The minimal representation-infinite algebras  which are special biserial

Ringel, Claus Michael

doi:10.4171/101-1/11

Cited by 15 publications

(16 citation statements)

References 20 publications

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“…We should stress that cluster additive functions are definitely also of interest when dealing with stable translation quivers which are not related to translation quivers of the form ZΔ with Δ a finite directed quiver. Examples of cluster additive functions on the translation quiver ZD ∞ (as well as on ZA ∞ ∞ ) have been exhibited in [12].…”

Section: Lemmamentioning

confidence: 99%

Cluster-additive functions on stable translation quivers

Ringel

2012

J Algebr Comb

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Additive functions on translation quivers have played an important role in the representation theory of finite dimensional algebras, the most prominent ones are the hammock functions introduced by S. Brenner. When dealing with cluster categories (and cluster-tilted algebras), one should look at a corresponding class of functions defined on stable translation quivers, namely the cluster-additive ones. We conjecture that the cluster-additive functions on a stable translation quiver of Dynkin type A n ,D n ,E 6 ,E 7 ,E 8 are non-negative linear combinations of cluster-hammock functions (with index set a tilting set). The present paper provides a first study of cluster-additive functions and gives a proof of the conjecture in the case A n .

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Section: Lemmamentioning

confidence: 99%

Cluster-additive functions on stable translation quivers

Ringel

2012

J Algebr Comb

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“…A different classification of τ -tilting finite special biserial algebras has recently been obtained in [15] based on the classification of minimal representation infinite algebras in [18]. We also note that gentle algebras are a subclass of special biserial algebras.…”

Section: Introductionmentioning

confidence: 93%

On band modules and $τ$-tilting finiteness

Schroll¹,

Treffinger²,

Valdivieso³

2019

Preprint

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“…A different classification of τ -tilting finite special biserial algebras has recently been obtained in [20] based on the classification of minimal representation infinite special biserial algebras in [25]. Gentle algebras are a subclass of special biserial algebras.…”

Section: Theorem 14 (Theorem 41) Let K Be An Algebraically Closed Field and Let A Be A Finite Dimensional K -Algebra Containing A Band Momentioning

confidence: 99%

On band modules and $$\tau $$-tilting finiteness

2021

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In this paper, motivated by a $$\tau $$ τ -tilting version of the Brauer-Thrall Conjectures, we study general properties of band modules and their endomorphisms in the module category of a finite dimensional algebra. As an application we describe properties of torsion classes containing band modules. Furthermore, we show that a special biserial algebra is $$\tau $$ τ -tilting finite if and only if no band module is a brick. We also recover a criterion for the $$\tau $$ τ -tilting finiteness of Brauer graph algebras in terms of the Brauer graph.

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The minimal representation-infinite algebras which are special biserial

Cited by 15 publications

References 20 publications

Cluster-additive functions on stable translation quivers

Cluster-additive functions on stable translation quivers

On band modules and $τ$-tilting finiteness

On band modules and $$\tau $$-tilting finiteness

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