A derived equivalence between cluster equivalent algebras

TL;DR

This paper proves that for co-$c$-sortable elements, certain algebras associated with acyclic quivers are derived equivalent, revealing a deep connection between different cluster-related categories.

Contribution

It establishes a derived equivalence between two classes of algebras linked to co-$c$-sortable elements, expanding understanding of their categorical relationships.

Findings

Algebras $ ext{ extGamma}_w$ and $A_w$ are derived equivalent for co-$c$-sortable $w$.

Uses 2-APR-tilting theory to prove the derived equivalence.

Connects generalized cluster categories with specific algebraic constructions.

Abstract

Let $Q$ be an acyclic quiver. Associated with any element $w$ of the Coxeter group of $Q$, triangulated categories $\underline{\Sub}\Lambda_w$ were introduced in \cite{Bua2}. There are shown to be triangle equivalent to generalized cluster categories $\Cc_{\Gamma_w}$ associated to algebras $\Gamma_w$ of global dimension $\leq 2$ in \cite{ART}. For $w$ satisfying a certain property, called co-$c$-sortable, other algebras $A_w$ of global dimension $\leq 2$ are constructed in \cite{AIRT} with a triangle equivalence $\Cc_{A_w}\simeq \underline{\Sub}\Lambda_w$. The main result of this paper is to prove that the algebras $\Gamma_w$ and $A_w$ are derived equivalent when $w$ is co-$c$-sortable. The proof uses the 2-APR-tilting theory introduced in \cite{IO}.