The article shows the relevance to find abelian representatives for a given equivalence class of crossed modules, that is crossed modules of the form \(\mu : V_2 \to V_3 \times_{\alpha} \mathfrak{g}\) which arise as connection between a short exact sequence of \(\mathfrak{g}\)-modules \(0 \to V_1 \to V_2 \mathop{\to}\limits^d V_3 \to 0\) and an abelian extension \(V_3 \times_{\alpha} \mathfrak{g}\) of \(\mathfrak{g}\) by \(V_3\) via a 2-cocycle \(\alpha\). Through the paper, authors almost always consider crossed modules where \(V_1 = \mathbb{C}\), the trivial \(\mathfrak{g}\)-module.
The explicit expression for the 2-cocycle \(\alpha\) in [F. Wagemann, Commun. Algebra 34, No. 5, 1699–1722 (2006; Zbl 1118.17004)] in order to construct an abelian representative for the Cartan cocycle corresponding to a general simple complex Lie algebra is not correct, so in the present paper the new expression for the 2-cocycle \(\alpha\) is proposed: \[ \widetilde{\alpha}(x,g_1,g_2):= \sum_{(x)} \int_0^1 dt f(pr(x^{(1)}), A_t (x^{(2)}, g_1), A_t(x^{(3)},g_2)) \] for all \(x \in U\mathfrak{g}\), \(g_1, g_2 \in \mathfrak{g}\), and \(f : \bigwedge^3 \mathfrak{g} \to \mathbb{C}\) is a Chevalley-Eilenberg 3-cochain of \(\mathfrak{g}\) with values in \(\mathbb{C}\). It works for any 3-cocycle with trivial coefficients.
The formula relies on an explicit formula for the homotopy contracting the Koszul complex. In order to render its expression explicit, one has to compute the Eulerian idempotent on the corresponding universal enveloping algebra, which is illustrated in the paper.
As an application of the proposed theory, authors show how to construct quasi-invariants tensors for a wide class of Lie algebras, including all simple complex Lie algebras.