Narrow Results

The main topic of this paper is two-fold. First, we compute the first relative cohomology group of the Lie algebra of smooth vector fields on the projective line, Vect(ℝℙ¹), with coefficients in the space of bilinear differential operators that act on tensor densities, , vanishing on the Lie algebra sl(2, ℝ). Second, we compute the first cohomology group of the Lie algebra sl(2, ℝ) with coefficients in .

We use an algorithm based on the notion of transvectants from classical invariant theory in determining the form of Stanley decomposition of the ring of invariants for the coupled Takens–Bogdanov systems when the Stanley decompositions of the Jordan blocks of the linear part are known at each stage. The set of systems of differential equations that are in normal form with respect to a particular linear part has the structure of a module of equivariants, and is best described by giving a Stanley decomposition of that module.