A duality between standard simplices and Stasheff polytopes (Q2745762)

Firstly, the authors describe how to associate an operad structure to the family of chain modules over the standard simplices, \({P}_\Delta(n)=C_*(\Delta^{n-1})\). The algebras over \({P}_\Delta\) are determined by 3 operations (called left operation, right operation and medium operation) and by 11 relations, and they are called associative trialgebras. Forgetting the medium operation, an associative trialgebra is simply an associative dialgebra [see \textit{J.-L. Loday, A. Frabetti, F. Chapoton} and \textit{F. Goichot}, ``Dialgebras and related operads'', Lect. Notes Math. 1763 (2001; Zbl 0970.00010)]. Identifying the three operations, an associative trialgebra is an associative algebra. NEWLINENEWLINENEWLINENext they show that the family of cochain modules of the Stasheff polytopes, \({P}^{K}(n)=C^*({K}^{n-1})\), has an operad structure. The algebras over \({P}^{K}\) are determined by 3 operations and 7 relations, and they are called dendriform trialgebras. Dendriform dialgebras and associative algebras are particular cases of dendriform trialgebras. NEWLINENEWLINENEWLINEIt is also proved that the introduced operads are dual to each other in the operadic sense [see \textit{V. Ginzburg} and \textit{M. Kapranov}, Duke Math. J. 76, No.~1, 203--272 (1994; Zbl 0855.18006)]. This fact is used to prove the main result of the paper: Both operads are Koszul operads, that is, their Koszul complexes are acyclic.