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Non-homogeneous Koszul duality in representation theory

Gwyn Bellamy, Simone Castellan, Isambard Goodbody

TL;DR

This work extends Koszul duality to non-homogeneous filtered algebras by encoding curvature in the Koszul dual curved dg-algebra $A^!$ and establishing an explicit derived-equivalence $D(U)\simeq K(\mathrm{Inj}\,A^!)/\mathcal{N}$ via a pair of adjoint functors. The quotient by the thick subcategory $\mathcal{N}$ captures the failure of injectivity in the curved setting, and if the associated graded algebra $A$ has finite global dimension, the quotient collapses to an equivalence $D(U)\simeq K(\mathrm{Inj}\,A^!)$, recovering and refining results of Positselski. The framework is illustrated with motivating examples from representation theory, including symplectic reflection algebras, deformed preprojective algebras, and graded Hecke algebras, and is shown to yield practical tools such as explicit resolutions, t-structures, and K-theory consequences. Overall, the paper provides a direct, elementary construction of derived equivalences in the non-homogeneous Koszul setting and demonstrates its relevance to key representation-theoretic algebras. The results offer new avenues to translate problems about those algebras into curved dg-module questions, with potential impact on Hochschild cohomology, localization, and categorical structures in geometric representation theory.

Abstract

Motivated by the representation theory of symplectic reflection algebras, deformed preprojective algebras, and graded Hecke algebras, we consider filtered algebras $U$ whose associated graded is Koszul. The Koszul dual of $U$, as defined by Positselski, is a curved dg-algebra. We establish an exact equivalence between the unbounded derived category of $U$ and an explicit quotient of the homotopy category of injective modules over the dual curved dg-algebra. This recovers a special case of a result of Positselski. In the case where $U$ has finite global dimension, the quotient is trivial and hence the unbounded derived category of $U$ is equivalent to the homotopy category of injective modules over the dual curved dg-algebra.

Non-homogeneous Koszul duality in representation theory

TL;DR

This work extends Koszul duality to non-homogeneous filtered algebras by encoding curvature in the Koszul dual curved dg-algebra and establishing an explicit derived-equivalence via a pair of adjoint functors. The quotient by the thick subcategory captures the failure of injectivity in the curved setting, and if the associated graded algebra has finite global dimension, the quotient collapses to an equivalence , recovering and refining results of Positselski. The framework is illustrated with motivating examples from representation theory, including symplectic reflection algebras, deformed preprojective algebras, and graded Hecke algebras, and is shown to yield practical tools such as explicit resolutions, t-structures, and K-theory consequences. Overall, the paper provides a direct, elementary construction of derived equivalences in the non-homogeneous Koszul setting and demonstrates its relevance to key representation-theoretic algebras. The results offer new avenues to translate problems about those algebras into curved dg-module questions, with potential impact on Hochschild cohomology, localization, and categorical structures in geometric representation theory.

Abstract

Motivated by the representation theory of symplectic reflection algebras, deformed preprojective algebras, and graded Hecke algebras, we consider filtered algebras whose associated graded is Koszul. The Koszul dual of , as defined by Positselski, is a curved dg-algebra. We establish an exact equivalence between the unbounded derived category of and an explicit quotient of the homotopy category of injective modules over the dual curved dg-algebra. This recovers a special case of a result of Positselski. In the case where has finite global dimension, the quotient is trivial and hence the unbounded derived category of is equivalent to the homotopy category of injective modules over the dual curved dg-algebra.

Paper Structure

This paper contains 36 sections, 53 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. The equivalence
  3. Applications
  4. Examples
  5. Structure of the paper
  6. Non-homogeneous Koszul algebras
  7. Basic definitions and notation
  8. Linear algebra
  9. Graded hom and tensor
  10. Homogeneous Koszul duality
  11. PBW Theorem for non-homogeneous Koszul algebras
  12. Non-homogeneous duality
  13. Motivating Examples
  14. Basic examples
  15. Koszul duality for the Symplectic Reflection Algebra
  16. ...and 21 more sections

Key Result

Theorem 1.1

The functors $G,F$ induce inverse equivalences \begin{tikzcd} D(U) \ar[r, shift left=1.0ex, "G"] & K(\Inj A^!)/\mathcal{N} \ar[l, shift left=1.0ex, "F"]. \end{tikzcd}

Theorems & Definitions (126)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 116 more