I define higher codimensional versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher codimensional versions of pre-contact structures as distributions on manifolds whose characteristic symmetries span a constant dimensional distribution. I call them pre-multicontact structures. Every distribution is almost everywhere, locally, a pre-multicontact structure. After showing that the standard symplectization of contact manifolds generalizes naturally to a (pre-)multisymplectization of (pre-)multicontact manifolds, I make use of results by C. Rogers and M. Zambon to associate a canonical L∞-algebra to any (pre-)multicontact structure. Such L∞-algebra is a multicontact version of the Jacobi bracket on a contact manifold. However, unlike the multisymplectic L∞-algebra of Rogers and Zambon, the multicontact L∞-algebra is always a homological resolution of a Lie algebra. Finally, I describe in local coordinates the L∞-algebra associated to the Cartan distribution on jet spaces.

L∞-algebras from multicontact geometry

VITAGLIANO, LUCA
2015

Abstract

I define higher codimensional versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher codimensional versions of pre-contact structures as distributions on manifolds whose characteristic symmetries span a constant dimensional distribution. I call them pre-multicontact structures. Every distribution is almost everywhere, locally, a pre-multicontact structure. After showing that the standard symplectization of contact manifolds generalizes naturally to a (pre-)multisymplectization of (pre-)multicontact manifolds, I make use of results by C. Rogers and M. Zambon to associate a canonical L∞-algebra to any (pre-)multicontact structure. Such L∞-algebra is a multicontact version of the Jacobi bracket on a contact manifold. However, unlike the multisymplectic L∞-algebra of Rogers and Zambon, the multicontact L∞-algebra is always a homological resolution of a Lie algebra. Finally, I describe in local coordinates the L∞-algebra associated to the Cartan distribution on jet spaces.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11386/4622257
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