arXiv subjects

Marco Zambon

Publications and source records attributed to Marco Zambon.

50 records · Page 3Linked to original sources

arXiv2012

Higher Lie algebra actions on Lie algebroids

We consider a simple instance of action up to homotopy. More precisely, we consider strict actions of DGLAs in degrees -1 and 0 on degree 1 NQ-manifolds. In a more conventional language this means: strict actions of Lie algebra crossed modules on Lie algebroids. When the action is strict, we show that it integrates to group actions in the categories of Lie algebroids and Lie groupoids (i.e. actions of LA-groups and 2-groups). We perform the integration in the framework of Mackenzie's doubles.

math.DG↗

arXiv2012

Distributions and quotients on degree 1 NQ-manifolds and Lie algebroids

It is well-known that a Lie algebroid A is equivalently described by a degree 1 Q-manifold M. We study distributions on M, giving a characterization in terms of A. We show that involutive Q-invariant distributions on M correspond bijectively to IM-foliations on A (the infinitesimal version of Mackenzie's ideal systems). We perform reduction by such distributions, and investigate how they arise from non-strict actions of strict Lie 2-algebras on M.

math.DG↗

arXiv2011

Smoothness of holonomy covers for singular foliations and essential isotropy

Given a singular foliation, we attach an "essential isotropy" group to each of its leaves, and show that its discreteness is the integrability obstruction of a natural Lie algebroid over the leaf. We show that a condition ensuring discreteness is the local surjectivity of a transversal exponential map associated with the maximal ideal of vector fields prescribed to be tangent to the foliation. The essential isotropy group is also shown to control the smoothness of the holonomy cover of the leaf (the associated fiber of the holonomy groupoid), as well as the {smoothness} of the associated isotropy group. Namely, the (topological) closeness of the essential isotropy group is a necessary and sufficient condition for the holonomy cover to be a smooth (finite-dimensional) manifold and the isotropy group to be a Lie group. These results are useful towards understanding the normal form of a singular foliation around a compact leaf. At the end of this article we briefly outline work of ours on this normal form, to be presented in a subsequent paper.

math.DG↗

arXiv2011

L-infinity algebras and higher analogues of Dirac structures and Courant algebroids

We define a higher analogue of Dirac structures on a manifold M. Under a regularity assumption, higher Dirac structures can be described by a foliation and a (not necessarily closed, non-unique) differential form on M, and are equivalent to (and simpler to handle than) the Multi-Dirac structures recently introduced in the context of field theory by Vankerschaver, Yoshimura and Marsden. We associate an L-infinity algebra of "observables" to every higher Dirac structure, extending work of Baez, Hoffnung and Rogers on multisymplectic forms. Further, applying a recent result of Getzler, we associate an L-infinity algebra to any manifold endowed with a closed differential form H, via a "higher analogue of Courant algebroid twisted by H". Finally, we study the relations between the L-infinity algebras appearing above.

math.SG↗

arXiv2010

Graded geometry and Poisson reduction

We previously extended the Marsden-Ratiu reduction theorem in Poisson geometry by means of graded geometry (see Part I of Arxiv:1009.0948) . In this note we provide the background material about graded geometry necessary for the proof. Further, we provide an alternative algebraic proof.

math.SG↗

arXiv2010

A construction for coisotropic subalgebras of Lie bialgebras

Given a Lie bialgebra (g,g*), we present an explicit procedure to construct coisotropic subalgebras, i.e. Lie subalgebras of g whose annihilator is a Lie subalgebra of g*. We write down families of examples for the case that g is a classical complex simple Lie algebra.

math.SG↗

arXiv2008

An extension of the Marsden-Ratiu reduction for Poisson manifolds

We propose a generalization of the reduction of Poisson manifolds by distributions introduced by Marsden and Ratiu. Our proposal overcomes some of the restrictions of the original procedure, and makes the reduced Poisson structure effectively dependent on the distribution. Different applications are discussed, as well as the algebraic interpretation of the procedure and its formulation in terms of Dirac structures.

math.SG↗

arXiv2008

Reduction of branes in generalized complex geometry

We show that certain submanifolds of generalized complex manifolds ("weak branes") admit a natural quotient which inherits a generalized complex structure. This is analog to quotienting coisotropic submanifolds of symplectic manifolds. In particular Gualtieri's generalized complex submanifolds ("branes") quotient to space-filling branes. Along the way we perform reductions by foliations (i.e. no group action is involved) for exact Courant algebroids - interpreting the reduced Ševera class - and for Dirac structures.

math.DG↗

arXiv2007

Pre-Poisson submanifolds

This is an expository and introductory note on some results obtained in "Coisotropic embeddings in Poisson manifolds" (ArXiv math/0611480). Some original material is contained in the last two sections, where we consider linear Poisson structures.

math.SG↗

arXiv2007

On the geometry of prequantization spaces

Given a Poisson (or more generally Dirac) manifold $P$, there are two approaches to its geometric quantization: one involves a circle bundle $Q$ over $P$ endowed with a Jacobi (or Jacobi-Dirac) structure; the other one involves a circle bundle with a (pre-) contact groupoid structure over the (pre-) symplectic groupoid of $P$. We study the relation between these two prequantization spaces. We show that the circle bundle over the (pre-) symplectic groupoid of $P$ is obtained from the groupoid of $Q$ via an $S^1$ reduction that preserves both the groupoid and the geometric structure.

math.DG↗

arXiv2005

Averaging of Legendrian submanifolds of contact manifolds

We give a procedure to ``average'' canonically $C^1$-close Legendrian submanifolds of contact manifolds. As a corollary we obtain that, whenever a compact group action leaves a Legendrian submanifold almost invariant, there is an invariant Legendrian submanifold nearby.

math.SG↗

arXiv2004

Variations on Prequantization

We extend known prequantization procedures for Poisson and presymplectic manifolds by defining the prequantization of a Dirac manifold P as a principal U(1)-bundle Q with a compatible Dirac-Jacobi structure. We study the action of Poisson algebras of admissible functions on P on various spaces of locally (with respect to P) defined functions on Q, via hamiltonian vector fields. Finally, guided by examples arising in complex analysis and contact geometry, we propose an extension of the notion of prequantization in which the action of U(1) on Q is permitted to have some fixed points.

math.SG↗

arXiv2004

Contact reduction and groupoid actions

We introduce a new method to perform reduction of contact manifolds that extends Willett's (math.SG/0104080) and Albert's results. To carry out our reduction procedure all we need is a complete Jacobi map $J$ from a contact manifold $M$ to a Jacobi manifold $Γ_0$. This naturally generates the action of the contact groupoid of $Γ_0$ on $M$, and we show that the quotients of fibers of $J$ by suitable Lie subgroups are either contact or locally conformal symplectic manifolds with structures induced by the one on $M$. We show that Willett's reduced spaces are prequantizations of our reduced spaces; hence the former are completely determined by the latter. Since a symplectic manifold is prequantizable iff the symplectic form is integral, this explains why Willett's reduction can be performed only at distinguished points. As an application we obtain Kostant's prequantizations of coadjoint orbits.

math.DG↗

arXiv2004

Submanifold averaging in riemannian and symplecitc geometry

We give a construction to obtain canonically an ``isotropic average'' of given $C^1$-close isotropic submanifolds of a symplectic manifold. To do so we use an improvement of Weinstein's submanifold averaging theorem (obtained in collaboration with H. Karcher) and apply ``Moser's trick''. We also present an application to Hamiltonian group actions.

math.DG↗