It is proved that uniformly bounded simple modules over higher-rank super-Virasoro algebras are modules of the intermediate series, and that simple modules with finite dimensional weight spaces are either modules ofthe intermediate series or generalized highest weightmodules

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.

Let B be the Lie algebra of Block type over C with basis {Lα,i, C | α, i ∈ Z, i−1} and relations [Lα,i,Lβ,j] = ((i +1)β −(j +1)α)Lα+β,i+j +αδα+β,0δi+j,−2C, [C,Lα,i] = 0. In this paper, it is proved that a quasifinite irreducible B-module is a highest or lowest weight module. Furthermore, the quasifinite irreducible highest weight modules are classified and the unitary ones are proved to be trivial. 

To any nonzero additive subgroup G of an algebraically closed %eld F of characteristic zero and s=0; 1, there corresponds a Lie algebra B(s;G) of Block type, with basis {xa; i|a ∈G, i ∈ Z+}, and relation [xa; i; xb; j]=s(b−a)xa+b; i+j +((a−1+s)j−(b−1+s)i)xa+b; i+j−1. In this paper, it is proved that B(s;G) has a nontrivial quasi%nite module if and only if s =1 and G is isomorphic to Z, and that a quasi%nite B(1; Z)-module is a highest or lowest weight module. Furthermore, the quasi%nite irreducible highest weight B(1; Z)-modules are classi%ed and the unitary ones are proved to be trivial.

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