1301.4613 (James Atkinson)

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

James Atkinson

A transformation is obtained which completes the unification of quadrirational Yang-Baxter maps and known integrable multi-quadratic quad equations. A generalised discrete dynamics emerges by combining theory from these two classes of quad-graph models, it 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, 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.

View original: http://arxiv.org/abs/1301.4613