Cluster algebras and integrable systems

• Grant number: 14/08512-3
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• Start date: August 18, 2014
• End date: August 17, 2015
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Igor Mencattini
• Grantee: Eber Daniel Chuno Vizarreta
• Supervisor: Michael Gekhtman
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Institution abroad: University of Notre Dame, United States
• Associated to the scholarship: 12/04707-9 - Clusters Algebras and Integrable Systems, BP.DR

Abstract

Since the inventions of cluster algebras (2001) by S. Fomin and A.Zelevinsky, motivated for the study of dual canonical bases and total positivity in semisimple groups, relations with Poisson geometry and Integrable systems were developed. The notion of Poisson bracket compatible with a cluster structure, introduced by M.Gekhtman, M.Shapiro and A.Vainshtein, was used to interpret cluster transformations and matrix mutations from a viewpoint of Poisson Geometry. Motivated by the Poisson properties of cluster algebras, Gekhtman-Shapiro-Vainhstein began the study of Poisson geometry of weighted directed networks in a disk and in an annulus. Then, Gekhtman-Shapiro-Vainhstein (2011) have given an interpretation, via cluster algebraic/weighted directed networks in an annulus, of the Generalized Bäcklund-Darboux transformations between different Coxeter-Toda lattices. Gekhtman and Faybusovich (2000) have defined a multi-Hamiltonian structure, via the Moser map, on particular Coxeter-Toda lattices called elementary Toda lattices which generalizes the relativistic Toda lattice and the non periodic Toda lattice. Our goal is to give a cluster algebraic/directed networks interpretation of the multi-Hamiltonian structure of the elementary Toda lattices. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

CHUNO, EBER. A NON-LOCAL POISSON BRACKET FOR COXETER-TODA LATTICES. Colloquium Mathematicum, v. 161, n. 1, p. 22-pg., 2020-01-01. (14/08512-3)