We prove that an irreducible quasifinite module over the central extension of the Lie algebra of N

For a nondegenerate additive subgroup G of the n-dimensional vector space Fn over an algebraically closed field F of characteristic zero, there is an associative algebra and a Lie algebra of Weyl typeWðG; nÞ spanned by all differential operators uDm1 1 ?Dmn n for uAF½G (the group algebra), and m1;y;mnX0; where D1;y;Dn are degree operators. In this paper, it is proved that an irreducible quasifinite WðZ; 1Þ-module is either a highest or lowest weight module or else a module of the intermediate series; furthermore, a classification of uniformly boundedWðZ; 1Þ-modules is completely given. It is also proved that an irreducible quasifinite WðG; nÞ-module is a module of the intermediate series and a complete classification of quasifinite WðG; nÞ-modules is also given, if G is not isomorphic to Z.

We classify the Harish-Chandra modules over the higher rank Virasoro and super-Virasoro algebras: It is proved that a Harish-Chandra module, i.e., an irreducible weight module with finite weight multiplicities, over a higher rank Virasoro or super-Virasoro algebra is either a module of the intermediate series, or a finitely-dense module. As an application, it is also proved that an indecomposable weight module with finite weight multiplicities over a generalized Witt algebra is either a uniformly bounded module (i.e., a module with weight multiplicities uniformly bounded) with all nonzero weights having the same multiplicity, or a finitely-dense module, as long as the generalizedWitt algebra satisfies one minor condition.

Abstract. For an (є; Ґ)-color-commutative associative algebra A with an identity element over a field F of characteristic not 2, and for a color-commutative color-derivation subalgebra D of A, denote by A[D] the associative subalgebra of End(A) generated by A (regarding as operators on A via multiplication) and D. It is proved that, as an associative algebra, A[D] is Ґ-graded simple if and only if A is Ґ-graded D-simple. Suppose A is Ґ-graded D-simple. Then, (a) A[D] is a free left A-module; (b) as a Lie color algebra, the subquotient [A[D];A[D]]=Z(A[D])\[A[D];A[D]] is simple (except one minor case), where Z(A[D]) is the color center of A[D]. The structure of this subquotient is explicitly described.

We classify all the pairs of a commutative associative algebra with an identity element and its finite-dimensional locally finite Abelian derivation subalgebra such that the commutative associative algebra is derivation-simple with respect to the derivation subalgebra over an algebraically closed field with characteristic 0. Such pairs are the fundamental ingredients for constructing simple Lie algebras of Cartan type. Moreover, we determine the isomorphism classes of the simple Lie algebras of Witt type. The structure space of these algebras is given explicitly.

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.

We compute the low dimensional cohomologies H q(gcN,C), Hq(gcN,C) of the infinite rank general Lie conformal algebras gcN with trivial coefficients for q<3, N51 or q<2, N>2. We also prove that the cohomology of gcN with coeffi-cients in its natural module is trivial, i.e., H*(gcN,C[э]N)=0, and thus partially solve an open problem of Bakalov-Kac-Voronov

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.

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.

Let B be the Lie algebra of Block type over C with basis {Lα,i, C | α, i ∈ Z, i−1} and relations [Lα,i,Lβ,j] = ((i +1)β −(j +1)α)Lα+β,i+j +αδα+β,0δi+j,−2C, [C,Lα,i] = 0. In this paper, it is proved that a quasifinite irreducible B-module is a highest or lowest weight module. Furthermore, the quasifinite irreducible highest weight modules are classified and the unitary ones are proved to be trivial. 