Crossed modules of Lie algebras
by
Friedrich Wagemann
Laboratoire de Mathématiques Jean Leray, Université de Nantes, France

The subject of my talk will concern the homological algebra of finite dimensional simple Lie algebras and the Lie algebra of vector fields on the line.

Exactly as central extensions, viewed as certain equivalence classes of short exact sequences

0 --> V --> m --> n --> g --> 0

are in one to one correspondence with classes in the second cohomology group H²(g, V) of g with values in the trivial module V, some equivalence classes of exact sequences

0 --> V --> m --> n --> g --> 0

are in one to one correspondence with classes in H³(g, V). This correspondence is somehow involved. [Hochschild gave an explicit construction of a crossed module corresponding to a given cohomology class inspired by the case of associated algebras, but it is rather complicated. Our construction is designed to simplify the matter. A construction due to Kassel and Loday concerns the relative case, and could be applied to the absolute case choosing generators and relations, but this is also too involved to be explicit.]

In our talk, we are going to construct an explicit crossed module representing the Godbillon-Vey cocycle \theta whose class generates H³(W₁, R) where W₁ is the Lie algebra of vector fields in one formal variable. More generally, we will show how to construct crossed modules from short exact coefficient sequences. In particular, we will have at hand a crossed module corresponding to the class of <[, ], > in H³(sl₂(C), C) where <, > is the Killing form of sl₂(C). Describing this last one in terms of duals of Verma modules, we will show how to generalize to an arbitrary simple complex finite dimensional Lie algebra g.

Further directions of research are a crossed module corresponding to the Gobillon-Vey cocycle of the universal covering group of the group of diffeomorphisms on the circle, and a q-deformed version of the crossed module corresponding to <[, ], > in relation to a q-deformed Wess-Zumino-Witten model (this last subject is joined work with Robert Wendt).

Date received: June 11, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cals-06.