  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 $\cal G$, and hence on the cohomology
properties of the Lie algebra considered.
