12,359 research outputs found
-Manifolds and Mackenzie Theory
Double Lie algebroids were discovered by Kirill Mackenzie from the study of
double Lie groupoids and were defined in terms of rather complicated conditions
making use of duality theory for Lie algebroids and double vector bundles. In
this paper we establish a simple alternative characterization of double Lie
algebroids in a supermanifold language. Namely, we show that a double Lie
algebroid in Mackenzie's sense is equivalent to a double vector bundle endowed
with a pair of commuting homological vector fields of appropriate weights. Our
approach helps to simplify and elucidate Mackenzie's original definition; we
show how it fits into a bigger picture of equivalent structures on `neighbor'
double vector bundles. It also opens ways for extending the theory to multiple
Lie algebroids, which we introduce here.Comment: This is a substantial re-work of our earlier paper
arXiv:math.DG/0608111. In particular, we included various details as well as
some new statements that may have independent interes
Invariant manifolds and the geometry of front propagation in fluid flows
Recent theoretical and experimental work has demonstrated the existence of
one-sided, invariant barriers to the propagation of reaction-diffusion fronts
in quasi-two-dimensional periodically-driven fluid flows. These barriers were
called burning invariant manifolds (BIMs). We provide a detailed theoretical
analysis of BIMs, providing criteria for their existence, a classification of
their stability, a formalization of their barrier property, and mechanisms by
which the barriers can be circumvented. This analysis assumes the sharp front
limit and negligible feedback of the front on the fluid velocity. A
low-dimensional dynamical systems analysis provides the core of our results.Comment: 14 pages, 11 figures. To appear in Chaos Focus Issue:
Chemo-Hydrodynamic Patterns and Instabilities (2012
Partial inner product spaces: Some categorical aspects
We make explicit in terms of categories a number of statements from the
theory of partial inner product spaces (PIP spaces) and operators on them.
In particular, we construct sheaves and cosheaves of operators on certain PIP
spaces of practical interest.Comment: 21 page
Hamiltonian Loop Group Actions and T-Duality for group manifolds
We carry out a Hamiltonian analysis of Poisson-Lie T-duality based on the
loop geometry of the underlying phases spaces of the dual sigma and WZW models.
Duality is fully characterized by the existence of equivariant momentum maps on
the phase spaces such that the reduced phase space of the WZW model and a pure
central extension coadjoint orbit work as a bridge linking both the sigma
models. These momentum maps are associated to Hamiltonian actions of the loop
group of the Drinfeld double on both spaces and the duality transformations are
explicitly constructed in terms of these actions. Compatible dynamics arise in
a general collective form and the resulting Hamiltonian description encodes all
known aspects of this duality and its generalizations.Comment: 34 page
Boolean Coverings of Quantum Observable Structure: A Setting for an Abstract Differential Geometric Mechanism
We develop the idea of employing localization systems of Boolean coverings,
associated with measurement situations, in order to comprehend structures of
Quantum Observables. In this manner, Boolean domain observables constitute
structure sheaves of coordinatization coefficients in the attempt to probe the
Quantum world. Interpretational aspects of the proposed scheme are discussed
with respect to a functorial formulation of information exchange, as well as,
quantum logical considerations. Finally, the sheaf theoretical construction
suggests an opearationally intuitive method to develop differential geometric
concepts in the quantum regime.Comment: 25 pages, Late
Explaining Gabriel-Zisman localization to the computer
This explains a computer formulation of Gabriel-Zisman localization of
categories in the proof assistant Coq. It includes both the general
localization construction with the proof of GZ's Lemma 1.2, as well as the
construction using calculus of fractions. The proof files are bundled with the
other preprint "Files for GZ localization" posted simultaneously
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