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。

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.

One of the four well-known series of simple Lie algebras of Cartan type is the series of Lie algebras of Special type, which are divergence-free Lie algebras associated with polynomial algebras and the operators of taking partial derivatives, connected with volume-preserving diffeomorphisms. In this paper, we determine the structure space of the divergence-free Lie algebras associated with pairs of a commutative associative algebra with an identity element and its finite-dimensional commutative locally finite derivation subalgebra such that the commutative associative algebra is derivation-simple with respect to the derivation subalgebra.

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.

In a recent paper by Xu, some simple Lie algebras of generalized Cartan type were constructed, using the mixtures of grading operators and down-grading operators. Among them are the simple Lie algebras of generalized Witt type, which are in general nongraded and have no torus. In this paper, some representations of these simple Lie algebras of generalized Witt type are presented.