Cluster Algebras of Finite Type and Positive Symmetrizable Matrices

By M. Barot, C. Geiss, Andrei Zelevinsky. 2004

87 citations recorded by the source corpus.

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 skew‐symmetrizable matrix gives rise to a cluster algebra of finite type.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d67323c82e1666bb5fa2443e9ee835186d39318f

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Triangulated Categories: Cluster algebras, quiver representations and triangulated categories (2008, unknown): “For a general quiver Q, a criterion forAQ to be cluster-finite in terms of quadratic forms was given in [ 14 ].”
Skew-symmetric cluster algebras of finite mutation type (2008, unknown): “Quivers of order 3 47 References 48”
Cluster algebras of finite mutation type via unfoldings (2010, background): “4 The results by Fomin and Zelevinsky were further developed in [16] and [1], where the effective criteria for cluster algebras of finite type were given.”