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Koszul dual $\mathcal{A}_{\infty}$-algebras from star-shaped diagrams -- part 2

Isabella Khan

Abstract

This paper proves a Koszul duality result between weighted $\mathcal{A}_{\infty}$-algebras constructed in the author's previous work. In the process, we construct a new box tensor product for weighted $\mathcal{A}_{\infty}$ bimodules, and verify a correspondence between weighted $\mathcal{A}_{\infty}$-algebra maps and a particular class of $\mathcal{A}_{\infty}$-bimodule. This paper is part 2 of arXiv:2408.01564.

Koszul dual $\mathcal{A}_{\infty}$-algebras from star-shaped diagrams -- part 2

Abstract

This paper proves a Koszul duality result between weighted -algebras constructed in the author's previous work. In the process, we construct a new box tensor product for weighted bimodules, and verify a correspondence between weighted -algebra maps and a particular class of -bimodule. This paper is part 2 of arXiv:2408.01564.
Paper Structure (18 sections, 12 theorems, 121 equations, 7 figures)

This paper contains 18 sections, 12 theorems, 121 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Geometric overview of the constructions from KhKos
  4. The structure of this paper
  5. Definitions
  6. Trees
  7. Weighted algebra diagonals
  8. Weighted bimodule diagonal primitives
  9. $\mathcal{A}_{\infty}$-algebras and -bimodules
  10. Box tensor products
  11. The $\mathcal{A}_{\infty}$-algebras $\mathcal{A}$ and $\mathcal{B}$
  12. The construction of $\mathcal{A}$
  13. The construction of $\mathcal{B}$
  14. The $\mathcal{A}_{\infty}$-bimodules
  15. The AA-bimodule $Y$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\mathcal{A}$ and $\mathcal{B}$ be the weighted $\mathcal{A}_{\infty}$-algebras defined in Section algs (and in KhKos). Then there exists a DD-bimodule $\:^{\mathcal{A}} X^{\mathcal{B}}$, constructed in Section dd and a weighted AA-bimodule $\:_{\mathcal{B}} Y_{\mathcal{A}}$ constructed in Secti and where $\:^{\mathcal{A}} \mathrm{id}_{\mathcal{A}}$ and $\:^{\mathcal{B}} \mathrm{id}_{\mathcal

Figures (7)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5: A $\mu_6$ and a $\mu_{10}$
  • ...and 2 more figures

Theorems & Definitions (38)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Proposition 2.8
  • ...and 28 more