Geometry of Yang--Baxter maps: pencils of conics and quadrirational mappings

2003/07/01 by V. E. Adler, A. I. Bobenko, Yu. B. Suris · 1 citation
Mathematics · Physics and Astronomy · #math.QA #math-ph #math.AG #math.MP #nlin.SI

paper · pdf

published as Commun. Anal. Geom., 2004, vol. 12, p. 967-1007 · LaTeX, 40pp, 3 Figs

arxiv created 2003/07/01 · arxiv updated 2009/11/30

Abstract

Birational Yang-Baxter maps (`set-theoretical solutions of the Yang-Baxter equation') are considered. A birational map (x,y)↦(u,v) is called quadrirational, if its graph is also a graph of a birational map (x,v)↦(u,y). We obtain a classification of quadrirational maps on \CP¹×\CP¹, and show that all of them satisfy the Yang-Baxter equation. These maps possess a nice geometric interpretation in terms of linear pencil of conics, the Yang-Baxter property being interpreted as a new incidence theorem of the projective geometry of conics.

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