32 research outputs found
Integrability of central extensions of the Poisson Lie algebra via prequantization
We present a geometric construction of central S^1-extensions of the quantomorphism group of a prequantizable, compact, symplectic manifold, and explicitly describe the corresponding lattice of integrable cocycles on the Poisson Lie algebra. We use this to find nontrivial central S^1-extensions of the universal cover of the group of Hamiltonian diffeomorphisms. In the process, we obtain central S^1-extensions of Lie groups that act by exact strict contact transformations
Geodesic Equations on Diffeomorphism Groups
We bring together those systems of hydrodynamical type that can be written as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups with right invariant L² or H¹ metrics. We present their formal derivation starting from Euler's equation, the first order equation satisfied by the right logarithmic derivative of a geodesic in Lie groups with right invariant metrics
Generalized Euler-Poincar\'e equations on Lie groups and homogeneous spaces, orbit invariants and applications
We develop the necessary tools, including a notion of logarithmic derivative
for curves in homogeneous spaces, for deriving a general class of equations
including Euler-Poincar\'e equations on Lie groups and homogeneous spaces.
Orbit invariants play an important role in this context and we use these
invariants to prove global existence and uniqueness results for a class of PDE.
This class includes Euler-Poincar\'e equations that have not yet been
considered in the literature as well as integrable equations like Camassa-Holm,
Degasperis-Procesi, CH and DP equations, and the geodesic equations
with respect to right invariant Sobolev metrics on the group of diffeomorphisms
of the circle
Vlasov moment flows and geodesics on the Jacobi group
By using the moment algebra of the Vlasov kinetic equation, we characterize
the integrable Bloch-Iserles system on symmetric matrices
(arXiv:math-ph/0512093) as a geodesic flow on the Jacobi group. We analyze the
corresponding Lie-Poisson structure by presenting a momentum map, which both
untangles the bracket structure and produces particle-type solutions that are
inherited from the Vlasov-like interpretation. Moreover, we show how the Vlasov
moments associated to Bloch-Iserles dynamics correspond to particular subgroup
inclusions into a group central extension (first discovered in
arXiv:math/0410100), which in turn underlies Vlasov kinetic theory. In the most
general case of Bloch-Iserles dynamics, a generalization of the Jacobi group
also emerges naturally.Comment: 45 page
Central extensions of groups of sections
If q : P -> M is a principal K-bundle over the compact manifold M, then any
invariant symmetric V-valued bilinear form on the Lie algebra k of K defines a
Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms
modulo exact forms. In the present paper we analyze the integrability of this
extension to a Lie group extension for non-connected, possibly
infinite-dimensional Lie groups K. If K has finitely many connected components
we give a complete characterization of the integrable extensions. Our results
on gauge groups are obtained by specialization of more general results on
extensions of Lie groups of smooth sections of Lie group bundles. In this more
general context we provide sufficient conditions for integrability in terms of
data related only to the group K.Comment: 54 pages, revised version, to appear in Ann. Glob. Anal. Geo
The Dynamics of a Rigid Body in Potential Flow with Circulation
We consider the motion of a two-dimensional body of arbitrary shape in a
planar irrotational, incompressible fluid with a given amount of circulation
around the body. We derive the equations of motion for this system by
performing symplectic reduction with respect to the group of volume-preserving
diffeomorphisms and obtain the relevant Poisson structures after a further
Poisson reduction with respect to the group of translations and rotations. In
this way, we recover the equations of motion given for this system by Chaplygin
and Lamb, and we give a geometric interpretation for the Kutta-Zhukowski force
as a curvature-related effect. In addition, we show that the motion of a rigid
body with circulation can be understood as a geodesic flow on a central
extension of the special Euclidian group SE(2), and we relate the cocycle in
the description of this central extension to a certain curvature tensor.Comment: 28 pages, 2 figures; v2: typos correcte
Shifted Symplectic Structures
This is the first of a series of papers about \emph{quantization} in the
context of \emph{derived algebraic geometry}. In this first part, we introduce
the notion of \emph{-shifted symplectic structures}, a generalization of the
notion of symplectic structures on smooth varieties and schemes, meaningful in
the setting of derived Artin n-stacks. We prove that classifying stacks of
reductive groups, as well as the derived stack of perfect complexes, carry
canonical 2-shifted symplectic structures. Our main existence theorem states
that for any derived Artin stack equipped with an -shifted symplectic
structure, the derived mapping stack is equipped with a
canonical -shifted symplectic structure as soon a satisfies a
Calabi-Yau condition in dimension . These two results imply the existence of
many examples of derived moduli stacks equipped with -shifted symplectic
structures, such as the derived moduli of perfect complexes on Calabi-Yau
varieties, or the derived moduli stack of perfect complexes of local systems on
a compact and oriented topological manifold. We also show that Lagrangian
intersections carry canonical (-1)-shifted symplectic structures.Comment: 52 pages. To appear in Publ. Math. IHE
Turbulence and Holography
We examine the interplay between recent advances in quantum gravity and the
problem of turbulence. In particular, we argue that in the gravitational
context the phenomenon of turbulence is intimately related to the properties of
spacetime foam. In this framework we discuss the relation of turbulence and
holography and the interpretation of the Kolmogorov scaling in the quantum
gravitational setting.Comment: 19 pages, LaTeX; version 2: reference adde
Geometric dynamics on the automorphism group of principal bundles: geodesic flows, dual pairs and chromomorphism groups
We formulate Euler-Poincar____'e equations on the Lie group Aut(P) of automorphisms of a principal bundle P. The corresponding flows are referred to as EPAut flows. We mainly focus on geodesic flows associated to Lagrangians of Kaluza-Klein type. In the special case of a trivial bundle P, we identify geodesics on certain infinite-dimensional semidirect-product Lie groups that emerge naturally from the construction. This approach leads naturally to a dual pair structure containing ____delta-like momentum map solutions that extend previous results on geodesic flows on the diffeomorphism group (EPDiff). In the second part, we consider incompressible flows on the Lie group of volume-preserving automorphisms of a principal bundle. In this context, the dual pair construction requires the definition of chromomorphism groups, i.e. suitable Lie group extensions generalizing the quantomorphism group