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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F47813059%3A19610%2F25%3AA0000184" target="_blank" >RIV/47813059:19610/25:A0000184 - isvavai.cz</a>

Výsledek na webu

<a href="https://www.sciencedirect.com/science/article/pii/S016727892500137X" target="_blank" >https://www.sciencedirect.com/science/article/pii/S016727892500137X</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.physd.2025.134658" target="_blank" >10.1016/j.physd.2025.134658</a>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

We consider the reduced quasi-classical self-dual Yang–Mills equation (rYME) and two recently found (Jahnová and Vojčák, 2024) invertible recursion operators Rq and Rm for its full-fledged (in a given differential covering) nonlocal symmetries. We introduce a Z-grading on the Lie algebra symLτ(E) of all nonlocal Laurent polynomial symmetries of the rYME and prove that both the operators Rq and Rm are Z-graded automorphisms of the underlying vector space on the set symLτ(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 symLτ(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 Rq and Rm. Finally, the linear dependence/independence of the (sub)set of generators of V is discussed.

Název v anglickém jazyce

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

Popis výsledku anglicky

We consider the reduced quasi-classical self-dual Yang–Mills equation (rYME) and two recently found (Jahnová and Vojčák, 2024) invertible recursion operators Rq and Rm for its full-fledged (in a given differential covering) nonlocal symmetries. We introduce a Z-grading on the Lie algebra symLτ(E) of all nonlocal Laurent polynomial symmetries of the rYME and prove that both the operators Rq and Rm are Z-graded automorphisms of the underlying vector space on the set symLτ(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 symLτ(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 Rq and Rm. Finally, the linear dependence/independence of the (sub)set of generators of V is discussed.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Physica D: Nonlinear Phenomena

ISSN

0167-2789

e-ISSN

1872-8022

Svazek periodika

476

Číslo periodika v rámci svazku

June

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

14

Strana od-do

„134658-14“-„134658-14“

Kód UT WoS článku

001475789500001

EID výsledku v databázi Scopus

2-s2.0-105002816631