On endomorphism algebras of silting complexes over hereditary abelian categories

TL;DR

This paper investigates the properties of endomorphism algebras of silting complexes over hereditary abelian categories, demonstrating their closure under various algebraic operations and extending known results to broader algebra classes.

Contribution

It establishes that the class of such endomorphism algebras is closed under idempotent quotients, subalgebras, and τ-reduction, and extends closure properties to several classic algebra classes.

Findings

The class of endomorphism algebras of silting complexes is closed under idempotent quotients, subalgebras, and τ-reduction.

Proper class of shod algebras is closed under these operations.

Several classic algebra classes, including laura, glued, and weakly shod algebras, are also closed under idempotent quotients.

Abstract

Let $\mathcal{E}$ be the class of finite-dimensional algebras isomorphic to endomorphism algebras of silting complexes over hereditary abelian categories. It is proved that the class $\mathcal{E}$ is closed under taking idempotent quotients, idempotent subalgebras and $\tau$-reduction. We also show that the proper class consisting of shod algebras is also closed under these operations. In addition, several classic classes of algebras -- including laura, glued, weakly shod algebras -- are proved to be closed under idempotent quotients, thereby generalizing a known result originally established for specific idempotents.