Transformation groups in difference spaces

V. A. Dorodnitsyn

Itogi Nauki i Tekhniki. Seriya Sovremennye Problemy Matematiki. Noveishie Dostizheniya, Tome 34 (1989), pp. 149-191.

Résumé

We consider formal groups of transformations on the space of differential and net (finite-difference) variables. We show that preservation of meaning of difference derivatives under transformations necessarily leads to Lie–Bäcklund group. We derive formulas for extension to net variables and formulate criteria for preservation of uniformity and invariance of differences of the network and a test for the invariance of difference equations. With the help of formal Newton series we construct the ideal of the algebra of all Lie–Bäcklund operators on a uniform network which is used to derive tests for the conservatism of difference equations on the basis of a discrete analog of Noether's identity.