Non-homogeneous Koszul duality in representation theory

TL;DR

This paper explores a generalized form of Koszul duality for filtered algebras related to symplectic reflection and graded Hecke algebras, establishing equivalences between derived categories and curved dg-algebras.

Contribution

It extends Koszul duality to non-homogeneous filtered algebras, connecting their derived categories with dual curved dg-algebras, generalizing Positselski's results.

Findings

Established an exact equivalence between derived categories of filtered algebras and curved dg-algebras.

Revealed that for algebras with finite global dimension, the duality simplifies to an equivalence with the homotopy category of injective modules.

Generalized Koszul duality framework applicable to algebras arising in 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.