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.

Xu introduced a class of nongraded Hamiltonian Lie algebras. These Lie algebras have a Poisson bracket structure. In this paper, the isomorphism classes of these Lie algebras are determined by employing a "sandwich" method and by studying some features of these Lie algebras. It is obtained that two Hamiltonian Lie algebras are isomorphic if and only if their corresponding Poisson algebras are isomorphic. Furthermore, the derivation algebras and the second cohomology groups are determined.

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.

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