The derived Hall algebra of a graded gentle one-cycle algebra I: the triangle structure
Hui Chen, Dong Yang
TL;DR
This work characterizes the perfect and finite-dimensional derived categories of graded gentle one-cycle algebras as twisted root categories of infinite quivers of type $A_\infty^\infty$, linking triangulated structures to the underlying additive data. It develops a comprehensive framework using triangulated orbit categories, pretriangulated dg enhancements, and twisted root categories, then applies it to two canonical quiver orientations, linear and generalized zigzag, to obtain explicit AR-structures and Serre functor descriptions. The main contributions include a complete classification of the derived categories in terms of twisted root categories associated to $\tilde{Q}$ and $Q_{p,q}$, and a demonstration that the triangle structures are uniquely determined by the additive category, yielding a derived-invariant perspective via the AG-invariant. These results bridge graded gentle one-cycle algebras with orbit-category techniques, enabling precise comparisons and equivalences between seemingly different algebras through their twisted-root realizations. The findings have potential implications for understanding derived Hall algebras and their invariants in the broader context of representation theory and higher-dimensional categories.
Abstract
Under a mild condition, the perfect derived category and the finite-dimensional derived category of a graded gentle one-cycle algebra are described as twisted root categories of certain infinite quivers of type $\mathbb{A}_\infty^\infty$. As a consequence, it is shown that the triangle structure of such derived categories is uniquely determined by the underlying additive category.