The appearence of Poisson–Lie groups [Drinfel’d 1983] and symplectic groupoids [Weinstein 1987] marks the origin of the currently rather active research about multiplicative structures on Lie groupoids and their infinitesimal counterparts on Lie algebroids. A geometric structure on a Lie groupoid (resp. algebroid) is multiplicative (resp. infinitesimally multiplicative (IM)) if it is “compatible” with the structure of Lie groupoid (resp. algebroid). Till now the research has been concentrated only on multiplicative structures of tensorial type, with very few exceptions. Motivated by Jacobi–Lie groups, contact groupoids, and potential applications to the geometric theory of PDEs, we have started (with Chiara Esposito and Luca Vitagliano) a systematic investigation of multiplicative multi-differential operators between sections of vector bundles, and their infinitesimal counterparts. Naturally we do not deal just with plain vector bundles but rather with VB-groupoids and VB-algebroids. Actually a VB-groupoid (resp. algebroid) is a vector bundle in the category of Lie groupoids (resp. algebroids). As starting steps of this ongoing research project we have studied multiplicative derivations.
A derivation of a vector bundle can be seen as an infinitesimal automorphism of the vector bundle. Hence we define a multiplicative (resp. IM) derivation of a VB-groupoid (Ω, E; G, M) (resp. algebroid (W, E; A, M)) as a derivation of Ω → G (resp. W → A) which generates a flow of VB-groupoid (resp. algebroid) automorphisms. We also provide algebraic characterizations of IM derivations in analogy with what known for IM vector fields on Lie algebroids [Mackenzie and Xu 1998]. Multiplicative derivations have IM derivations as their infinitesimal counterparts: the Lie functor maps the former to the latter. Further we establish an integration theorem: under a certain topological condition the above map is one-to-one. Finally, after having singled out the special class of source-trivial VB-groupoids and algebroids, we prove that any source-trivial VB-groupoid (resp. algebroid) canonically determines a second VB-groupoid (resp. algebroid) such that multiplicative (resp. IM) derivations of the former identify exactly with morphic sections of the latter.