Fachbereich Mathematik 
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Cohomology of Lie algebras



Credits:   6LP (to obtain the credits, a talk is required; other participants are welcome as well)
Lecturer:   JProf. Dr. Christoph Wockel
Target:   Students in the master's program of Mathematics and Mathematical Physics
Prerequisite:   Linear algebra
Language:   English (people who don't want to give a talk in English may give their talk in German)

Background: Lie algebras occur in Mathematics and in Mathematical Physics as infinitesimal objects of continuous symmetry groups. An example are the (n x n)-matrices, which are the Lie algebra of the Lie group of invertible (n x n)-matrices.

Motivation: The cohomology of a Lie algebra describes algebraic invariants of it. These invariants are frequently related to other, algebraic or geometric objects, assigned to this Lie algebra. For instance, the first cohomology describes crossed homomorphisms, the second cohomology abelian extensions and the third cohomology crossed modules. Moreover, the cohomology of the Lie algebra g of a compact Lie group G is closely related to the de Rham cohomology of G. In Mathematical Physics, Lie algebra cohomology enters via the theory of Kac-Moody algebras, the BRST formalism or categorical Lie algebras.

Topics for the talks:
  • Recap of the basic notions and properties of Lie algebras (1 talk)
  • Definition of cohomology of Lie algebras (1-2 talks)
  • Calculations in easy examples and vanishing theorems (1 talk)
  • Relation to Cohomology of Lie groups (1-2 talks)
  • Extensions and crossed modules of Lie algebras (1-2 talks)
  • Kac-Moody algebras (2 talks)
  • BRST cohomology (2 talks)
  • Categorical Lie algebras (1-2 talks)
Several of the above topics may be continued to themes for a master's thesis.
  Seitenanfang  Impress 2010-07-19, Christoph Wockel