Exact Borel subalgebras of path algebras of quivers of Dynkin type A

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, ... , n}, we compute the quiver and relations for the 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 Konig and we show that there is always a regular exact Borel subalgebra containing the idempotents e1, . . . , en and find a minimal generating set for it. For a quiver Q and a deconcatenation Q = Q1 U Q2 of Q at a sink or source v, we describe the Ext-algebra of standard modules over KQ, up to an isomorphism of associative algebras, in terms of that over KQ1 and KQ2. Moreover, we determine necessary and sufficient conditions for KQ to admit a regular exact Borel subalgebra, provided that KQ1 and KQ2 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. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).

Automated summary

Hereditary algebras are quasi-hereditary and exhibit certain regularity properties with respect to adapted partial orders. This article investigates the Ext-algebra of standard modules over path algebras of linear quivers and provides necessary and sufficient conditions for regular exact Borel subalgebras. The findings have implications for the understanding of linear quivers with arbitrary orientations.