arXiv subjects

Markus Thuresson

Publications and source records attributed to Markus Thuresson.

3 recordsLinked to original sources

arXiv2023

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

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 $\{1,\dots, n\}$, 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 König 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.

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arXiv2022

Tilting modules and exceptional sequences for a family of dual extension algebras

We provide a classification of generalized tilting modules and full exceptional sequences for the dual extension algebra of the path algebra of a uniformly oriented linear quiver modulo the ideal generated by paths of length two with its opposite algebra. For the classification of generalized tilting modules we develop a combinatorial model for the poset of indecomposable self-orthogonal modules with standard filtration with respect to the relation arising from higher extensions.

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arXiv2021

The $\operatorname{Ext}$-algebra of standard modules over dual extension algebras

We exhibit an isomorphism of associative algebras between the $\operatorname{Ext}$-algebra $\operatorname{Ext}_Λ^\ast(Δ,Δ)$ of standard modules over the dual extension algebra $Λ$ of two directed algebras $B$ and $A$ and the dual extension algebra of the $\operatorname{Ext}$-algebra $\operatorname{Ext}_B^\ast(\mathbb{L},\mathbb{L})$ with $A$. There are natural $A_\infty$-structures on these $\operatorname{Ext}$-algebras, and, under certain technical assumptions on $B$, we describe that on $\operatorname{Ext}_Λ^\ast(Δ,Δ)$ completely in terms of that on $\operatorname{Ext}_B^\ast(\mathbb{L},\mathbb{L})$. As an example, we compute these $A_\infty$-structures explicitly in the case where $B=A=K\mathbb{A}_n /(\operatorname{rad}K\mathbb{A}_n)^\ell$.

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