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Configuration spaces, props and wheel-free deformation quantizationBackman, Theo January 2016 (has links)
The main theme of this thesis is higher algebraic structures that come from operads and props. The first chapter is an introduction to the mathematical framework needed for the content of this thesis. The chapter does not contain any new results. The second chapter is concerned with the construction of a configuration space model for a particular 2-colored differential graded operad encoding the structure of two A∞ algebras with two A∞ morphisms and a homotopy between the morphisms. The cohomology of this operad is shown to be the well-known 2-colored operad encoding the structure of two associative algebras and of an associative algebra morphism between them. The third chapter is concerned with deformation quantization of (potentially) infinite dimensional (quasi-)Poisson manifolds. Our proof employs a variation on the transcendental methods pioneered by M. Kontsevich for the finite dimensional case. The first proof of the infinite dimensional case is due to B. Shoikhet. A key feature of the first proof is the construction of a universal L∞ structure on formal polyvector fields. Our contribution is a simplification of B. Shoikhet proof by considering a more natural configuration space and a simpler choice of propagator. The result is also put into a natural context of the dg Lie algebras coming from graph complexes; the L∞ structure is proved to come from a Maurer-Cartan element in the oriented graph complex. The fourth chapter also deals with deformation quantization of (quasi-)Poisson structures in the infinite dimensional setting. Unlike the previous chapter, the methods used here are purely algebraic. Our main theorem is the possibility to deformation quantize quasi-Poisson structures by only using perturbative methods; in contrast to the transcendental methods employed in the previous chapter. We give two proofs of the theorem via the theory of dg operads, dg properads and dg props. We show that there is a dg prop morphism from a prop governing star-products to a dg prop(erad) governing (quasi-)Poisson structures. This morphism gives a theorem about the existence of a deformation quantization of (quasi-)Poisson structure. The proof proceeds by giving an explicit deformation quantization of super-involutive Lie bialgebras and then lifting that to the dg properad governing quasi-Poisson structures. The prop governing star-products was first considered by S.A. Merkulov, but the properad governing quasi-Poisson structures is a new construction. The second proof of the theorem employs the Merkulov-Willwacher polydifferential functor to transfer the problem of finding a morphism of dg props to that of finding a morphism of dg operads.We construct an extension of the well known operad of A∞ algebras such that the representations of it in V are equivalent to an A∞ structure on V[[ħ]]. This new operad is also a minimal model of an operad that can be seen as the extension of the operad of associative algebras by a unary operation. We give an explicit map of operads from the extended associative operad to the operad we get when applying the Merkulov-Willwacher polydifferential functor to the properad of super-involutive Lie bialgebras. Lifting this map so as to go between their respective models gives a new proof of the main theorem.
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[pt] A GEOMETRIA DE ESPAÇOS DE POLÍGONOS GENERALIZADOS / [en] THE GEOMETRY OF GENERALIZED POLYGON SPACESRAIMUNDO NETO NUNES LEAO 17 June 2021 (has links)
[pt] Espaços de moduli de polígonos em R(3) com comprimento dos lados fixados é um exemplo amplamente estudado de variedade simplética. Esses
espaços podem ser descritos como quociente simplético de um número finito
de órbitas coadjuntas pelo grupo SU(2). Nesta tese esses espaços de moduli
são identificados como folhas simpléticas de uma variedade de Poisson que
pode ser construída como quociente. Essa construção é a seguir generalizada
ao caso de um produto de um número finito de órbitas coadjuntas pelo grupo
SU(n), e o resultado principal desse trabalho de tese descreve como esses
espaços de moduli de polígonos generalizados formam uma folheação em
folhas simpléticas de uma variedade de Poisson. / [en] Moduli spaces of polygons in R(3)with fixed sides length are a
widely studied example of symplectic manifold that can be described as the
symplectic quotient of a finite number of SU(2)−coadjoint orbits by the
diagonal action of the group SU(2). In this thesis these spaces are identified
as the symplectic leaves of a Poisson manifold, that can itself be obtained
by a quotient procedure. The construction is then generalized to the case of
the quotient of a product of finitely many SU(n)−coadjoint orbits by the
diagonal action of SU(n), and the main result of this thesis describes how
these moduli spaces of generalized polygons fit together as the symplectic
leaves of a quotient Poisson manifold.
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