Maximal Green Sequences of Exceptional Finite Mutation Type Quivers

Download
2014-01-01
Maximal green sequences are particular sequences of mutations of quivers which were introduced by Keller in the context of quantum dilogarithm identities and independently by Cecotti-Cordova-Vafa in the context of supersymmetric gauge theory. The existence of maximal green sequences for exceptional finite mutation type quivers has been shown by Alim-Cecotti-Cordova-Espahbodi-Rastogi-Vafa except for the quiver X-7. In this paper we show that the quiver X-7 does not have any maximal green sequences. We also generalize the idea of the proof to give sufficient conditions for the non-existence of maximal green sequences for an arbitrary quiver.
SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS

Suggestions

Maximal green sequences of skew-symmetrizable 3 x 3 matrices
Seven, Ahmet İrfan (2014-01-01)
Maximal green sequences are particular sequences of mutations of skew-symmetrizable matrices which were introduced by Keller in the context of quantum dilogarithm identities and independently by Cecotti-Cordova-Vafa in the context of supersymmetric gauge theory. In this paper we study maximal green sequences of skew-symmetrizable 3 x 3 matrices. We show that such a matrix with a mutation-cyclic diagram does not have any maximal green sequence. We also obtain some basic properties of maximal green sequences ...
Mutation Classes of 3 x 3 Generalized Cartan Matrices
Seven, Ahmet İrfan (2012-01-01)
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 different areas of mathematics. A particular analogy exists 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 Caftan matrices. In this paper, we...
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...
All timelike supersymmetric solutions of three-dimensional half-maximal supergravity
DEĞER, NİHAT SADIK; Moutsopoulos, George; Samtleben, Henning; Sarıoğlu, Bahtiyar Özgür (2015-06-22)
We first classify all supersymmetric solutions of the 3-dimensional half-maximal ungauged supergravity that possess a timelike Killing vector by considering their identification under the complexification of the local symmetry of the theory. It is found that only solutions that preserve 16/2(n), 1 <= n <= 3 real supersymmetries are allowed. We then classify supersymmetric solutions under the real local symmetry of the theory and we are able to solve the equations of motion for all of them. It is shown that ...
Quivers of finite mutation type and skew-symmetric matrices
Seven, Ahmet İrfan (Elsevier BV, 2010-11-01)
Quivers of finite mutation type are certain directed graphs that first arised in Fomin-Zelevinsky's theory of cluster algebras. It has been observed that these quivers are also closely related with different areas of mathematics. In fact, main examples of finite mutation type quivers are the quivers associated with triangulations of surfaces. In this paper, we study structural properties of finite mutation type quivers in relation with the corresponding skew-symmetric matrices. We obtain a characterization ...
Citation Formats
A. İ. Seven, “Maximal Green Sequences of Exceptional Finite Mutation Type Quivers,” SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS, pp. 0–0, 2014, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/47545.