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# Unoriented and oriented Kontsevich graph cocycles: Finding infinitesimal deformations of Poisson structures

Rutten, N. J. (2018) Unoriented and oriented Kontsevich graph cocycles: Finding infinitesimal deformations of Poisson structures. Bachelor's Thesis, Mathematics.

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## Abstract

This bachelor thesis is concerned with finding infinitesimal deformations of Poisson structures by using the unoriented and oriented graph complexes introduced by M. Kontsevich. We first give a short historical introduction to deformation theory and its development in general. The thesis is organized in four chapters. Chapter 1 consists of the published paper (joint with R.Buring and A.V.Kiselev) about the unoriented graph complex and cocycles therein. In Chapter 2 we give rigorous proofs of several statements that have already been used in the first chapter (as well as in the literature it is based on). Chapter 3 consists of the paper (joint with R.Buring and A.V.Kiselev) where we present many algorithms that are used in the search for cocycles in the oriented graph complex. Chapter 4 contains another joint paper where the cocycle Or(gamma5) in the oriented graph complex is obtained and the respective deformation is given explicitly. Definitions of the unoriented and oriented graph complexes with their differentials are recalled in Chapters 1 and 3. In Chapter 2 we first show that the differential d of the unoriented graph complex satisfies the defining property of a differential: d d = 0. Secondly we show that the differential applied to a "zero" graph is zero. Also the definition of the Lie bracket on the unoriented graph complex is recalled. We prove that the Lie bracket applied to a "zero" graph is zero. This result is a necessary condition for the Lie bracket to be a well defined map on the quotient space of of formal sums of graphs with an ordered set of edges modulo the equivalence relation induced by the wedge product.

Item Type: Thesis (Bachelor's Thesis) Mathematics Bachelor's Thesis English 15 Feb 2018 08:35 15 Feb 2018 08:35 https://fse.studenttheses.ub.rug.nl/id/eprint/16423

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