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The properad of quadratic Poisson structures is Koszul

Anton Khoroshkin

TL;DR

The paper develops a practical criterion for Koszulness of the properadic envelope of a quadratic dioperad by passing to a 2-coloured twisted associative algebra $ abla_{ ext{in}} abla_{ ext{out}}\mathcal{D}$. If this twisted algebra is quadratic and Koszul (and certain module conditions hold for the envelope), then the original dioperad and its properadic envelope are Koszul, enabling explicit minimal resolutions. Applying the framework to the dioperad of quadratic Poisson structures, the authors prove that its properad is Koszul and provide a minimal resolution description via higher-arity corollas; they also treat quadratic–linear Poisson structures and outline additional Koszul examples. The results yield computable homological models for deformation theory of quadratic Poisson structures and suggest broad applicability to other properads, enhancing tractability of obstruction theories. The approach connects dioperadic data to twisted associative algebra structures, offering a concrete path to verify Koszulness in settings previously out of reach.

Abstract

In this paper, we suggest a sufficient condition on the properadic envelope of a quadratic dioperad to be Koszul in terms of twisted associative algebras. As a particular new example, we show that the properad of quadratic Poisson structures is Koszul.

The properad of quadratic Poisson structures is Koszul

TL;DR

The paper develops a practical criterion for Koszulness of the properadic envelope of a quadratic dioperad by passing to a 2-coloured twisted associative algebra . If this twisted algebra is quadratic and Koszul (and certain module conditions hold for the envelope), then the original dioperad and its properadic envelope are Koszul, enabling explicit minimal resolutions. Applying the framework to the dioperad of quadratic Poisson structures, the authors prove that its properad is Koszul and provide a minimal resolution description via higher-arity corollas; they also treat quadratic–linear Poisson structures and outline additional Koszul examples. The results yield computable homological models for deformation theory of quadratic Poisson structures and suggest broad applicability to other properads, enhancing tractability of obstruction theories. The approach connects dioperadic data to twisted associative algebra structures, offering a concrete path to verify Koszulness in settings previously out of reach.

Abstract

In this paper, we suggest a sufficient condition on the properadic envelope of a quadratic dioperad to be Koszul in terms of twisted associative algebras. As a particular new example, we show that the properad of quadratic Poisson structures is Koszul.
Paper Structure (14 sections, 13 theorems, 64 equations)

This paper contains 14 sections, 13 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Recollection on Dioperads, Properads and Koszul duality
  4. Definition of a properad
  5. Koszul duality
  6. From properads to twisted associative algebras and back
  7. Twisted associative algebras
  8. A sufficient condition of Koszul property for dioperads
  9. A sufficient condition for the Koszul property of the properadic envelope
  10. Properad of quadratic Poisson structures and other examples
  11. The description of the dioperad and the properadic envelope
  12. Twisted Associative algebra $\partial_{\mathsf{in}}\partial_{\mathsf{out}}\mathcal{QP}\mathit{ois}^{!}$
  13. Properad of quadratic--linear Poisson structures
  14. Other known examples of Koszul properads

Key Result

Theorem A

Let $\mathcal{D}$ be a quadratic dioperad generated by an ${\mathbb S}\times{\mathbb S}$--collection ${\mathcal{E}}$ such that ${\mathcal{E}}(m,n)=0$ for $m+n<3$.

Theorems & Definitions (41)

  • Theorem A
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Proposition 1.5
  • Remark 1.6
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 31 more