Exact Borel subalgebras of quasi-hereditary monomial algebras
Anna Rodriguez Rasmussen
TL;DR
The paper develops a path-centric framework for exact Borel subalgebras in monomial quasi-hereditary algebras, proving that every basic monomial quasi-hereditary algebra $A$ admits a Reedy decomposition $A \cong C \otimes_L B$ with a path-basis exact Borel $B$ and a path-basis $\Delta$-subalgebra $C$. A concrete, explicit construction $B_{\min}$ is provided, and $B_{\min}$ is shown to be the unique path-basis exact Borel subalgebra (with $C_{\min}$ playing the dual role) up to conjugacy in the appropriate settings; normality of $B_{\min}$ is demonstrated via a splitting. The authors give a technical criterion (via Conde) for when $B_{\min}$ is regular, with a simpler hereditary criterion, and explore behavior under idempotents and quotients. They then count quasi-hereditary structures on path algebras of quivers of type $A$, $D$, and $E$ that admit regular exact Borels, obtaining Catalan-number expressions and asymptotic proportions (notably a limit of $3/5$ in certain families). Overall, the work clarifies when standard/costandard module filtrations align with path-based Borels and provides practical tools for constructing and classifying such structures.
Abstract
Green and Schroll give an easy criterion for a monomial algebra $A$ to be quasi-hereditary with respect to some partial order $\leq_A$. A natural follow-up question is under which conditions a monomial quasi-hereditary algebra $(A, \leq_A)$ admits an exact Borel subalgebra in the sense of König. In this article, we show that it always admits a Reedy decomposition consisting of an exact Borel subalgebra $B$, which has a basis given by paths, and a dual subalgebra. Moreover, we give an explicit description of $B$ and show that it is the unique exact Borel subalgebra of $A$ with a basis given by paths. Additionally, we give a criterion for when $B$ is regular, using a criterion by Conde.