REFLECTION GROUP RELATIONS ARISING FROM CLUSTER ALGEBRAS

Download

index.pdf

Date

2016-11-01

Author

Seven, Ahmet İrfan

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

82
views

0
downloads

There is a well-known analogy between cluster algebras and Kac-moody algebras: roughly speaking, Kac-Moody algebras are associated with symmetrizable generalized Cartan matrices while cluster algebras correspond to skew-symmetrizable matrices. In this paper, we study an interplay between these two classes of matrices. We obtain relations in the Weyl groups of Kac-Moody algebras that come from mutation classes of skew-symmetrizable matrices. More precisely, we establish a set of relations satisfied by the reflections of the so-called companion bases; these include c-vectors, which parametrize coefficients in a cluster algebra with principal coefficients. These relations generalize the relations obtained by Barot and Marsh for finite type. For affine type, we also show that the reflections of the companion bases satisfy the relations obtained by Felikson and Tumarkin. As an application, we obtain some combinatorial properties of the mutation classes of skew-symmetrizable matrices.

Subject Keywords

Applied Mathematics, General Mathematics

URI

https://hdl.handle.net/11511/46711

Journal

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY

DOI

https://doi.org/10.1090/proc/13157

Collections

Department of Mathematics, Article

Suggestions

OpenMETU
Core

CLUSTER ALGEBRAS AND SEMIPOSITIVE SYMMETRIZABLE MATRICES Seven, Ahmet İrfan (American Mathematical Society (AMS), 2011-05-01) There is a particular analogy between combinatorial aspects of cluster algebras and Kac-Moody algebras: roughly speaking, cluster algebras are associated with skew-symmetrizable matrices while Kac-Moody algebras correspond to (symmetrizable) generalized Cartan matrices. Both classes of algebras and the associated matrices have the same classification of finite type objects by the well-known Cartan-Killing types. In this paper, we study an extension of this correspondence to the affine type. In particular, w...
CLUSTER ALGEBRAS AND SYMMETRIZABLE MATRICES Seven, Ahmet İrfan (American Mathematical Society (AMS), 2019-07-01) In the structure theory of cluster algebras, principal coefficients are parametrized by a family of integer vectors, called c-vectors. Each c-vector with respect to an acyclic initial seed is a real root of the corresponding root system, and the c-vectors associated with any seed defines a symmetrizable quasi-Cartan companion for the corresponding exchange matrix. We establish basic combinatorial properties of these companions. In particular, we show that c-vectors define an admissible cut of edges in the a...
Concrete description of CD0(K)-spaces as C(X)-spaces and its applications Ercan, Z (American Mathematical Society (AMS), 2004-01-01) We prove that for a compact Hausdorff space K without isolated points, CD0(K) and C(K x {0, 1}) are isometrically Riesz isomorphic spaces under a certain topology on K x {0, 1}. Moreover, K is a closed subspace of K x {0, 1}. This provides concrete examples of compact Hausdorff spaces X such that the Dedekind completion of C(X) is B(S) (= the set of all bounded real-valued functions on S) since the Dedekind completion of CD0(K) is B(K) (CD0(K, E) and CDw (K, E) spaces as Banach lattices).
Legendrian realization in convex Lefschetz fibrations and convex stabilizations Akbulut, Selman; Arıkan, Mehmet Fırat (Walter de Gruyter GmbH, 2015-05-01) We show that, up to a Liouville homotopy and a deformation of compact convex Lefschetz fibrations on W, any Lagrangian submanifold with trivial first de Rham cohomology group, embedded on a (symplectic) page of the (induced) convex open book on partial derivative W, can be assumed to be Legendrian in partial derivative W with the induced contact structure. This can be thought as the extension of Giroux's Legendrian realization (which holds for contact open books) for the case of convex open books. We also s...
A formula for the joint local spectral radius Emel'yanov, EY; Ercan, Z (American Mathematical Society (AMS), 2004-01-01) We give a formula for the joint local spectral radius of a bounded subset of bounded linear operators on a Banach space X in terms of the dual of X.

Citation Formats

A. İ. Seven, “REFLECTION GROUP RELATIONS ARISING FROM CLUSTER ALGEBRAS,” PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, pp. 4641–4650, 2016, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/46711.