Structure of Algebras of Weyl Type

In a recent paper by the authors, the associative and the Lie algebras of Weyl type A[D] = A⊗F[D] were introduced, where A is a commutative associative algebra with an identity element over a field F of any characteristic, and F[D] is the polynomial algebra of a commutative derivation subalgebra D of A. In the present paper, a class of the above associative and Lie algebras A[D] with F being a field of characteristic 0 and D consisting of locally finite derivations of A, is studied. The isomorphism classes of these associative and Lie algebras are determined. The structure of these algebras is described explicitly.