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  SS 2007
    Lineare Algebra II
    Integr. Systeme
    Rep. theory
    Sem. alg. Geom.
  WS 2006/2007

Topics in representation theory

Titel: Topics in representation theory
Dozent(in): Dr. Igor Mencattini
Termin: Do. 10:00 - 11:30 (T 2001)
Gebäude/Raum: T 2001
Ansprechpartner: Dr. Igor Mencattini

Inhalt der Lehrveranstaltung:

The goal of this course is to provide an introduction to some geometric constructions in representation theory. During the first part of the lectures, I will review the necessary background from the theory of algebraic groups and I will discuss some geometry and representation theory related to flag varieties and other varieties associated to semisipmle groups. During the second part we will study the geometry of the nilpotent cone and of the Steinberg's variety. Applications to the Springer's representation of Weyl groups and to the geometry of the kleinian singularities will be discussed as well.

Tentative plan:
1. Symplectic structures, moment maps and coadjoint orbits.
2. Review of the theory of algebraic groups: Borel subgroups, maximal tori, Weyl groups, Bruhat's decomposition, etc.
3. Universal resolution of $g$, semisimple and nilpotent elements, regular elements. Chevalley theorem.
4. Nilpotent cone and Steinberg's variety. Lagrangian construction of $U (g)$ and applications to the representation theory of the Weyl groups.
5. Kleinian singularities.

Vorkenntnis für die Lehrveranstaltung:

algebraic topology and (some) algebraic geometry; theory of Lie groups and Lie algebras.

Literatur zur Lehrveranstaltung:

J.E. Humphreys: Linear algebraic groups
N. Chriss and V. Ginzburg: Representation theory and complex geometry
P. Slodowy: Four lectures on simple groups and singularities
J.P. Serre: Algebres de Lie semi-simples complexes

weitere Informationen zu der Lehrveranstaltung:

empfohlenes Studiensemester der Lehrveranstaltung: ab dem 6. Semester
Fachrichtung Lehrveranstaltung: Mathematik
Nummer der Lehrveranstaltung: 06062
Dauer der Lehrveranstaltung: 2 SWS
Typ der Lehrveranstaltung: V - Vorlesung
Leistungspunkte: 6 LP für Vortrag
Bereich: Geometrie
Lehrveranstaltungspflicht: Wahl
Begleitende Lehrveranstaltung(en): 06063
Semester: SS 2007

[Institut für Mathematik]   [Universität Augsburg]   [Math.-Nat. Fakultät]   [Impressum]