For a simple module M over the positive part of the Virasoro algebra (actually for any simple module over some finite dimensional solvable Lie algebras a r ) and any α ∈ C, a class of weight modules N (M, α) over the Virasoro algebra are constructed. The necessary and sufficient condition for N (M, α) to be simple is obtained. We also determine the necessary and sufficient conditions for two such irreducible Virasoro modules to be isomorphic. Many examples for such irreducible Virasoro modules with different features are provided. In particular the irreducible weight Virasoro modules Γ(α 1 , α 2 , λ 1 , λ 2 ) are defined on the polynomial algebra C[x] ⊗ C[t, t −1 ] for any α 1 , α 2 , λ 1 λ 2 ∈ C with λ 1 or λ 2 nonzero. By twisting the weight modules N (M, α) we also obtain nonweight simple Virasoro modules N (M, β) for any β ∈ C[t, t −1 ].

In this paper, we studied the jet modules for the centerless Virasoro-like algebra which is the Lie algebra of the Lie group of the area-preserving diffeomorphisms of a 2-torus. The jet modules are certain natural modules over the Lie algebra of semi-direct product of the centerless Virasoro-like algebra and the Laurent polynomial algebra in two variables. We reduce the irreducible jet modules to the finite-dimensional irreducible modules over some infinite-dimensional Lie algebra and then characterize the irreducible jet modules with irreducible finite dimensional modules over sl 2 . To determine the indecomposable jet modules, we use the technique of polynomial modules in the sense of [BB, BZ]. Consequently, indecomposable jet modules are described using modules over the algebra B + , which is the "positive part" of a Block type algebra studied first by [DZ] and recently by [IM, I]).

Let Λ=K[x,y]/(xp,yq,xy) be a Gelfand–Ponomarev algebra, where p,q ∈ ℕ, and [Formula: see text] the matrix problem of Λ. In the present paper, we determine the canonical forms of indecomposable modules of Λ, and give a one-to-one correspondence between the iso-classes of the modules over Λ and the canonical forms over [Formula: see text]. Finally, we discuss some properties of the module variety of Λ using the canonical forms.

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Let d be a positive integer, q=(qij)d×d be a d×d matrix, ℂq be the quantum torus algebra associated with q. We have the semidirect product Lie algebra $\mathfrak{g}$=Der(ℂq)⋉Z(ℂq), where Z(ℂq) is the centre of the rational quantum torus algebra ℂq. In this paper, we construct a class of irreducible weight $\mathfrak{g}$-modules $\mathcal{V}$α (V,W) with three parameters: a vector α∈ℂd, an irreducible $\mathfrak{gl}$d-module V and a graded-irreducible $\mathfrak{gl}$N-module W. Then, we show that an irreducible Harish Chandra (uniformaly bounded) $\mathfrak{g}$-module M is isomorphic to $\mathcal{V}$α(V,W) for suitable α, V, W, if the action of Z(ℂq) on M is associative (respectively nonzero).