On a theorem of Kontsevich

By James F. Conant, Karen Vogtmann. 2002

148 citations recorded by the source corpus.

In [12] and [13], M. Kontsevich introduced graph homology as a tool to compute the homology of three infinite dimensional Lie algebras, associated to the three operads "commutative," "associative" and "Lie." We generalizehis theorem to all cyclic operads, in the process giving a more careful treatment of the construction than in Kontsevich's original papers. We also give a more explicit treatment of the isomorphisms of graph homologies with the homology of moduli space and Out(F r ) outlined by Kontsevich. In [4] we defined a Lie bracket and cobracket on the commutative graph complex, which was extended in [3] to the case of all cyclic operads. These operations form a Lie bi-algebra on a natural subcomplex. We show that in the associative and Lie cases the subcomplex on which the bi-algebra structure exists carries all of the homology, and we explain why the subcomplex in the commutative case does not.

Subfields: Algebra and Number Theory, Mathematical Physics, Pathology and Forensic Medicine

View Paper: https://www.semanticscholar.org/paper/28f273b813579c706af11afe4cb43f90ba5857f1

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In Algebra and Number Theory

In Mathematical Physics

In Pathology and Forensic Medicine

Citations in the corpus

Hodge‐type decomposition in the homology of long knots (2010, background): “Similar graph-complexes appear in the study of the homology of the outer spaces [11, 12, 17, 19, 24].”
Cohomological structure of the mapping class group and beyond (2005, background): “See also the paper [9] by Conant and Vogtmann for a detailed proof as well as discussion of this theorem in the context of cyclic operads.”
Hairy graphs and the unstable homology of Mod(g, s), Out(Fn) and Aut(Fn) (2011, methodology): “In [3] Conant and Vogtmann generalized Kontsevich’s construction to define a Lie algebra of symplectic derivations and an associated O-graph homology for any cyclic operad O of finite-dimensional vector spaces.”
On some anticyclic operads (2004, background): “One interesting point with cyclic and anticyclic operads is that they allow the building of Lie algebras and graph complexes [6, 4, 12].”
Wheeled PROPs, graph complexes and the master equation (2006, unknown): “2 in [4] for full details and pictures); • or(G) = (or1(G) ∧ e)⊗ or2(G).”