TY - BOOK

T1 - Discrete integrable systems associated with deformations of cluster maps

AU - Kim, Wookyung

PY - 2024

Y1 - 2024

N2 - In this thesis, we study a discrete dynamical system provided by the deformation of cluster mutations associated with cluster algebras of finite Dynkin diagrams of type A_{2N} (N ≥ 3), C_2(∼= B_2), B_3, B_4, D_4 and D_6. In the first part of the thesis, we show that the corresponding cluster algebras exhibit a remarkable periodicity phenomenon, known as Zamolodchikov periodicity. We present a particular deformation, which is a novel approach introduced by Hone and Kouloukas. This procedure modifies cluster mutation in such a way that it preserves the natural presymplectic form in cluster algebras. For the cases of type C_2, B_3, D_4, we construct Liouville integrable maps defined by the specific composition of deformed cluster mutations and show that the new cluster algebras emerge by considering Laurentification, which is a lifting of the map into a higher-dimensional space where the Laurent property is exhibited. The corresponding deformed integrable maps are also closely related to Somos-type recurrences.In particular, we prove the integrability of the birational map defined by a sequence of mutations in cluster algebra of type A_{2N} . We examine the deformation of discrete dynamics in cluster algebra of type A_6 and compare the result associated with the type A_4 case. Following this, we introduce a local expansion operation on quivers, which provides a special family of quivers corresponding to specific types of cluster algebras. We demonstrate that these cluster algebras are obtained by lifting the deformation of type A_{2N} maps via Laurentification. We also show that the discrete dynamics in the new cluster algebras, induced from cluster algebras of type A_{2N}, B_4 and D_6 via deformation, admit the tropical (max-plus) analogue of the system of homogeneous recurrences. This allows us to calculate the exact degree growth of the discrete dynamical system.

AB - In this thesis, we study a discrete dynamical system provided by the deformation of cluster mutations associated with cluster algebras of finite Dynkin diagrams of type A_{2N} (N ≥ 3), C_2(∼= B_2), B_3, B_4, D_4 and D_6. In the first part of the thesis, we show that the corresponding cluster algebras exhibit a remarkable periodicity phenomenon, known as Zamolodchikov periodicity. We present a particular deformation, which is a novel approach introduced by Hone and Kouloukas. This procedure modifies cluster mutation in such a way that it preserves the natural presymplectic form in cluster algebras. For the cases of type C_2, B_3, D_4, we construct Liouville integrable maps defined by the specific composition of deformed cluster mutations and show that the new cluster algebras emerge by considering Laurentification, which is a lifting of the map into a higher-dimensional space where the Laurent property is exhibited. The corresponding deformed integrable maps are also closely related to Somos-type recurrences.In particular, we prove the integrability of the birational map defined by a sequence of mutations in cluster algebra of type A_{2N} . We examine the deformation of discrete dynamics in cluster algebra of type A_6 and compare the result associated with the type A_4 case. Following this, we introduce a local expansion operation on quivers, which provides a special family of quivers corresponding to specific types of cluster algebras. We demonstrate that these cluster algebras are obtained by lifting the deformation of type A_{2N} maps via Laurentification. We also show that the discrete dynamics in the new cluster algebras, induced from cluster algebras of type A_{2N}, B_4 and D_6 via deformation, admit the tropical (max-plus) analogue of the system of homogeneous recurrences. This allows us to calculate the exact degree growth of the discrete dynamical system.

U2 - 10.17635/lancaster/thesis/2499

DO - 10.17635/lancaster/thesis/2499

M3 - Doctoral Thesis

PB - Lancaster University

ER -