Tensor product of filtered $A_\infty$-algebras

TL;DR

This paper introduces a new definition for the tensor product of filtered $A_ abla$-algebras, explores its properties, and connects it to classical cases, with applications to Fukaya algebras and Lagrangian submanifolds.

Contribution

It defines the tensor product for filtered $A_ abla$-algebras, compares it to classical definitions, and provides criteria for quasi-isomorphisms, enabling future K"unneth theorems in symplectic geometry.

Findings

The tensor product of filtered $A_ abla$-algebras is well-defined and has specific properties.

The new definition recovers classical $A_ abla$-algebra tensor products by Markl and Shnider.

A criterion for $A_ abla$-algebras to be quasi-isomorphic to tensor products of subalgebras is established.

Abstract

We define the tensor product of filtered $A_\infty$-algebras. establish some of its properties and give a partial description of the space of bounding cochains in the tensor product. Furthermore we show that in the case of classical $A_\infty$-algebras our definition recovers the one given by Markl and Shnider. We also give a criterion that implies that a given $A_\infty$-algebra is quasi-isomorphic to the tensor product of two subalgebras. This will be used in a sequel to prove a K\"unneth Theorem for the Fukaya algebra of a product of Lagrangian submanifolds.