Simple and collective twisted symmetries

TL;DR

This paper reviews recent advances in the theory of twisted symmetries, providing a unifying geometric framework that relates various types of these symmetries to classical Lie symmetries through Frobenius reduction.

Contribution

It offers a comprehensive geometric unification of different twisted symmetries, extending previous surveys and clarifying their relation to standard symmetries via gauge transformations.

Findings

Unified geometric description of twisted symmetries.

Relation of twisted symmetries to gauge transformations.

Application of Frobenius reduction to symmetry distributions.

Abstract

After the introduction of $\lambda$-symmetries by Muriel and Romero, several other types of so called "twisted symmetries" have been considered in the literature (their name refers to the fact they are defined through a deformation of the familiar prolongation operation); they are as useful as standard symmetries for what concerns symmetry reduction of ODEs or determination of special (invariant) solutions for PDEs and have thus attracted attention. The geometrical relation of twisted symmetries to standard ones has already been noted: for some type of twisted symmetries (in particular, $\lambda$ and $\mu$-symmetries), this amounts to a certain kind of gauge transformation. In a previous review paper [G. Gaeta, "Twisted symmetries of differential equations", {\it J. Nonlin. Math. Phys.}, {\bf 16-S} (2009), 107-136] we have surveyed the first part of the developments of this theory; in the present paper we review recent developments. In particular, we provide a unifying geometrical description of the different types of twisted symmetries; this is based on the classical Frobenius reduction applied to distribution generated by Lie-point (local) symmetries.