Reference : Prop of ribbon hypergraphs and strongly homotopy involutive Lie bialgebras
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
Prop of ribbon hypergraphs and strongly homotopy involutive Lie bialgebras
Merkulov, Sergei mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
International Mathematics Research Notices
Oxford University Press
Yes (verified by ORBilu)
United Kingdom
[en] Lie bialgebra ; string topology ; graph complexes
[en] For any integer d we introduce a prop RHrad of d-oriented ribbon hypergraphs (in which "edges" can connect more than two vertices) and prove that there exists a canonical morphism Holieb⋄d⟶RHrad from the minimal resolution Holieb⋄d of the (degree shifted) prop of involutive Lie bialgebras into the prop of ribbon hypergraphs which is non-trivial on each generator of Holieb⋄d. As an application we show that for any graded vector space W equipped with a family of cyclically (skew)symmetric higher products the associated vector space of cyclic words in elements of W has a combinatorial Holieb⋄d-structure. As an illustration we construct for each natural number N≥1 an explicit combinatorial strongly homotopy involutive Lie bialgebra structure on the vector space of cyclic words in N graded letters which extends the well-known Schedler's necklace Lie bialgebra structure from the formality theory of the Goldman-Turaev Lie bialgebra in genus zero.

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