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,... ,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 König and we show that there is always a regular exact Borel subalgebra containg the idempotents e 1 ,... ,e n and find a minimal generating set for it. For a quiver Q and a deconcatenation Q = Q 1 ⊔Q 2 of Q at a sink or source v , we describe the Ext-algebra of standard modules over K Q, up to an isomorphism of associative algebras, in terms of that over K Q 1 and K Q 2 . Moreover, we determine necessary and sufficient conditions for K Q to admit a regular exact Borel subalgebra, provided that K Q 1 and K Q 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.