Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

Lie groupoids are to be found throughout differential geometry including the theory of group actions, foliations, Poisson geometry, connection theory and the study of singular spaces such as orbifolds. Over the years there has been much interest in 'categorified 'objects in the category of Lie groupoids; this has been spurred on by the original ideas of Mackenzie such as 'double Lie groupoids' etc. Very recently there has been a resurgence of interest in VB-groupoids and VB-algebroids, which are vector bundles in the category of Lie groupoids and Lie algebroids respectively. Part of this resurgence is motivated by the representation theory of Lie groupoids and algebroids. However, the original definitions of 'VB-objects' are very complicated and not obvious. Bursztyn, Cabrera and de Hoyo realised last year that VB-groupoids and VB-algebroids can be neatly described using regular actions of the multiplicative monoid of real numbers a la Grabowski and Rotkiewicz. In this talk I will highlight how, rather naturally, one can generalise and simplify the notion of 'VB-objects' by using homogeneity structures; that is smooth actions of the multiplicative monoid of reals. It is known, via Grabowski and Rotkiewicz that homogeneity structures always lead to what they called graded bundles; these are manifolds with a non-negative grading on their structure sheaf and give particularly nice examples of polynomial bundles. Graded bundles can be viewed as a very natural generalisation of a vector bundle. Thus, we can pass from VB-groupoids to weighted Lie groupoids which we understand as graded bundles in the category of Lie groupoids, or indeed vice-versa. We will highlight natural examples of this very rich geometric theory and briefly describe the Lie theory relating weighted Lie groupoids and weighted Lie algebroids. Time permitting I will also describe weighted Poisson-Lie groupoids and weighted Courant algebroids. This talk is based on joint work with K. Grabowska and J. Grabowski.