On Lie algebra crossed modules
Friedrich Wagemann
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 sl2(R), where the generator is usually called Godbillon-Vey class.
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