We discuss actions of Lie n-groups and the corresponding action Lie n-groupoids; discuss actions of Lie n-algebras (L∞-algebras) and the corresponding action Lie n-algebroids; and discuss the relation between the two by integration and differentiation. As an example of interest, we discuss the BRST complex that appears in quantum field theory. We describe it as the Chevalley-Eilenberg algebra o...

We define and study multiplicative connections in the tangent bundle of a Lie groupoid. Multiplicative are linear satisfying an appropriate compatibility with groupoid structure. Our definition is natural sense that connection on if only its torsion tensor Bursztyn–Drummond [Lie theory tensors, Mat. Ann. 375 (2019) 1489–1554, arXiv:1705.08579] geodesic spray vector field. identify obstruction t...

In this work we deal with coverings and actions of Lie groupgroupoids being a sort of the structured Lie groupoids. Firstly, we define an action of a Lie group-groupoid on some Lie group and the smooth coverings of Lie group-groupoids. Later, we show the equivalence of the category of smooth actions of Lie group-groupoids on Lie groups and the category of smooth coverings of Lie group-groupoids.

It is well known that a measured groupoid G defines a von Neumann algebra W ∗(G), and that a Lie groupoid G canonically defines both a C∗-algebra C∗(G) and a Poisson manifold A∗(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C∗-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of obj...

This paper studies the construction of geometric integrators for nonholonomic systems. We derive the nonholonomic discrete Euler-Lagrange equations in a setting which permits to deduce geometric integrators for continuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifying a discrete Lagrangian and a constraint submanifold on it. Additionally, it ...

We present some features of the smooth structure and of the canonical stratification on the orbit space of a proper Lie groupoid. One of the main features is that of Morita invariance of these structures-it allows us to talk about the canonical structure of differentiable stratified space on the orbispace (an object analogous to a separated stack in algebraic geometry) presented by the proper L...

We show that the path construction integration of Lie algebroids by Lie groupoids is an actual equivalence from the of integrable Lie algebroids and complete Lie algebroid comorphisms to the of source 1-connected Lie groupoids and Lie groupoid comorphisms. This allows us to construct an actual symplectization functor in Poisson geometry. We include examples to show that the integrability of com...

The root systems appearing in the theory of Lie superalgebras and Nichols algebras admit a large symmetry extending properly the one coming from the Weyl group. Based on this observation we set up a general framework in which the symmetry object is a groupoid. We prove that in our context the groupoid is generated by reflections and Coxeter relations. This answers a question of Serganova. Our w...