Shanghai Jiaotong University

A classification of the pairs $(A,D)$ is obtained, where $A$ is
a

commutative associative algebra with an identity element over an

algebraically closed field of characteristic zero and $D$ is a finite

dimensional subspace of locally-finite commuting derivations of

$A$ such that $A$ is $D$-simple. Such pairs $(A,D)$ are the fundamental

ingredients of constructing Lie algebras of generalized Cartan type.

>From the such pairs $(A,D)$, some new infinite-dimensional simple
Lie

algebras can be constructed, which are in general not finitely graded.