25 research outputs found
On the integrability of subalgebroids
Let G be a Lie groupoid with Lie algebroid g. It is known that, unlike in the
case of Lie groups, not every subalgebroid of g can be integrated by a
subgroupoid of G. In this paper we study conditions on the invariant foliation
defined by a given subalgebroid under which such an integration is possible. We
also consider the problem of integrability by closed subgroupoids, and we give
conditions under which the closure of a subgroupoid is again a subgroupoid
Multiplicative bijections between algebras of differentiable functions
We show that any multiplicative bijection between the algebras of
differentiable functions, defined on differentiable manifolds of positive
dimension, is an algebra isomorphism, given by composition with a unique
diffeomorphism
On the developability of subalgebroids
In this paper, the Almeida-Molino obstruction to developability of
transversely complete foliations is extended to Lie groupoids
On the universal enveloping algebra of a Lie-Rinehart algebra
We review the extent to which the universal enveloping algebra of a
Lie-Rinehart algebra resembles a Hopf algebra, and refer to this structure as a
Rinehart bialgebra. We then prove a Cartier-Milnor-Moore type theorem for such
Rinehart bialgebras
Stacky Lie groups
Presentations of smooth symmetry groups of differentiable stacks are studied
within the framework of the weak 2-category of Lie groupoids, smooth principal
bibundles, and smooth biequivariant maps. It is shown that principality of
bibundles is a categorical property which is sufficient and necessary for the
existence of products. Stacky Lie groups are defined as group objects in this
weak 2-category. Introducing a graphic notation, it is shown that for every
stacky Lie monoid there is a natural morphism, called the preinverse, which is
a Morita equivalence if and only if the monoid is a stacky Lie group. As
example we describe explicitly the stacky Lie group structure of the irrational
Kronecker foliation of the torus.Comment: 40 pages; definition of group objects in higher categories added;
coherence relations for groups in 2-categories given (section 4
Quantized reduction as a tensor product
Symplectic reduction is reinterpreted as the composition of arrows in the
category of integrable Poisson manifolds, whose arrows are isomorphism classes
of dual pairs, with symplectic groupoids as units. Morita equivalence of
Poisson manifolds amounts to isomorphism of objects in this category.
This description paves the way for the quantization of the classical
reduction procedure, which is based on the formal analogy between dual pairs of
Poisson manifolds and Hilbert bimodules over C*-algebras, as well as with
correspondences between von Neumann algebras. Further analogies are drawn with
categories of groupoids (of algebraic, measured, Lie, and symplectic type). In
all cases, the arrows are isomorphism classes of appropriate bimodules, and
their composition may be seen as a tensor product. Hence in suitable categories
reduction is simply composition of arrows, and Morita equivalence is
isomorphism of objects.Comment: 44 pages, categorical interpretation adde