Non-weight modules over the affine-Virasoro algebra of type A1

Abstract

In this paper, we study a class of non-weight modules over the affine-Virasoro algebra of type A 1 , which are free modules of rank one when restricted to the Cartan subalgebra (modulo center). We give the classification of such modules. Moreover, the simplicity and the isomorphism classes of these modules are determined.

A new class of irreducible modules over the affine-Virasoro algebra of type $A_1$

“…These modules were introduced in [7] to characterize the U(h)-free modules of rank one for L. It is worthwhile to point out that C[s, t] in each case has the same module structure over the subalgebra span{h i , d i , C | i ∈ Z}. For later use, we need the following known result on conditions for irreducibility and a classification of isomorphism classes for the modules constructed above.…”

“…Next we assume that m ≥ 2 in the following. We need to show that the tensor product module T M is neither isomorphic to a Whittaker module nor isomorphic to a U(h)-free modules of rank one in [7]. For that, let W be a Whittaker module, then W is isomorphic to a quotient of 3), we see that T M is not isomorphic to the irreducible non-weight modules defined in [7].…”

“…In this section, we compare the tensor product modules constructed in the present paper with all other known non-weight L-modules, i.e., U(h)-free modules of rank one and Whittaker modules (cf. [7]). We fix an irreducible tensor module T M as defined in (3.1), where…”

“…Since then, this kind of non-weight modules, which many authors call U(h)-free modules, have been extensively studied. Especially, the authors classified the U(h)-free modules of rank one for L in [7]. Moreover, the irreducibility and isomorphism classes of these modules were determined therein.…”

“…[5,6], [14], [27], [28], [31]). As a follow-up of our previous paper [8], the purpose of the present paper is to construct new irreducible non-weight L-modules by taking tensor products of a finite number of irreducible modules defined in [7] with irreducible highest weight modules.…”