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...
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...
Hamilton-Jacobi theory of discrete, regular constrained systems
Güler, Y. (Springer Science and Business Media LLC, 1987-8)
The Hamilton-Jacobi differential equation of a discrete system with constraint equationsG α=0 is constructed making use of Carathéodory’s equivalent Lagrangian method. Introduction of Lagrange’s multipliersλ˙α as generalized velocities enables us to treat the constraint functionsG α as the generalized momenta conjugate toλ˙α. Canonical equations of motion are determined.
Citation Formats
K. Velioğlu, “Quasi-Cartan companions of elliptic cluster algebras,” Ph.D. - Doctoral Program, Middle East Technical University, 2016.