Fusion and fission in graph complexes

Details Fusion and fission in graph complexes Fusion and fission in graph complexes

2002-08-12

arxiv.org/abs/math/0208093v3

Pac. J. Math, Vol. 209, No. 2, (2003) 219-230

Mathematics

Quantum Algebra

This is the final version. The published version, which is slightly different, is available at http://nyjm.albany.edu:8000/Pac

Scientific paper

We analyze a functor from cyclic operads to chain complexes first considered by Getzler and Kapranov and also Markl. This functor is a generalization of the graph homology considered by Kontsevich, which was defined for the three operads Comm, Assoc, and Lie. More specifically we show that these chain complexes have a rich algebraic structure in the form of families of operations defined by fusion and fission. These operations fit together to form uncountably many Lie-infinity and co-Lie-infinity structures. In particular, the chain complexes have a bracket and cobracket which are compatible in the Lie bialgebra sense on a certain natural subcomplex.

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