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), t hat contain the elements partial derivative/partial derivative x(i) an d x(i)(partial derivative/partial derivative x(i)) for i = 1...n were studied. To any Lie algebra l of this class, an N-valued n X n matrix A and a set of special elements l subset of{1,...,n} are associated. I t is proven that the pair (A,j) necessarily satisfies two properties. Conversely, to any pair (A,j) satisfying those two properties is assoc iated a Lie algebra l(A,j), such that l(A,j) is maximal in the class o f all l with matrix A and special elements l. For the Lie algebras l(A ,j) the possible extensions to first order differential operators, and its modules of C-infinity functions are discussed. (C) 1994 American Institute of Physics.