Deformations of semi-direct products

TL;DR

This paper constructs a contraction from the direct product of a Lie algebra with itself to a semi-direct product, revealing a non-trivial cohomology class and generalizing to crossed modules with numerous examples.

Contribution

It introduces a new contraction method from direct to semi-direct products of Lie algebras and extends the framework to crossed modules, providing explicit examples.

Findings

Constructed a contraction from g⊕g to g⋉g.

Identified a non-zero second cohomology class.

Generalized results to crossed modules of Lie algebras.

Abstract

We exhibit in this article a contraction of the direct product Lie algebra $g\oplus g$ of a finite-dimensional complex Lie algebra $g$ onto the semi-direct product Lie algebra $g\rtimes g$, where the first factor $g$ is viewed as a trivial Lie algebra and as the adjoint $g$-module. This contraction gives rise to a non-zero cohomology class in the second cohomology space. We generalize to the setting of $h\oplus g$ and $h\rtimes g$ with respect to a given crossed module of Lie algebras $h\to g$. We give many examples to illustrate our results.