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A note on higher structures on the complexes associated to quiver algebras with applications to toupie algebras

Yuming Liu, Bohan Xing

TL;DR

The paper develops a general method to elevate DG structures to $A_inf$- and related higher structures on complexes tied to the reduced bar resolution of quiver algebras by leveraging algebraic Morse theory and SDR data. This framework is then specialized to toupie algebras to obtain explicit $A_inf$-structures on Tor and Ext complexes, yielding concrete descriptions of Yoneda algebras and their multiplications. A key outcome is that, for certain special toupie algebras, the double homological duals are isomorphic to the associated graded algebras, with the associated graded algebras often Koszul; these results tie quasi-hereditary structure to homological duality. The approach unifies transfer theory with Morse-theoretic reductions, enabling precise higher operations and shedding light on the interplay between $A_inf$-structures, Yoneda algebras, and radical filtrations in the toupie setting.

Abstract

In this paper, we summarize a general method of transforming DG structures into higher structures on the various complexes related to the reduced bar resolution of a given quiver algebra using algebraic Morse theory. As an application, we describe the $A_\infty$-structures of toupie algebras. Additionally, for certain special toupie algebras, we also prove that their double homological duals are isomorphic to their associated graded algebras.

A note on higher structures on the complexes associated to quiver algebras with applications to toupie algebras

TL;DR

The paper develops a general method to elevate DG structures to - and related higher structures on complexes tied to the reduced bar resolution of quiver algebras by leveraging algebraic Morse theory and SDR data. This framework is then specialized to toupie algebras to obtain explicit -structures on Tor and Ext complexes, yielding concrete descriptions of Yoneda algebras and their multiplications. A key outcome is that, for certain special toupie algebras, the double homological duals are isomorphic to the associated graded algebras, with the associated graded algebras often Koszul; these results tie quasi-hereditary structure to homological duality. The approach unifies transfer theory with Morse-theoretic reductions, enabling precise higher operations and shedding light on the interplay between -structures, Yoneda algebras, and radical filtrations in the toupie setting.

Abstract

In this paper, we summarize a general method of transforming DG structures into higher structures on the various complexes related to the reduced bar resolution of a given quiver algebra using algebraic Morse theory. As an application, we describe the -structures of toupie algebras. Additionally, for certain special toupie algebras, we also prove that their double homological duals are isomorphic to their associated graded algebras.
Paper Structure (12 sections, 15 theorems, 70 equations)

This paper contains 12 sections, 15 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Higher structures and transfer theories
  3. $A_{\infty}$-structures and transfer theories
  4. $L_{\infty}$-structure and transfer theory
  5. Higher structures on the complexes associated to quiver algebras
  6. An SDR datum from algebraic Morse theory
  7. DG structures on the complexes associated to quiver algebras
  8. Applications to toupie algebras
  9. Basic definitions
  10. The minimal resolution of toupie algebras
  11. $A_\infty$-structure on $\mathbb{T}_*$
  12. More properties on Yoneda algebras

Key Result

Theorem 2.4

$(MM)$ Let $(A,m_1,m_2)$ be a DG associative algebra and consider a homotopy retract datum \begin{tikzcd} A \arrow[r, "p", shift left=2] \arrow["h"', loop, distance=2em, in=215, out=145] & A' \arrow[l, "i", shift left] \end{tikzcd}There exists an $A_\infty$-algebra structure on $A'$ which is gi with the convention that $hm_1=\mathrm{Id}$.

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Definition 2.6
  • Theorem 2.7
  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 21 more