Construction of first-order invariant differential operators (Q6638441)

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.