446 research outputs found
On the deformation quantization of symplectic orbispaces
In the first part of this article we provide a geometrically oriented
approach to the theory of orbispaces which originally had been introduced by
Chen. We explain the notion of a vector orbibundle and characterize the good
sections of a reduced vector orbibundle as the smooth stratified sections. In
the second part of the article we elaborate on the quantizability of a
symplectic orbispace. By adapting Fedosov's method to the orbispace setting we
show that every symplectic orbispace has a deformation quantization. As a
byproduct we obtain that every symplectic orbifold possesses a star product
Holomorphic deformation of Hopf algebras and applications to quantum groups
In this article we propose a new and so-called holomorphic deformation scheme
for locally convex algebras and Hopf algebras. Essentially we regard converging
power series expansion of a deformed product on a locally convex algebra, thus
giving the means to actually insert complex values for the deformation
parameter. Moreover we establish a topological duality theory for locally
convex Hopf algebras. Examples coming from the theory of quantum groups are
reconsidered within our holomorphic deformation scheme and topological duality
theory. It is shown that all the standard quantum groups comprise holomorphic
deformations. Furthermore we show that quantizing the function algebra of a
(Poisson) Lie group and quantizing its universal enveloping algebra are
topologically dual procedures indeed. Thus holomorphic deformation theory seems
to be the appropriate language in which to describe quantum groups as deformed
Lie groups or Lie algebras.Comment: 40 page
The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs
In this paper, we study the formal solution space of a nonlinear PDE in a
fiber bundle. To this end, we start with foundational material and introduce
the notion of a pfd structure to build up a new concept of profinite
dimensional manifolds. We show that the infinite jet space of the fiber bundle
is a profinite dimensional manifold in a natural way. The formal solution space
of the nonlinear PDE then is a subspace of this jet space, and inherits from it
the structure of a profinite dimensional manifold, if the PDE is formally
integrable. We apply our concept to scalar PDEs and prove a new criterion for
formal integrability of such PDEs. In particular, this result entails that the
Euler-Lagrange equation of a relativistic scalar field with a polynomial
self-interaction is formally integrable
Phase Space Reduction of Star Products on Cotangent Bundles
In this paper we construct star products on Marsden-Weinstein reduced spaces
in case both the original phase space and the reduced phase space are
(symplectomorphic to) cotangent bundles. Under the assumption that the original
cotangent bundle carries a symplectique structure of form
with a closed two-form on , is
equipped by the cotangent lift of a proper and free Lie group action on and
by an invariant star product that admits a -equivariant quantum momentum
map, we show that the reduced phase space inherits from a star product.
Moreover, we provide a concrete description of the resulting star product in
terms of the initial star product on and prove that our reduction scheme
is independent of the characteristic class of the initial star product. Unlike
other existing reduction schemes we are thus able to reduce not only strongly
invariant star products. Furthermore in this article, we establish a relation
between the characteristic class of the original star product and the
characteristic class of the reduced star product and provide a classification
up to -equivalence of those star products on , which
are invariant with respect to a lifted Lie group action. Finally, we
investigate the question under which circumstances `quantization commutes with
reduction' and show that in our examples non-trivial restrictions arise
Differentiable stratified groupoids and a de Rham theorem for inertia spaces
We introduce the notions of a differentiable groupoid and a differentiable
stratified groupoid, generalizations of Lie groupoids in which the spaces of
objects and arrows have the structures of differentiable spaces, respectively
differentiable stratified spaces, compatible with the groupoid structure. After
studying basic properties of these groupoids including Morita equivalence, we
prove a de Rham theorem for locally contractible differentiable stratified
groupoids. We then focus on the study of the inertia groupoid associated to a
proper Lie groupoid. We show that the loop and the inertia space of a proper
Lie groupoid can be endowed with a natural Whitney B stratification, which we
call the orbit Cartan type stratification. Endowed with this stratification,
the inertia groupoid of a proper Lie groupoid becomes a locally contractible
differentiable stratified groupoid
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