Recognizing Cluster Algebras of Finite Type

Seven, Ahmet

doi:10.37236/922

Cited by 23 publications

(23 citation statements)

References 13 publications

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“…Combining the above theorem with Theorem 2.6 we deduce the construction of the quiver of a cluster-tilted algebra. This allows, for instance, as done in [12], to relate the list of tame concealed algebras of Happel and Vossieck [19] with Seven's list of minimal infinite cluster quivers [24]. This paper consists of two sections.…”

Section: Theorem 11 An Algebrac Is Cluster-tilted If and Only If Thmentioning

confidence: 99%

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Cluster-tilted algebras as trivial extensions

Assem¹,

Brüstle²,

Schiffler³

2008

Bulletin of the London Mathematical Society

116

266

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Given a finite dimensional algebra C (over an algebraically closed field) of global dimension at most two, we define its relation-extension algebra to be the trivial extension C ⋉ Ext 2 C (DC, C) of C by the C-C-bimodule Ext 2 C (DC, C). We give a construction for the quiver of the relationextension algebra in case the quiver of C has no oriented cycles. Our main result says that an algebraC is cluster-tilted if and only if there exists a tilted algebra C such thatC is isomorphic to the relation-extension of C.

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Section: Theorem 11 An Algebrac Is Cluster-tilted If and Only If Thmentioning

confidence: 99%

“…(a) The construction of Corollary 3.5 is easily seen to generalise the one in [12, Proposition 4.1] and, thus, can be used to relate the Happel-Vossieck list of tame concealed algebras [19] with Seven's list of minimal infinite cluster quivers [24].…”

Section: Remarks and Examplesmentioning

confidence: 99%

Cluster-tilted algebras as trivial extensions

Assem¹,

Brüstle²,

Schiffler³

2008

Bulletin of the London Mathematical Society

116

266

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“…The problem makes sense since the mutations are hard to control, so each of the conditions (2) and (3) in Theorem 1.1 is hard to check in general. A nice solution of the problem was obtained by Seven [6]. His answer is given in terms of 'forbidden minors' of B.…”

Section: Introductionmentioning

confidence: 99%

Cluster Algebras of Finite Type and Positive Symmetrizable Matrices

Barot

Geiß

Zelevinsky

2006

J. London Math. Soc.

160

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The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However, the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to skew-symmetrizable matrices. We study an interplay between the two classes of matrices, in particular, establishing a new criterion for deciding whether a given skewsymmetrizable matrix gives rise to a cluster algebra of finite type.

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“…In their classification Fomin and Zelevinsky introduced mutations on diagrams, and Seven classified all minimal 2-infinite diagrams in [9]. The work by Seven on minimal 2-infinite diagrams inspired the study of minimal mutation-infinite quivers and this paper builds on work done by Felikson, Shapiro and Tumarkin in [4,Section 7] proving a number of useful results about minimal mutation-infinite quivers.…”

Section: Introductionmentioning

confidence: 98%