiten, von denen aus dem Universitätsnetz die jeweiligen Referenzen zugänglich sind. Falls kein Link gesetzt ist, finden Sie die Referenz in der Bibliothek des Mathematischen Institutes Freiburg.

• T. Bröcker, T. Dieck, Representations of Compact Lie Groups, Springer-Verlag, 2003
• J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1974
• Brian Hall, Lie Groups, Lie Algebras, and Representations : An Elementary Introduction, Springer-Verlag, 2015
• A. Knapp, Lie groups Beyond an Introduction, Birkhäuser, 2002
• W. Rossmann, Lie groups, Oxford Graduate Texts in Mathematics, Oxford University Press, 2002

Vortragsprogamm:
Das Vortragsprogramm finden Sie hier.
Die Vorträge können auf Deutsch oder auf Englisch präsentiert werden.

Tutorium: Dr. Santosh Kandel