In this note a functorial approach to the integration problem of an LAgroupoid to a double Lie groupoid is discussed. To do that, we study the notions of fibered products in the categories of Lie groupoids and Lie algebroids, giving necessary and sufficient conditions for the existence of such. In particular, it turns out, that the fibered product of Lie algebroids along integrable morphisms is...

<p style='text-indent:20px;'>We survey recent results on the local and global integrability of a Lie algebroid, as well infinitesimal multiplicative geometric structures it.</p>

We introduce the notion of -algebroid, generalising both Lie and Courant algebroids, as well algebroids used in exceptional generalised geometry for . Focusing on case, we prove a classification “exact” translate related Leibniz parallelisable spaces into tractable algebraic problem. After discussing general Poisson–Lie duality, show that U-duality is compatible with equations motion supergravi...

<p style='text-indent:20px;'>The core diagram of a double Lie algebroid consists the algebroid, together with two core-anchor maps to sides algebroid. If these core-anchors are surjective, then and its called <i>transitive</i>. This paper establishes an equivalence between transitive algebroids, diagrams over fixed base manifold. In other words, it proves that is completely de...

We give an integrability criterion for a projective limit of Banach distributions on Fr\'echet manifold which is manifolds. This leads to result involutive bundles sequence can be seen as version Frobenius Theorem in setting. As consequence, we obtain the third Lie theorem Fr\'echet-Lie group submersive groups. also application prolongations algebroid.

In order to obtain a framework in which both non-holonomic mechanical systems and non-holonomic mechanical systems with symmetry can be described, we introduce in this paper the notion of a Lagrangian system on a subbundle of a Lie algebroid.

Main ideas of the differential geometry on affine bundles are presented. Affine counterparts of Lie algebroid and Poisson structures are introduced and discussed. The developed concepts are applied in a frame-independent formulation of the time-dependent and the Newtonian mechanics. MSC (2000): 37J05, 53A40, 53D99, 55R10, 70H99.