Geometric algebra and quadrilateral lattices

TL;DR

This paper explores quadrilateral lattices within projective spaces over division rings, extending classical geometric algebra results to noncommutative settings and analyzing their transformations and permutability properties.

Contribution

It introduces noncommutative discrete Darboux equations and develops a framework for fundamental transformations and permutability theorems in noncommutative quadrilateral lattices.

Findings

Noncommutative discrete Darboux equations formulated

Fundamental transformation and permutability theorems established

Potential for noncommutative B-(Moutard) quadrilateral lattices explored

Abstract

Motivated by the fundamental results of the geometric algebra we study quadrilateral lattices in projective spaces over division rings. After giving the noncommutative discrete Darboux equations we discuss differences and similarities with the commutative case. Then we consider the fundamental transformation of such lattices in the vectorial setting and we show the corresponding permutability theorems. We discuss also the possibility of obtaining in a similar spirit a noncommutative version of the B-(Moutard) quadrilateral lattices.