Combinatorics of quasi-hereditary structures

Flores, Manuel; Kimura, Yuta; Rognerud, Baptiste

doi:10.1016/j.jcta.2021.105559

Cited by 4 publications

(11 citation statements)

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“…This article is organized in the following way: In Section 2 we give the necessary background on quasi-hereditary algebras and fix some notation. In Section 3, we recall the results of [FKR22] on the quasi-hereditary structures of A n and expand upon them, proving (A). In Subsection 3.1, we compute the quiver and relations of the Ringel dual of A n .…”

Section: Introductionmentioning

confidence: 89%

“…, n} such that (A n , ) is quasi-hereditary, we construct a graded quiver Q and an admissible ideal I ⊂ K Q, such that there is an isomorphism of graded associative algebras Ext * A n (∆, ∆) ∼ = K Q/I . According to results from [FKR22], each quasi-hereditary structure corresponds to a unique binary tree, and we show that this tree encodes all necessary information about extensions between the standard modules. For all details, we refer to Section 3.…”

Section: Introductionmentioning

confidence: 93%

“…In a recent paper by Flores, Kimura and Rognerud, [FKR22], the authors showed that when A is the path algebra of a uniformly oriented linear quiver, the different quasi-hereditary structures are counted by the Catalan numbers. Moreover, they counted the quasi-hereditary structures of path algebras of more complicated quivers by means of a combinatorial process called "deconcatenation".…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, they counted the quasi-hereditary structures of path algebras of more complicated quivers by means of a combinatorial process called "deconcatenation". The main goal of the present article is to use the combinatorial techniques of Flores, Kimura and Rognerud to further study the quasi-hereditary structures of some of the algebras considered in [FKR22]. In particular, we want to describe the Ext-algebras of their standard modules and their regular exact Borel subalgebras.…”

Section: Introductionmentioning

confidence: 99%

“…A n (∆, ∆). In Section 4, we give the background on deconcatenations from [FKR22] and prove (C). Subsection 4.1 discusses how regular exact Borel subalgebras behave under deconcatenations and contains the proof of (D).…”

Section: Introductionmentioning

confidence: 99%

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

Thuresson¹

2023

Preprint

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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.

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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