Construction of first-order invariant differential operators
type: Item
instance of: scholarly article
title: Construction of first-order invariant differential operators
author: O. L. Kurnyavko
author: I. V. Shirokov
published in: Russian Mathematics
publication date: 2024-11-14T00:00:00Z
review text: The paper investigates the problem of constructing invariant systems of vector fields on Lie groups. The authors focus on invariant differential operators, a critical topic in mathematical physics and geometry. The study of the paper highlights a special class of Lie algebras, termed ``Integrable Lie algebras'' for which the considered problem can be solved using elementary methods. In Section 1, the authors introduced the formulation of the considered problem in the paper. Namely, they defined the problem of constructing invariant vector fields, introduced the necessary mathematical preliminaries and stated the challenges associated with non-canonical coordinate systems. The main results of the paper are presented in Sections 2 and 3. First of all, Section 2, titled ``Invariant differential operators'', introduced the first result of the paper which is a method to construct left- and right-invariant vector fields using canonical coordinates of the second kind. The authors were identified that integrable Lie algebras are a special class for which the problem is tractable. Next, Section 3, titled ``An example of invariant differential operators on five-dimensional Lie group'', introduced the second result of the paper. Specifically, this is a practical example involving a five-dimensional unsolvable Lie algebra, detailing the construction of invariant vector fields and their commutation relations.
reviewed by: Le Anh Vu
Mathematics Subject Classification ID: 22E05
Mathematics Subject Classification ID: 17B66
zbMATH DE Number: 7944480
zbMATH Keywords: Lie algebra
zbMATH Keywords: Lie group
zbMATH Keywords: invariant differential operator
zbMATH Keywords: left-invariant vector field
zbMATH Keywords: right-invariant vector field
zbMATH Keywords: invariant differentiation operator
MaRDI profile type: Publication
cites work: Computation of composition functions and invariant vector fields in terms of structure constants of associated Lie algebras
cites work: Q3545540
cites work: Q3311238
cites work: Q3715336
cites work: Differential invariants: Theorem of commutativity.
DOI: 10.3103/S1066369X24700336
description: scientific article; zbMATH DE number 7944480