Andrew Hone (U of Kent)
Advanced Graduate Course of 4 two-hour lectures
The purpose of these lectures is to give a brief introduction to cluster algebras, and explain some of the connections with discrete integrable systems. The following is a rough outline of the particular topics to be covered:
- Background and examples of cluster algebras: Somos sequences in number theory; Laurent property; Abel pentagon identity, Lyness map and the dilogarithm; Zamolodchikov Y-systems; Plucker coordinates in Grassmanians; discrete Hirota equations.
- Cluster algebras without coefficients: quivers and quiver mutation; exchange matrices and matrix mutation; cluster variables and cluster mutation.
- Poisson and symplectic structures: Poisson brackets; symplectic forms; Gekhtman-Shapiro-Vainshtein Poisson structure for cluster algebras; examples of noninvariant symplectic leaves; compatible presymplectic forms and reduction to symplectic coordinates.
- Cluster mutation-periodiscity: Mutation-periodic quivers; Fordy & Marsh classification of period 1 and recurrence relations; primitives and affine Dynkin diagrams; Dodgson condensation; linear relations for cluster variables.
- Tropical relations and algebraic entropy: Growth of denominators; max-plus tropical algebra; dynamics of tropical maps; algebraic entropy; experimental classification.
- Discrete integrable systems: Affine A-type cluster algebras and dressing chain - monodromy matrix and Lenard-Magri chain; discrete Hirota and reduction to Somos/Gale-Robinson; connection with QRT maps.
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