Tensor Products of $A_\infty$-algebras with Homotopy Inner Products

TL;DR

This paper demonstrates that the tensor product of cyclic $A_ abla$-algebras results in an $A_ abla$-algebra with homotopy inner products, not necessarily cyclic, and provides an explicit combinatorial construction for this tensor product.

Contribution

It introduces a combinatorial diagonal on pairahedra to define a categorically closed tensor product for $A_ abla$-algebras with homotopy inner products.

Findings

Tensor product of cyclic $A_ abla$-algebras is generally not cyclic.

Constructs an explicit combinatorial diagonal on pairahedra.

Defines a categorically closed tensor product for $A_ abla$-algebras with homotopy inner products.

Abstract

We show that the tensor product of two cyclic $A_\infty$-algebras is, in general, not a cyclic $A_\infty$-algebra, but an $A_\infty$-algebra with homotopy inner product. More precisely, we construct an explicit combinatorial diagonal on the pairahedra, which are contractible polytopes controlling the combinatorial structure of an $A_\infty$-algebra with homotopy inner products, and use it to define a categorically closed tensor product. A cyclic $A_\infty$-algebra can be thought of as an $A_\infty$-algebra with homotopy inner products whose higher inner products are trivial. However, the higher inner products on the tensor product of cyclic $A_\infty$-algebras are not necessarily trivial.