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

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.

n one of our recent papers, the associative and the Lie algebras of Weyl type A[D] = AUF[D] were de ned and studied, where A is a commutative associative algebra with an identity element over a eld 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 eld of characteristic 0, D consisting of locally nite but not locally nilpotent derivations of A, are studied. The isomorphism classes and automorphism groups of these associative and Lie algebras are determined.

The present paper is another step toward the classification of Z × Z–graded Lie algebras: we classify Z × Z–graded Lie algebras A =⊕i,j∈Z Ai,j over a field F of characteristic 0 with dimAi,j ≤ 1 for i, j ∈ Z, satisfying: (1) A0 = ⊕t∈Z Ai,0 Vir, the centerless Virasoro algebra, (2) A0,−1, A0,0, A0,1 span the 3-dimensional simple Lie algebra sl2, (3) A is generated by the centerless Virasoro algebra and sl2. These algebras include some rank 2 centerless Virasoro algebras, and some rank 2 Block algebras and their extensions.

We classify the pairs (A,D) consisting of an (e,ſ)-color-commutative associative algebra A with an identity element over an algebraically closed field F of characteristic zero and a finite dimensional subspace D of (e,ſ)-color-commutative locally finite color-derivations of A such that A is G-graded D-simple and the eigenspaces for elements of D are G-graded. Such pairs are the important ingredients in constructing some simple Lie color algebras which are in general not finitely-graded. As some applications, using such pairs, we construct new explicit simple Lie color algebras of generalized Witt type, Weyl type.