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Diagonals and A-infinity Tensor Products

Robert Lipshitz, Peter Ozsváth, Dylan Thurston

Abstract

Extending work of Saneblidze-Umble and others, we use diagonals for the associahedron and multiplihedron to define tensor products of A-infinity algebras, modules, algebra homomorphisms, and module morphisms, as well as to define a bimodule analogue of twisted complexes (type DD structures, in the language of bordered Heegaard Floer homology) and their one- and two-sided tensor products. We then give analogous definitions for 1-parameter deformations of A-infinity algebras; this involves another collection of complexes. These constructions are relevant to bordered Heegaard Floer homology.

Diagonals and A-infinity Tensor Products

Abstract

Extending work of Saneblidze-Umble and others, we use diagonals for the associahedron and multiplihedron to define tensor products of A-infinity algebras, modules, algebra homomorphisms, and module morphisms, as well as to define a bimodule analogue of twisted complexes (type DD structures, in the language of bordered Heegaard Floer homology) and their one- and two-sided tensor products. We then give analogous definitions for 1-parameter deformations of A-infinity algebras; this involves another collection of complexes. These constructions are relevant to bordered Heegaard Floer homology.

Paper Structure

This paper contains 68 sections, 169 theorems, 537 equations, 49 figures.

Table of Contents

  1. Introduction
  2. Context and motivation
  3. Statement of results in the unweighted case
  4. Statement of results in the weighted case
  5. Additional topics
  6. Diagonals: definitions, existence
  7. Trees and the associahedron
  8. Algebra diagonals
  9. Module diagonals
  10. Module diagonal primitives
  11. Multiplihedra and multiplihedron diagonals
  12. Module-map diagonals
  13. Module-map primitives and partial module-map diagonals
  14. DADD diagonals
  15. Applications of diagonals
  16. ...and 53 more sections

Key Result

Theorem 1.1

There exist associahedron diagonals ${\boldsymbol{\Gamma}}$, which can be used to define a tensor product $\mathcal{A}_1\otimes_{{\boldsymbol{\Gamma}}}\mathcal{A}_2$ of $A_\infty$-algebras $\mathcal{A}_1$ and $\mathcal{A}_2$, with the following properties:

Figures (49)

  • Figure 1: Some trees. At the left is the corolla $\Psi_{4}$ with four inputs, which corresponds to the parenthesization $(1,2,3,4)$. To its right is the right-associated binary tree with four inputs, which corresponds to the parenthesization $(1,(2,(3,4)))$.
  • Figure 2: Examples of compositions of trees. An illustration of the stacking of trees $S$ and $T$, $\phi^7_{4,6}(S\otimes T)$.
  • Figure 3: From an associahedron tree diagonal to an associahedron diagonal. (a) The first few terms in a particular associahedron tree diagonal. (b) Substituting these terms into a more complicated tree.
  • Figure 4: A profile tree. The tree on the left has $n=7$. At the right we have drawn the profile tree corresponding to the triple of inputs $1$, $4$, and $7$.
  • Figure 5: The first three terms in a module diagonal associated to a module diagonal primitive. This is the module diagonal associated to the primitive in Figure \ref{['fig:mprim-terms']}.
  • ...and 44 more figures

Theorems & Definitions (470)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Theorem 1.4
  • Definition 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Lemma 1.9
  • Theorem 1.10
  • Definition 1.11
  • ...and 460 more