Related papers: Pfaffian groupoids
Symmetry lies at the heart of todays theoretical study of particle physics. Our manuscript is a tutorial introducing foundational mathematics for understanding physical symmetries. We start from basic group theory and representation theory.…
We solve the differentiation problem for Lie $\infty$-groups. Our approach builds on a classical version of Cartier duality which canonically identifies the Hopf algebra of point distributions supported at the identity of a Lie group with…
In the paper we study the algebroid A of the groupoid of partially invertible elements over the lattice of orthogonal projections of a $W^*$-algebra. In particular the complex analytic manifold structure of these objects is investigated.…
In this article we endow the group of bisections of a Lie groupoid with compact base with a natural locally convex Lie group structure. Moreover, we develop thoroughly the connection to the algebra of sections of the associated Lie…
We describe a local model for any Singular Riemannian Foliation in a neighbourhood of a closed saturated submanifold of a regular stratum. Moreover we construct a Lie groupoid which controls the transverse geometry of the linear…
The article is devoted to the investigation of groups of diffeomorphisms and loops of manifolds over ultra-metric fields of zero and positive characteristics. Different types of topologies are considered on groups of loops and…
In recent years methods for the integration of Poisson manifolds and of Lie algebroids have been proposed, the latter being usually presented as a generalization of the former. In this note it is shown that the latter method is actually…
These are notes for a Ph.D.\ course I held at SISSA, Trieste, in the Winter 2025. We review well-known topics in Riemannian geometry where Lie groups play a fundamental role. Part of the theory of compact connected Lie groups, their…
Using an algebraic point of view we present an introduction to the groupoid theory, that is, we give fundamental properties of groupoids as, uniqueness of inverses and properties of the identities, and study subgroupoids, wide subgroupoids…
Starting with some motivating examples (classical atlases for a manifold, space of leaves of a foliation, group orbits), we propose to view a Lie groupoid as a generalized atlas for the "virtual structure" of its orbit space, the…
The purpose of the paper is to study the relationship between differential equations, Pfaffian systems and geometric structures, via the method of moving frames of E.Cartan. We show a local structure theorem. The Lie algebra aspects…
In the first section we recall some basic notions on Lie algebras. In a second time we study the algebraic variety of complex $n$-dimensional Lie algebras. We present different notions of deformations : Gerstenhaber deformations,…
We outline the construction of the holonomy groupoid of a locally Lie groupoid and the monodromy groupoid of a Lie groupoid. These specialise to the well known holonomy and monodromy groupoids of a foliation, when the groupoid is just an…
This paper deals essentially with affine or projective transformations of Lie groups endowed with a flat left invariant affine or projective structure. These groups are called flat affine or flat projective Lie groups. Our main results…
This work presents a geometrical formulation of the Clairin theory of conditional symmetries for higher-order systems of partial differential equations (PDEs). We devise methods for obtaining Lie algebras of conditional symmetries from…
For a Lie groupoid there is an analytic index morphism which takes values in the $K-$theory of the $C^*$-algebra associated to the groupoid. This is a good invariant but extracting numerical invariants from it, with the existent tools, is…
We demonstrate that the proper general setting for contrast (potential) functions in statistical and information geometry is the one provided by Lie groupoids and Lie algebroids. The contrast functions are defined on Lie groupoids and give…
Although contemporary model theory has been called "algebraic geometry minus fields", the formal methods of the two fields are radically different. This dissertation aims to shrink that gap by presenting a theory of logical schemes,…
We extend the characterization of Lie bialgebroids via Manin triples to the context of double structures over Lie groupoids. We consider Lie bialgebroid groupoids, given by LA-groupoids in duality, and establish their correspondence with…
We consider the Lie group of smooth diffeomorphisms Diff$(M)$ of a simple polytope $M$ in the euclidean space. Simple polytopes are special cases of manifolds with corners. The geometric setting allows to study in particular, the subgroup…