On a non-abelian Poincaré lemma (Q2845560)

In this paper the author is proving two interesting results. First, by considering odd forms \(\omega\) with values in a Lie superalgebra \(\mathfrak{g}\) (whose corresponding Lie supergroup is denoted by \(G\)) instead of the usual 1-forms, he derives a non abelian homologous result of Poincaré lemma, namely: If \(d\omega+ \omega^{2}=0\), then there exists a necessarily even \(G\)-valued form \(g\) such that \( \omega=gCg^{-1}-dgg^{-1}\) (where \(C\) is a constant odd element of \(\mathfrak{g}\) satisfying \(C^{2}=0\)). Second, theorem 5.2 states that if \(\mathfrak{g}\) is a dLie algebra and \(\mathfrak{G}\) is its corresponding dLie group which is connected, then two odd elements in \(\mathfrak{g}\) are homotopic iff they are gauge equivalent. Many very interesting discussions, examples and connected results are also presented.