A CLASS OF GRADED LIE-ALGEBRAS OF VECTOR-FIELDS AND FIRST-ORDER DIFFERENTIAL-OPERATORS

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