A class of graded Lie algebras of vector fields and first order differential operators

Abstract

Finite-dimensional Lie algebras of polynomial vector fields on Rn, that contain the elements ∂/∂xi and xi(∂/∂xi) for i=1...n were studied. To any Lie algebra 𝔏 of this class, an N-valued n×n matrix A and a set of special elements 𝒮⊆{1,...,n} are associated. It is proven that the pair (A,𝒮) necessarily satisfies two properties. Conversely, to any pair (A,𝒮) satisfying those two properties is associated a Lie algebra 𝔏(A,𝒮), such that 𝔏(A,𝒮) is maximal in the class of all 𝔏 with matrix A and special elements 𝒮. For the Lie algebras 𝔏(A,𝒮) the possible extensions to first order differential operators, and its modules of C∞ functions are discussed. © 1994 American Institute of Physics.