Lie theory of multiplicative tensors

Abstract

We study tensors on Lie groupoids suitably compatible with the groupoid structure, called multiplicative. Our main result gives a complete description of these objects only in terms of infinitesimal data. Special cases include the infinitesimal counterparts of multiplicative forms, multivector fields and holomorphic structures, obtained through a unifying and conceptual method. We also give a full treatment of multiplicative vector-valued forms, particularly Nijenhuis operators and related structures. 10 4. Proof of the Theorems 17 4.

Differential forms with values in VB-groupoids and its Morita invariance

“…Our next result concerns some equivariance properties of the map F. It should be seen as a generalization of Proposition 4.1 in [10] (when applied to Example 5.3).…”

“…with values in T G are also known as vector valued forms. The Lie theory of multiplicative vector valued forms were studied in [10] (see also [9]). The lift operation ϑ → ϑ takes multiplicative vector valued forms to multiplicative forms on the jet groupoid J 1 G with values in the adjoint representation t * A.…”

“…These objects were introduced in [10] as the infinitesimal data associated to multiplicative vector valued forms on Lie groupoids.…”

“…Let us begin by explaining the general philosophy which will guide us through the proofs of Theorem 3.9 and 4.4. The same strategy used in [10] to study multiplicative tensor fields on a Lie groupoid will give us the proper viewpoint to tackle the correspondence between multiplicative differential forms with values in VB-groupoids and IM-forms with values in VB-algebroids. The presence of coefficients encoded in the VB-groupoid will force us to adapt (in a non-trivial way) most of the techniques used there as one can note by comparing this section with [10, Section 4].…”

“…Our approach to prove Theorem 1.1 is built on ideas developed in [6,10]. Namely, the multiplicativity of ϑ ∈ Ω q (G, V) can be characterized by a cocycle equation for the corresponding function on the big groupoid G = T G ⊕ · · · ⊕ T G ⊕ V * and the corresponding IM q-form on A with values in v can be obtained by taking Lie derivatives along right-invariant vector fields of G coming from linear and other special sections of its Lie algebroid known as core sections.…”