Tensor Products of $A_{\infty }$-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.

arXiv comment45 pages, 22 figures, typos corrected, some additional explanations added, this version is accepted for publication in Trans. Amer. Math. Soc

Cited by in corpus (1)

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