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Operad profiles of Nijenhuis structures
Stockholm University, Faculty of Science, Department of Mathematics.
2009 (English)In: Differential geometry and its applications (Print), ISSN 0926-2245, E-ISSN 1872-6984, Vol. 27, no 6, 780-792 p.Article in journal (Refereed) Published
Abstract [en]

Recently S. Merkulov [S.A. Merkulov, Operads, deformation theory and F-manifolds, in: Frobenius manifolds, in: Aspects Math., vol. E36, Vieweg, Wiesbaden, 2004. pp. 213-251; S.A. Merkulov, Nijenhuis infinity and contractible differential graded manifolds, Compos. Math. 141 (5) (2005) 1238-1254; S.A. Merkulov, Prop profile of Poisson geometry, Comm. Math. Phys. 262 (2006) 117-135] established a new link between differential geometry and homological algebra by giving descriptions of several differential geometric structures in terms of algebraic operads and props. In particular he described Nijenhuis structures as corresponding to representations of the cobar construction on the Koszul dual of a certain quadratic operad. In this paper we prove, using the PBW-basis method of E. Hoffbeck [E. Hoffbeck, A Poincare-Birkhoff-Witt criterion for Koszul operads, arXiv:0709.2286v3 [math.AT], 2008], that the operad governing Nijenhuis structures is Koszul, thereby showing that Nijenhuis structures correspond to representations of the minimal resolution of this operad. We also construct an operad such that representations of its minimal resolution in a vector space V are in one-to-one correspondence with pairs of compatible Nijenhuis structures on the formal manifold associated to V. 

Place, publisher, year, edition, pages
2009. Vol. 27, no 6, 780-792 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-59650DOI: 10.1016/j.difgeo.2009.03.014ISI: 000271806700010OAI: oai:DiVA.org:su-59650DiVA: diva2:432257
Note
authorCount :1Available from: 2011-08-02 Created: 2011-07-05 Last updated: 2017-12-08Bibliographically approved

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