The Ext-algebra of standard modules over dual extension algebras

Thuresson, Markus

doi:10.1016/j.jalgebra.2022.05.011

Cited by 2 publications

(2 citation statements)

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“…When the algebra in question admits a duality on its module category which preserves simple modules, the structure is essentially unique, as shown by Coulembier in [Cou20]. This contrasts the algebras studied in this article with the examples from [KS12;Thu22], as the algebras appearing there admit a simple-preserving duality.…”

Section: Introductionmentioning

confidence: 84%

“…One way to do this involves determining an A ∞ -structure on Ext * A (∆, ∆), the algebra of extensions between standard modules over A, which can be arduous. Some examples of where this has been done can be found in [KS12;Thu22]. However, in the situations studied in this paper, the aforementioned A ∞ -structures can be easily computed.…”

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: 84%

Section: Introductionmentioning

confidence: 99%