Seminarium KMMF "Teoria Dwoistości"

The tangent bundle of a Lie group G bears a "symmetry". The reduced space, TG/G, is naturally identified with T_eG, the tangent space at the identity element of G, is not merely a vector space but has an algebraic (Lie algebra) structure inherited from the group multiplication in G. Similarly, the reduction of the tangent bundle of a Lie groupoid leads to a rich algebro-geometric structure called a Lie algebroid. Reductions of higher tangent bundles of Lie groupoids provide natural examples of structures which we would like to call higher algebroids. The basic problem is: what is the algebraic structure on the reduced bundle inherited from the groupoid multiplication? The key difficulty is that there is no bracket operation on the space of sections of a higher tangent bundle. The basic idea which allowed us to approach the problem is a reformulation of the definition of an algebroid in terms of a relation which can be obtained by a reduction of the canonical involution of TTG if the algebroid integrates to a Lie groupoid G. Thus a higher algebroid is, in principle, a graded bundle (a natural generalization of the notion of a vector bundle) equipped with a relation of a special kind. Such a point of view is natural in Geometric Mechanics as higher-order systems with internal symmetries can be reduced to systems defined on such higher algebroids. Based on a joint paper with Michał Jóźwikowski.