Idempotent biquadratics, Yang-Baxter maps and birational representations of Coxeter groups

Abstract

A transformation is obtained which completes the unification of quadrirational Yang-Baxter maps and known integrable multi-quadratic quad equations. By combining theory from these two classes of quad-graph models we find an extension of the known integrability feature, and show how this leads subsequently to a natural extension of the associated lattice geometry. The extended lattice is encoded in a birational representation of a particular sequence of Coxeter groups. In this setting the usual quad-graph is associated with a subgroup of type BC_n, and is part of a larger and more symmetric ambient space. The model also defines, for instance, integrable dynamics on a triangle-graph associated with a subgroup of type A_n, as well as finite degree-of-freedom dynamics, in the simplest cases associated with affine-E6 and affine-E8 subgroups. Underlying this structure is a class of biquadratic polynomials, that we call idempotent, which express the trisection of elliptic function periods algebraically via the addition law.

40 pages, 9 figures, 2 tables

References in corpus (2)

• Yang-Baxter maps: dynamical point of view
• A multidimensionally consistent version of Hirota's discrete KdV equation

Cited by in corpus (9)

• On non-abelian quadrirational Yang-Baxter maps
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• Quadrirational Yang-Baxter maps and the elliptic Cremona system
• On the lattice-geometry and birational group of the six-point multi-ratio equation
• Integrable multi-component difference systems of equations
• Reflection groups and discrete integrable systems
• Coble's group and the integrability of the Gosset-Elte polytopes and tessellations
• $2^n-$rational maps