Geometry of Mutation Classes of Rank 3 Quivers

@article{Felikson2019GeometryOM,
  title={Geometry of Mutation Classes of Rank 3 Quivers},
  author={A. A. Felikson and Pavel Tumarkin},
  journal={Arnold Mathematical Journal},
  year={2019},
  pages={1-19}
}
We present a geometric realization for all mutation classes of quivers of rank 3 with real weights. This realization is via linear reflection groups for acyclic mutation classes and via groups generated by $$\pi $$π-rotations for the cyclic ones. The geometric behavior of the model turns out to be controlled by the Markov constant $$p^2+q^2+r^2-pqr$$p2+q2+r2-pqr, where p, q, r are the weights of arrows in a quiver. We also classify skew-symmetric mutation-finite real $$3\times 3$$3×3 matrices… 
Exchange graphs for mutation-finite non-integer quivers of rank 3
Skew-symmetric non-integer matrices with real entries can be viewed as quivers with non-integer weights of arrows. One can mutate such quivers according to usual rules of quiver mutation. Felikson
CLUSTER ALGEBRAS AND COXETER GROUPS
Coxeter groups are classical objects appearing in connection with symmetry groups of regular polytopes and various tessellations. Cluster algebras were introduced by Fomin and Zelevinsky in 2002 and
Mutation-finite quivers with real weights
We classify all mutation-finite quivers with real weights. We show that every finite mutation class not originating from an integer skew-symmetrizable matrix has a geometric realization by
Discrete Dynamical Systems From Real Valued Mutation
We introduce a family of discrete dynamical systems which includes, and generalizes, the mutation dynamics of rank two cluster algebras. These systems exhibit behavior associated with integrability,

References

SHOWING 1-10 OF 30 REFERENCES
Mutation Classes of Skew-Symmetric 3 × 3-Matrices
In this article, we establish a bijection between the set of mutation classes of mutation-cyclic skew-symmetric integral 3 × 3-matrices and the set of triples of integers (a, b, c) such that 2 ≤ a ≤
Mutation Classes of 3×3 Generalized Cartan Matrices
One of the recent developments in representation theory has been the introduction of cluster algebras by Fomin and Zelevinsky. It is now well known that these algebras are closely related with
Cluster-Cyclic Quivers with Three Vertices and the Markov Equation
Acyclic cluster algebras have an interpretation in terms of tilting objects in a Calabi-Yau category defined by some hereditary algebra. For a given quiver Q it is thus desirable to decide if the
Skew-symmetric cluster algebras of finite mutation type
In the famous paper [FZ2] Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster
Moduli spaces of local systems and higher Teichmüller theory
Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S
Cluster algebras and Weil-Petersson forms
In our previous paper we have discussed Poisson properties of cluster algebras of geometric type for the case of a nondegenerate matrix of transition exponents. In this paper we consider the case of
Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller
Cluster Algebras of Finite Type and Positive Symmetrizable Matrices
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
The Geometry of Discrete Groups
Describing the geometric theory of discrete groups and the associated tesselations of the underlying space, this work also develops the theory of Mobius transformations in n-dimensional Euclidean
...
1
2
3
...