Prop of Ribbon Hypergraphs and Strongly Homotopy Involutive Lie Bialgebras

Abstract

Abstract For any integer $d$, we introduce a prop ${{\mathcal {R}}}{{\mathcal {H}}} ra_d$ of oriented ribbon hypergraphs (in which “edges” can connect more than two vertices) and prove that it admits a canonical morphism of props, $$ \begin{align*} & {\mathcal{H}}\textit{olieb}_{d}^\diamond \longrightarrow {{\mathcal{R}}}{{\mathcal{H}}} ra_d, \end{align*}$$${\mathcal {H}}\textit {olieb}_{d}^\diamond $ being the (degree shifted) minimal resolution of prop of involutive Lie bialgebras, which is nontrivial on every generator of ${\mathcal {H}}\textit {olieb}_{d}^\diamond $. We obtain two applications of this general construction. First we show that for any graded vector space $W$ equipped with a family of cyclically (skew)symmetric higher products, $$ \begin{align*} & \Theta_n: (\otimes^n W[d])_{{{\mathbb{Z}}}_n} \longrightarrow {{\mathbb{K}}}[1+d], \ \ \ \ \ n\geq 1, \end{align*}$$the associated vector space of cyclic words $Cyc(W)=\oplus _{n\geq 0} (\otimes ^n W)_{{{\mathbb {Z}}}_n} $ has a combinatorial ${\mathcal {H}}\textit {olieb}_{d}^\diamond $-structure. As an illustration, we construct for each natural number $N\geq 1$ 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. Second, we introduced new (in general, nontrivial) operations in string topology. Given any closed connected and simply connected manifold $M$ of dimension $\geq 4$. We show that the reduced equivariant homology $\bar {H}_{\bullet }^{S^1}(LM)$ of the spacej $LM$ of free loops in $M$ carries a canonical representation of the dg prop ${\mathcal {H}}\textit {olieb}^\diamond _{2-n}$ on $\bar {H}_{\bullet }^{S^1}(LM)$ controlled by four ribbon hypergraphs explicitly shown in this paper.