Seminarium KMMF "Teoria Dwoistości"

A fundamental problem in defining compound geometrical structures is the compatibility of the ingredients. For instance, VB-groupoids and VB-algebroids are vector bundles endowed with compatible structures of a Lie groupoid and Lie algebroid, respectively. Canonical examples emerge from the tangent lifts of Lie groupoids and Lie algebroids. More generally, weighted structures are geometric structures on even N-graded bundles that are compatible with the graded structure. They lead to VB-structures if the graded bundle is of degree one, i.e., it is actually a vector bundle.We will consider compatibility conditions for a large class of geometric structures, e.g., those represented by tensors fields (Poisson, symplectic, Nijenhuis, etc.) as well as many others, like distributions, foliations, Ehresmann connections, principal bundles, etc. These compatibility conditions are expressed in the language of the homogeneity associated with N-graded bundles, understood as a smooth action of the multiplicative monoid of real numbers. An intelligent guess of what the compatibility means in each case comes from canonical examples of higher tangent lifts of the structure in question, and is based on the belief that the lifts are automatically compatible with the canonical graded bundle structures of higher tangent bundles.