Prop profiles of compatible Poisson and Nijenhuis structures
A prop profile of a differential geometric structure is a minimal resolution of an algebraic prop such that representations of this resolution are in one-to-one correspondence with structures of the given type. We begin this thesis with a detailed account of the algebraic tools necessary to construct prop profiles; we treat operads and props, and resolutions of these through Koszul duality.
Our main results can be summarized as follows.
Firstly, we contribute to the work of S.A. Merkulov on the prop profiles of Poisson and Nijenhuis structures. We prove that the operad of the latter prop profile is Koszul by showing that it has a PBW-basis, and we provide a geometrical interpretation of the former in terms of an L-infinity structure on the structure sheaf of a manifold.
Secondly, we construct prop profiles of compatible Poisson and Nijenhuis structures. Representations of minimal resolutions of props correspond to Maurer-Cartan elements of certain Lie algebras associated to the resolved props. Also the differential geometric structures are defined as solutions of Maurer-Cartan equations. We show the correspondence between props and differential geometry by providing explicit isomorphisms between these Lie algebras.
Thirdly, in order to construct the prop profiles of compatible Poisson and Nijenhuis structures we study operads of compatible algebraic structures. By studying Cohen-Macaulay properties of posets associated to such operads we prove the Koszulness of a large class of operads of compatible structures.
Source Type:Doctoral Dissertation
Keywords:MATHEMATICS; Koszul duality; resolutions; props; operads; compatible structures; Mathematics; matematik
Date of Publication:01/01/2009