The B-quadrilateral lattice, its transformations and the algebro-geometric construction

Abstract

Abstract. The B-quadrilateral lattice (BQL) provides geometric interpretation of Miwa's discrete BKP equation within the quadrialteral lattice (QL) theory. After discussing the projectivegeometric properties of the lattice we give the algebro-geometric construction of the BQL ephasizing the role of Prym varieties and the corresponding theta functions. We also present the reduction of the vectorial fundamental transformation of the QL to the BQL case.

Generalized isothermic lattices

“…The additional assumption about coplanarity of the points x 0 , x 12 , x 13 , x 23 on the sphere is then equivalent to the additional assumption about concircularity of the corresponding points s 0 , s 12 , s 13 , s 23 . In view of Lemma 2 we obtain therefore another configuration of circles, the so called Clifford configuration, which can be described as follows.…”

“…where f is an appropriate function [46,23]. Because of the additional quadratic constraints the function is given by (compare also equations (3.16) and (3.16)) (4.14) Figure 8.…”

“…The corresponding reduction of the quadrilateral lattice was called in [23], because of its connection with the discrete BKP equation, the B-quadrilateral lattice (BQL). In fact, research in this direction prevented me from publication of the above mentioned generalization of discrete isothermic surfaces, announced however in my talk during the Workshop "Geometry and Integrable Systems" (Berlin, 3-7 June 2005).…”

Integrable lattices and their sublattices. II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices

“…Remark 1.1. In [14] it was shown that the discrete Moutard system (1.1) characterizes algebraically (up to a gauge transformation) special quadrilateral lattices x = [ψ] : Z N → P(V) in the projectivization of V, which satisfy the following additional local linear constraint: any point x of the lattice and its neighbours x ij , x ik and x jk , for i, j, k distinct, are coplanar.…”

“…These two relations, which give the geometric characterization of the B-quadrilateral lattice [14], can be interpreted as the following linear self-adjoint system of equations on the honeycomb white lattice (see Figure 3) 1…”