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

V. E. Adler, A. I. Bobenko, Yu B. Suris

Research output: Contribution to journalArticlepeer-review

61 Scopus citations


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 ℂℙ 1 × 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.

Original languageEnglish
Pages (from-to)967-1007
Number of pages41
JournalCommunications in Analysis and Geometry
Issue number5
StatePublished - Dec 2004
Externally publishedYes


  • 3D-consistency
  • Quadrirational map
  • Set-theoretical solution
  • Yang-Baxter equation
  • Yang-Baxter map


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