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Title:
Prop of ribbon hypergraphs and strongly homotopy involutive Lie bialgebras
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Abstract:
We introduce a new prop of ribbon hypergraphs (in which "edges” can connect more than two vertices), HGra, and prove that there exists a canonical morphism from the minimal resolution of the prop of Lie bialgebras into HGra which is non-trivial on all generators of the latter. We show that for any given a graded vector space W equipped with a family (for all n greater than or equal to 2) of homogeneous cyclically symmetric ``n-valent” scalar products there is an associated representation of HGra in the vector space of ``cyclic words” in elements of W. The latter observation implies, in particular, the following result: given any graded vector space V with a fixed basis, there is a family of strongly homotopy Lie bialgebra structures (with all higher strongly homotopy involutive Lie bialgebra operations non-trivial in general) on certain vector space of cyclic words built from elements V. This result generalizes a well-known necklace Lie bialgebra construction by T. Schedler which plays an important role in the theory of free homotopy classes of loops on Riemann surfaces with punctures.
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