The C-(symmetric) quadrilateral lattice, its transformations and the algebro-geometric construction

TL;DR

This paper explores the geometric and algebraic properties of the C-quadrilateral lattice, highlighting its role in discrete integrable systems, and provides an algebro-geometric construction and transformation reduction.

Contribution

It introduces the symmetric C-quadrilateral lattice, analyzes its geometric properties, and develops an algebro-geometric construction and transformation framework.

Findings

The C-quadrilateral lattice interprets the discrete CKP equation geometrically.

The Gallucci theorem is key to the lattice's multidimensional consistency.

A reduction of the vectorial fundamental transformation to the CQL is established.

Abstract

The C-quadrilateral lattice (CQL), called also the symmetric lattice, provides geometric interpretation of the discrete CKP equation within the quadrilateral lattice (QL) theory. We discuss affine-geometric properties of the lattice emphasizing the role of the Gallucci theorem in the multidimensional consistency of the CQL. Then we give the algebro-geometric construction of the lattice. We also present the reduction of the vectorial fundamental transformation of the QL to the CQL case. In the Appendix we show a relation between the QL and the so called Darboux maps.