Material groupoids and algebroids

Hopf Algebroids and quantum groupoids

We introduce the notion of Hopf algebroids, in which neither the total algebras nor the base algebras are required to be commutative. We give a class of Hopf algebroids associated to module algebras of the Drinfeld doubles of Hopf algebras when the Rmatrices act properly. When this construction is applied to quantum groups, we get examples of quantum groupoids, which are semi-classical limits o...

Groupoids and the Integration of Lie Algebroids

We show that a Lie algebroid on a stratified manifold is integrable if, and only if, its restriction to each strata is integrable. These results allow us to construct a large class of algebras of pseudodifferential operators. They are also relevant for the definition of the graph of certain singular foliations of manifolds with corners and the construction of natural algebras of pseudodifferent...

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

Morita Theory for Hopf Algebroids and Presheaves of Groupoids

Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff , the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of preshe...