Tensor products of \(A_\infty\)-algebras with homotopy inner products (Q2847191)

The authors consider the following problem: whether the tensor product of two cyclic \(A_\infty\)-algebras is also cyclic. They show that the answer is in general negative, but the tensor product of two such algebras is an \(A_\infty\)-algebra with homotopy inner product.NEWLINENEWLINEFirst of all they show that a cyclic \(A_\infty\)-algebra can be considered as an \(A_\infty\)-algebra with homotopy inner products, defined by \textit{T. Tradler} [J. Homotopy Relat. Struct. 3, No. 1, 245--271 (2008; Zbl 1243.16008); erratum ibid. 6, No. 1, 113--114 (2011)], whose higher homotopy inner products are trivial. The authors construct two three-colored operads and operadic maps between them, which define a quasi-isomorphism. Both operads are defined in terms of three types of planar diagrams: planar trees, module trees and inner product diagrams. Planar diagrams are generated by three families of corollas, described in the paper. The authors adapt Boardman and Vogt's \(\mathbf W\)-construction [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Mathematics. 347. York: Springer-Verlag (1973; Zbl 0285.55012)] in order to obtain the second three-colored operad noted above, which is a cubical decomposition of the first constructed operad. The main result in the solution of the problem is a construction of diagonals on the constructed operads. Here the authors use the construction of the Serre diagonal on the cellular chains of the \(n\)-cube, which induces a coassociative diagonal on the second operad constructed in the paper. The constructions of the diagonals are used to define a categorical closed tensor product. The authors make explicit computations and remarks about the diagonals in special cases, in particular on pairahedra, which are contractible polytopes controlling, in the terminology of the paper, the combinatorial structure of an \(A_\infty\)- algebra with homotopy inner products.