In this paper, we determine the isomorphism classes of contact simple Lie algebras constructed from locally-finite commuting derivations, introduced earlier by the second author. These algebras are in general not finitely graded.

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.

确定了特征0的代数闭域上与局部有限导子相关的中心单Poisson代数的结构。这些Poisson代数的Lie代数结构一般来说不是有限阶化的。

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.