On Lie algebra crossed modules

Abstract

This article constructs a crossed module corresponding to the generator of the third cohomology group with trivial coefficients of a complex simple Lie algebra. This generator reads as <[,],>, constructed from the Lie bracket [,] and the Killing form <,>. The construction is inspired by the corresponding construction for the Lie algebra of formal vector fields in one formal variable on R, and its subalgebra sl_2(R), where the generator is usually called Godbillon-Vey class.