Tensor Products of A_∞-algebras with Homotopy Inner Products

2011/02/01 by Tradler, Thomas, Umble, Ronald
#18D50 #52B05 #55S15 #55U99 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1102.0047

Abstract

We show that the tensor product of two cyclic A_∞-algebras is, in general, not a cyclic A_∞-algebra, but an A_∞-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_∞-algebra with homotopy inner products, and use it to define a categorically closed tensor product. A cyclic A_∞-algebra can be thought of as an A_∞-algebra with homotopy inner products whose higher inner products are trivial. However, the higher inner products on the tensor product of cyclic A_∞-algebras are not necessarily trivial.

Related

• Tensor product of A_∞-categories
• Tensor products of representations up to homotopy
• Associahedra, cellular W-construction and products of\n A_\∞-algebras
• Strong homotopy inner product of an A-infinity algebra
• Diagonals and A-infinity Tensor Products