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arXiv · 1603.08325

Mutations of the cluster algebra of type $\boldsymbol{A^{(1)}_1}$ and the periodic discrete Toda lattice

Atsushi Nobe
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Abstract

A direct connection between two sequences of points, one of which is generated by seed mutations in the cluster algebra of type $A^{(1)}_1$ and the other by time evolutions of the periodic discrete Toda lattice, is explicitly given. In this construction, each of them is realized as an orbit of a QRT map and specialization of the parameters in the maps and appropriate choices of the initial points relate them. The connection with the periodic discrete Toda lattice enables us a geometric interpretation of the seed mutations in the cluster algebra of type $A^{(1)}_1$ as addition of points on an elliptic curve arising as the spectral curve of the Toda lattice.

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Atsushi Nobe. 2018-02-13. Mutations of the cluster algebra of type $\boldsymbol{A^{(1)}_1}$ and the periodic discrete Toda lattice. https://doi.org/10.1088/1751-8113%2F49%2F28%2F285201

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