The completely separating incidence algebras of tame representation type

Leszczyński, Zbigniew

doi:10.4064/cm94-2-7

Cited by 7 publications

(17 citation statements)

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“…A general description of the representation-finite case has been made in [Lou75b,Lou75a]. The tame case has been considered in [Les03,Les02].…”

Section: Introductionmentioning

confidence: 99%

On the Wildness of Cambrian Lattices

Rognerud

2018

Algebr Represent Theor

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In this note, we investigate the representation type of the cambrian lattices and some other related lattices. The result is expressed as a very simple trichotomy. When the rank of the underlined Coxeter group is at most 2, the lattices are of finite representation type. When the Coxeter group is a reducible group of type A 3 1 , the lattices are of tame representation type. In all the other cases they are of wild representation type. * Cet auteur a bénéficié d'une aide de l'Agence Nationale de la Recherche (projet Carma, référence ANR-12-BS01-0017). † Cet auteur a bénéficié d'un financement de l'IDEX BMM/PN/AM/N o 2016-096c.

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“…A general description of the representation-finite case has been made in [Lou75b,Lou75a]. The tame case has been considered in [Les03,Les02].…”

Section: Introductionmentioning

confidence: 99%

On the Wildness of Cambrian Lattices

Rognerud

2018

Algebr Represent Theor

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“…in [U], see also [L] and [Wi]). In particular, it was shown in [HV] that there are only four families of critical algebras of type D n , given by the quivers (1)-(4) in [Le;p. 247].…”

Section: Preliminariesmentioning

confidence: 99%

“…Similarly to [Le;Lemma 4.5], we can degenerate the algebras Q 1 , Q 3 (see [G]), and then degenerate the whole algebra A 0 to a biserial algebra. Hence,…”

Section: Proposition 33 If A(q) ∈ (B) and Contains No Convex Subcatmentioning

confidence: 99%

“…In particular, pgcritical incidence algebras of finite posets are listed in [Le;Lemma 2.4] (see also [NS;Theorem 3.2…”

Section: Assume That B Is the Incidence Algebra Of A Poset Andmentioning

confidence: 99%

“…Then, according to [Le;Lemma 2.7], the universal Galois covering (1.3) is an isomorphism and, in view of (iii) and [Le; Theorem], A(Q) is of tame representation type.…”

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Representation-tame incidence algebras of finite posets

Leszczyński

2003

Colloq. Math.

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Abstract. Continuing the paper [Le], we give criteria for the incidence algebra of an arbitrary finite partially ordered set to be of tame representation type. This completes our result in [Le], concerning completely separating incidence algebras of posets.1. Introduction. Throughout, we assume that K is an algebraically closed field. We continue the study of representation-tame incidence Kalgebras of finite posets, that is, partially ordered sets, started in [Le]. We use the terminology and notation introduced in [Le]. In particular, given a finite-dimensional K-algebra A, we denote by mod A the category of all finite-dimensional right A-modules. The algebra A is said to be of tame representation type (or representation-tame) if, for each dimension d, the isomorphism classes of indecomposable modules in mod A of dimension d form at most finitely many one-parameter families. The reader is referred to [Dr] and [S; Sections 14.2-4] for precise definitions of representation-tame and representation-wild algebras.Throughout, we denote by Q = (Q 0 , Q 1 ) a finite connected quiver with Q 0 being the set of vertices and Q 1 the set of arrows. We assume that Q has no oriented cycles and no arrows having the same starting and ending vertex with another path. In particular Q has no multiple arrows. We view the quiver Q as a poset with respect to the partial order relation on Q 0 defined by the formula:

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Integral bilinear forms, Coxeter transformations and Coxeter polynomials of finite posets

Simson

2010

Linear Algebra and its Applications

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The completely separating incidence algebras of tame representation type

Abstract: Abstract. We prove that a completely separating incidence algebra of a partially ordered set is of tame representation type if and only if the associated Tits integral quadratic form is weakly non-negative.

Cited by 7 publications

References 3 publications

On the Wildness of Cambrian Lattices

On the Wildness of Cambrian Lattices

Representation-tame incidence algebras of finite posets

Integral bilinear forms, Coxeter transformations and Coxeter polynomials of finite posets

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