A straightforward construction of $\Bbb Z$-graded Lie algebras of full-fledged nonlocal symmetries via recursion operators

TL;DR

This paper introduces a novel $Z$-grading on the Lie algebra of nonlocal symmetries for the reduced quasi-classical self-dual Yang-Mills equation, enabling a structured construction of a Lie subalgebra containing key symmetries and their hierarchies.

Contribution

It establishes a $Z$-grading automorphism for the symmetry Lie algebra and constructs a Lie subalgebra of nonlocal symmetries using seed generators, a novel approach in the field.

Findings

$Z$-grading automorphisms of symmetry Lie algebra established

Construction of a Lie subalgebra containing known nonlocal symmetries

Analysis of symmetry hierarchies and generator independence

Abstract

We consider the reduced quasi-classical self-dual Yang-Mills equation (rYME) and two recently found (Jahnov\'{a} and Voj\v{c}\'{a}k, 2024) invertible recursion operators $\mathcal{R}^q$ and $\mathcal{R}^m$ for its full-fledged (in a given differential covering) nonlocal symmetries. We introduce a $\mathbb{Z}$-grading on the Lie algebra $\mathrm{sym}_{\mathcal{L}}^{\tau^W} (\mathcal E)$ of all nonlocal Laurent polynomial symmetries of the rYME and prove that both the operators $\mathcal{R}^q$ and $\mathcal{R}^m$ are $\mathbb{Z}$-graded automorphisms of the underlying vector space on the set $\mathrm{sym}_{\mathcal{L}}^{\tau^W} (\mathcal E)$. This inter alia implies that all its vector subspaces formed by all homogeneous elements of a given fixed degree (i.e. a weight in the context below) are mutually isomorphic, and thus each of them can be uniquely reconstructed from the vector space of all homogeneous symmetries of the zero degree. To the best of our knowledge, such a result is unparalleled in the current body of literature. The obtained results are used for the construction of a Lie subalgebra $V$ of $\mathrm{sym}_{\mathcal{L}}^{\tau^W} (\mathcal E)$ which contains all known to us nonlocal Laurent polynomial symmetries of the rYME. The Lie algebra $V$ is subsequently described as the linear span of the orbits of a set of selected zero-weight symmetries - we refer to them as to the seed generators of $V$. Further, we study the hierarchies of symmetries related to the seed generators under the action of the group of recursion operators generated by $\mathcal{R}^q$ and $\mathcal{R}^m$. Finally, the linear dependence/independence of the (sub)set of generators of $V$ is discussed.