Quasi-Cartan companions of elliptic cluster algebras

Download
2016
Velioğlu, Kutlucan
There is an analogy between combinatorial aspects of cluster algebras and diagrams corresponding to skew-symmetrizable matrices. In this thesis, we study quasi-Cartan companions of skew-symmetric matrices in the mutation-class of exceptional elliptic diagrams. In particular, we establish the existence of semipositive admissible quasi-Cartan companions for these matrices and exhibit some other invariant properties.

Suggestions

CLUSTER ALGEBRAS AND SYMMETRIC MATRICES
Seven, Ahmet İrfan (2015-02-01)
In the structural theory of cluster algebras, a crucial role is played by a family of integer vectors, called c-vectors, which parametrize the coefficients. It has recently been shown that each c-vector with respect to an acyclic initial seed is a real root of the corresponding root system. In this paper, we obtain an interpretation of this result in terms of symmetric matrices. We show that for skew-symmetric cluster algebras, the c-vectors associated with any seed defines a quasi-Cartan companion for the ...
MUTATION CLASSES OF SKEW-SYMMETRIZABLE 3 x 3 MATRICES
Seven, Ahmet İrfan (2013-05-01)
Mutation of skew-symmetrizable matrices is a fundamental operation that first arose in Fomin-Zelevinsky's theory of cluster algebras; it also appears naturally in many different areas of mathematics. In this paper, we study mutation classes of skew-symmetrizable 3 x 3 matrices and associated graphs. We determine representatives for these classes using a natural minimality condition, generalizing and strengthening results of Beineke-BrustleHille and Felikson-Shapiro-Tumarkin. Furthermore, we obtain a new num...
Classification of skew-symmetric forms corresponding to cluster algebras with principal coefficients
Mazı, Sedanur; Seven, Ahmet İrfan; Department of Mathematics (2016)
In this thesis, we study algebraic and combinatorial properties of the skew-symmetric forms that correspond to cluster algebras with principal coefficients. We obtain a classification of these forms under congruence and compute the Arf invariants for finite types. 
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...
Shape models based on elliptic PDES, associated energies, and their applications in 2D and 3D
Gençtav, Aslı; Tarı, Zehra Sibel; Can, Tolga; Department of Computer Engineering (2018)
By using an elliptic PDE or its modifications, we develop implicit shape representations and demonstrate their two- and three-dimensional applications. In the first part of the thesis, we present a novel shape characterization field that provides a local measure of roundness at each shape point. The field is computed by comparing the solution of the elliptic PDE on the shape domain and the solution of the same PDE on the reference disk. We demonstrate its potential via illustrative applications including gl...
Citation Formats
K. Velioğlu, “Quasi-Cartan companions of elliptic cluster algebras,” Ph.D. - Doctoral Program, Middle East Technical University, 2016.