Modules For The Virasoro-like Algebra And Its Q-analogue

Let d be an integer. Witt algebra is the derivation algebra of the Laurent polynomial ring Ad= C[t1±1,t2±1…,td±1] in commuting variables, denoted by Wd. Rational quantum tori Cq are very similar to Ad, which are the unital associative algebra, and denote their derivation Lie algebras by Der(Cq). When d=2, Witt algebra W2 has natural subalgebras L=L2 and L= L2,which called the Virasoro-like algebra by abusing language, but L2 does not has a Cartan subalgebra, at the same time,we call the algebras L(q)=L2(q) and L(q)=L2(q) which are the subalgebras of Der(Cq) be q-Virasoro-like algebra.In 1986, Guangyu Shen defined a class of weight modules Fbα(V) over Witt algebra for α∈C, b∈C and a gld-module V on which the identity matrix acts as the scalar b. Let Fbα(V)= V(?)Ad, then Fbα(V) became a natural Wd-module. Later, these modules as well as the functor Fbα,known as Larsson funcror, were also studied by Rao. When d=2, these modules Fbα(V) can be realized as modules over Virasoro-like algebra. Similarly, we also define the modules Fqα(V) over q-Virasoro-like algebra. In this paper, we determine the irreducibility these modules using a unified method,and our results are as follows: (1) We determine the irreducibility of Fbα(V), especially in case of the weight spaces are infinite dimensional, these modules provide new irreducible weight modules with infinite dimensional weight spaces for Virasoro-like algebra; (2) However the weight spaces V are finite or infinite dimensional, these modules Fqα(V) are the sum of two submodules, where one of the submodules is irreducible, and we also determine the irreducibility of the other submodule.In addition, we characterize the modules over L which are free C[(?)1,(?)2]-modules of rank 1, these modules are isomorphic to Ω(A2,α).