Diagram rewriting for orthogonal matrices: A study of critical peaks

Abstract

Orthogonal diagrams represent decompositions of isometries of ℝn into symmetries and rotations. Some convergent (that is noetherian and confluent) rewrite system for this structure was introduced by the first author. One of the rules is similar to Yang-Baxter equation. It involves a map h : ]0, π[3 →]0, π[3. In order to obtain the algebraic properties of h, we study the confluence of critical peaks (or critical pairs) for our rewrite system. For that purpose, we introduce parametric diagrams describing the calculation of angles of rotations generated by rewriting. In particular, one of those properties is related to the tetrahedron equation (also called Zamolodchikov equation). © 2008 Springer-Verlag.