A family of non-restricted D=11D=11 geometric supersymmetries

We construct a two parameter family of eleven-dimensional indecomposable Cahen-Wallach spaces with irreducible, non-flat, non-restricted geometric supersymmetry of fraction $\nu=\tfrac{3}{4}$. Its compactified moduli space can be parametrized by a compact interval with two points corresponding to two non-isometric, decomposable spaces. These singular spaces are associated to a restricted $N=4$ geometric supersymmetry with $\nu=\tfrac{1}{2}$ in dimension six and a non-restricted $N=2$ geometric supersymmetry with $\nu=\tfrac{3}{4}$ in dimension nine.Comment: 26pp, v3: compared with the published version we clarified the content of Section 7 and corrected one of the interpretation

Polynomial poly-vector fields

In this text we give a decomposition result on polynomial poly-vector fields generalizing a result on the decomposition of homogeneous Poisson structures. We discuss consequences of this decomposition result in particular for low dimensions and low degrees. We provide the tools to calculate simple cubic Poisson structures in dimension three and quadratic Poisson structures in dimension four. Our decomposition result has a nice effect on the relation between Poisson structures and Jacobi structures.Comment: 20 pages. v4: sections rearranged and formulations clarifie

A note on the regularity of matrices with uniform polynomial entries

In this text we study the regularity of matrices with special polynomial entries. Barring some mild conditions we show that these matrices are regular if a natural limit size is not exceeded. The proof draws connections to generalized Vandermonde matrices and Schur polynomials that are discussed in detail

SUSY structures on deformed supermanifolds

We construct a geometric structure on deformed supermanifolds as a certain subalgebra of the vector fields. In the classical limit we obtain a decoupling of the infinitesimal odd and even transformations, whereas in the semiclassical limit the result is a representation of the supersymmetry algebra. In the case of mass preserving structure we describe all high energy corrections to this algebra.Comment: 20 pages. v2 coincides with the version published in Differential Geometry and its Application