Hidden supersymmetries in supersymmetric quantum mechanics

We discuss the appearance of additional, hidden supersymmetries for simple 0+1 $Ad(G)$-invariant supersymmetric models and analyse some geometrical mechanisms that lead to them. It is shown that their existence depends crucially on the availability of odd order invariant skewsymmetric tensors on the (generic) compact Lie algebra $\mathcal{G}$, and hence on the cohomology properties of the Lie algebra considered.