geometrical categories of generalized lie groups and lie group-groupoids

Lie Groupoids as Generalized Atlases

Starting with some motivating examples (classsical 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 equivalence between atlases being here the smooth Morita equivalence. This “structure” keeps memory of the isotropy groups and of the smoothness as well. To take the ...

Generalized Lie Bialgebras and Jacobi Structures on Lie Groups

We study generalized Lie bialgebroids over a single point, that is, generalized Lie bialgebras. Lie bialgebras are examples of generalized Lie bialgebras. Moreover, we prove that the last ones can be considered as the infinitesimal invariants of Lie groups endowed with a certain type of Jacobi structures. We also propose a method to obtain generalized Lie bialgebras. It is a generalization of t...

Lie Algebras and Lie Brackets of Lie Groups–matrix Groups

The goal of this paper is to study Lie groups, specifically matrix groups. We will begin by introducing two examples: GLn(R) and SLn(R). Then in each section we will prove basic results about our two examples and then generalize these results to general matrix groups.

Lie local subgroupoids and their holonomy and monodromy Lie groupoids

The notion of local equivalence relation on a topological space is generalized to that of local subgroupoid. The main result is the construction of the holonomy and monodromy groupoids of certain Lie local subgroupoids, and the formulation of a monodromy principle on the extendability of local Lie morphisms.  2001 Elsevier Science B.V. All rights reserved. AMS classification: 58H05; 22A22; 18F20

Lie Groupoids and Lie algebroids in physics and noncommutative geometry

Groupoids generalize groups, spaces, group actions, and equivalence relations. This last aspect dominates in noncommutative geometry, where groupoids provide the basic tool to desingularize pathological quotient spaces. In physics, however, the main role of groupoids is to provide a unified description of internal and external symmetries. What is shared by noncommutative geometry and physics is...

Lie, symplectic and Poisson groupoids and their Lie algebroids