Construction of some operads and bialgebras associated to Stasheff polytopes and hypercubes (Q2750943)

Homotopy associative algebras are described by the operad \(A_{\infty}\), whose underlying complex is the sum of the Stasheff polytopes (associahedra). In dimension zero, the cells of the Stasheff polytope are planar binary rooted trees. These trees index a basis of the free dendriform algebra with one generator. The operad of dendriform algebras had been defined by \textit{J.-L. Loday} [Lect. Notes Math. 1763, 7--66 (2001; Zbl 0999.17002)], and \textit{J.-L. Loday} and \textit{M. O. Ronco} [Adv. Math. 139, 293--309 (1998; Zbl 0926.16032)] have shown that the free algebras in this operad are bialgebras.NEWLINENEWLINEIn earlier work [\textit{F. Chapoton}, Ann. Inst. Fourier 50, 1127--1153 (2000; Zbl 0963.16032); Adv. Math. 150, 264--275 (2000; Zbl 0958.16038)] the author of the article under review has constructed bialgebras with bases indexed by (not necessarily binary) planar rooted trees, which are also indexing the higher-dimensional cells of the Stasheff polytope.NEWLINENEWLINEThe article under review constructs for each of these bialgebras an operad such that the bialgebras are the free algebras over these operads. Other bialgebras are constructed in a similar way with hypercubes instead of Stasheff polytopes. Here, Solomon's descent algebra [see \textit{C. Malvenuto} and \textit{C. Reutenauer}, J. Algebra 177, 967--982 (1995; Zbl 0838.05100) for the Hopf algebra structure] replaces the bialgebra of planar binary trees.