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Configuration spaces, props and wheel-free deformation quantizationPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2016 (English)Doctoral thesis, monograph (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Stockholm: Department of Mathematics, Stockholm University , 2016. , p. 139
##### Keywords [en]

operad, properad, prop, deformation quantization, configuration space, Lie bialgebra, Poisson manifold
##### National Category

Geometry Algebra and Logic
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:su:diva-136025ISBN: 978-91-7649-635-0 (print)ISBN: 978-91-7649-636-7 (print)OAI: oai:DiVA.org:su-136025DiVA, id: diva2:1050523
##### Public defence

2017-01-20, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
##### Opponent

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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt530",{id:"formSmash:j_idt530",widgetVar:"widget_formSmash_j_idt530",multiple:true}); Available from: 2016-12-28 Created: 2016-11-29 Last updated: 2016-12-29Bibliographically approved

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

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

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 *s**uper-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

isbn
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