For a nondegenerate additive subgroup G of the n-dimensional vector space Fn over an algebraically closed field F of characteristic zero, there is an associative algebra and a Lie algebra of Weyl typeWðG; nÞ spanned by all differential operators uDm1 1 ?Dmn n for uAF½G (the group algebra), and m1;y;mnX0; where D1;y;Dn are degree operators. In this paper, it is proved that an irreducible quasifinite WðZ; 1Þ-module is either a highest or lowest weight module or else a module of the intermediate series; furthermore, a classification of uniformly boundedWðZ; 1Þ-modules is completely given. It is also proved that an irreducible quasifinite WðG; nÞ-module is a module of the intermediate series and a complete classification of quasifinite WðG; nÞ-modules is also given, if G is not isomorphic to Z.

We classify the Harish-Chandra modules over the higher rank Virasoro and super-Virasoro algebras: It is proved that a Harish-Chandra module, i.e., an irreducible weight module with finite weight multiplicities, over a higher rank Virasoro or super-Virasoro algebra is either a module of the intermediate series, or a finitely-dense module. As an application, it is also proved that an indecomposable weight module with finite weight multiplicities over a generalized Witt algebra is either a uniformly bounded module (i.e., a module with weight multiplicities uniformly bounded) with all nonzero weights having the same multiplicity, or a finitely-dense module, as long as the generalizedWitt algebra satisfies one minor condition.

Abstract. For an (є; Ґ)-color-commutative associative algebra A with an identity element over a field F of characteristic not 2, and for a color-commutative color-derivation subalgebra D of A, denote by A[D] the associative subalgebra of End(A) generated by A (regarding as operators on A via multiplication) and D. It is proved that, as an associative algebra, A[D] is Ґ-graded simple if and only if A is Ґ-graded D-simple. Suppose A is Ґ-graded D-simple. Then, (a) A[D] is a free left A-module; (b) as a Lie color algebra, the subquotient [A[D];A[D]]=Z(A[D])\[A[D];A[D]] is simple (except one minor case), where Z(A[D]) is the color center of A[D]. The structure of this subquotient is explicitly described.

In a study of finite-dimensional modules of simple Lie superalgebras, Kac introduced certain indecomposable modules, now known as Kac-modules V

We prove that an irreducible quasifinite module over the central extension of the Lie algebra of N