CLUSTER MUTATION VIA QUIVER REPRESENTATIONS

By Aslak Bakke Buan, B. Marsh, I. Reiten. 2004

198 citations recorded by the source corpus.

Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary alge- bras. Using this, we obtain a representation theoretic interpretation of cluster mutation in case of acyclic cluster algebras of finite type.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/dd9332625ce9371b8094121f439c849e9f38aeff

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. B. Keller
2. I. Reiten
3. Osamu Iyama
4. Ralf Schiffler
5. Bernard Leclerc
6. Aslak Bakke Buan
7. C. Geiss
8. J. Schröer

In Geometry and Topology

1. B. Keller
2. I. Reiten
3. Osamu Iyama
4. Bernard Leclerc
5. Ralf Schiffler
6. Aslak Bakke Buan
7. C. Geiss
8. J. Schröer

In Statistical and Nonlinear Physics

1. B. Keller
2. I. Reiten
3. Osamu Iyama
4. Bernard Leclerc
5. Ralf Schiffler
6. Aslak Bakke Buan
7. C. Geiss
8. J. Schröer

Citations in the corpus

• Quivers with Relations and Cluster Tilted Algebras (2004, background): “In [ 3 ], Buan, Marsh and Reiten obtained a representation theoretical interpretation of matrix mutation for the hereditary case.”
• Cluster-tilted algebras of finite representation type (2005, methodology): “BMR2 ]. The cluster category has also motivated a Hall-algebra type definition of a cluster algebra of finite type [CC, CK].”
• Calabi–Yau triangulated categories (2008, result): “The theorem is proved in [16] on the basis of the previous results obtained by many authors notably Buan-Marsh-Reiten-Todorov [13], Buan-Marsh-Reiten [4], Buan-Marsh-Reineke-Reiten-Todorov [3], Caldero-Chapoton [14], Marsh-ReinekeZelevinsky [46], .”
• Mutation in triangulated categories and rigid Cohen–Macaulay modules (2006, methodology): “We refer the reader to [21, 22 , 26, 49, 51, 13, 69, 63, 62] for more recent developments in the study of cluster categories and their generalizations.”
• Acyclic Calabi–Yau categories (2006, unknown): “This was used to prove significant new results on cluster algebras.”
• From triangulated categories to cluster algebras (2005, background): “Note that in this particular case, this formula was a conjecture of [5] and is since a theorem of [9,4].”
• Derived equivalence classification for cluster-tilted algebras of type An (2007, background): “By [ BMR2 ], the cluster-tilted algebras of type An are exactly the algebras given by quivers obtained from An-quivers by mutation, an operation which will be described in Section 2, with certain relations determined by the quiver [BMR3].”
• A geometric model for cluster categories of type Dn (2006, background): “Since then cluster categories have been the subject of many investigations; see, for instance, [1, 2, 7–10, 12–14, 21–23, 26].”
• Cluster structures from 2-Calabi–Yau categories with loops (2008, unknown): “Moreover, by [6],”
• Cluster‐tilted algebras as trivial extensions (2006, background): “Since then, these algebras have been the subject of many investigations, see, for instance, [7, 8, 9, 10, 11, 12, 13, 21].”
• Tilting theory and cluster combinatorics (2004, background): “A local version of Conjecture 9.3 is proved in [ BMR04b ].”
• Rigid modules over preprojective algebras (2005, background): “Then T = HomΛ(T2, T1) is a tilting module over EndΛ(T1), and we have EndEndΛ(T1)(T ) ∼= EndΛ(T2) op.”