Exact Borel subalgebras of path algebras of quivers of Dynkin type $\mathbb{A}$

Abstract

Hereditary algebras are quasi-hereditary with respect to any adapted partial order on the indexing set of the isomorphism classes of their simple modules. For any adapted partial order on $\{1,\dots, n\}$, we compute the quiver and relations for the $\operatorname{Ext}$-algebra of standard modules over the path algebra of a uniformly oriented linear quiver with $n$ vertices. Such a path algebra always admits a regular exact Borel subalgebra in the sense of K\"onig and we show that there is always a regular exact Borel subalgebra containg the idempotents $e_1,\dots, e_n$ and find a minimal generating set for it. For a quiver $Q$ and a deconcatenation $Q=Q^1\sqcup Q^2$ of $Q$ at a sink or source $v$, we describe the $\operatorname{Ext}$-algebra of standard modules over $KQ$, up to an isomorphism of associative algebras, in terms of that over $KQ^1$ and $KQ^2$. Moreover, we determine necessary and sufficient conditions for $KQ$ to admit a regular exact Borel subalgebra, provided that $KQ^1$ and $KQ^2$ do. We use these results to obtain sufficient and necessary conditions for a path algebra of a linear quiver with arbitrary orientation to admit a regular exact Borel subalgebra.