Tensor product weight modules over the Virasoro algebra

Abstract

Abstract. The tensor product of highest weight modules with intermediate series modules over the Virasoro algebra was discussed by Zhang [Z] in 1997. Since then the irreducibility problem for the tensor products has been open. In this paper, we determine the necessary and sufficient conditions for these tensor products to be simple. From non-simple tensor products, we can get other interesting simple Virasoro modules. We also obtain that any two such tensor products are isomorphic if and only if the corresponding highest weight modules and intermediate series modules are isomorphic respectively. Our method is to develop a "shifting technique" and to widely use Feigin-Fuchs' Theorem on singular vectors of Verma modules over the Virasoro algebra.

Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications

“…Similar irreducibility problems for the Virasoro algebra and the twisted Heisenberg-Virasoro algebra were studied in the papers [8,17,19,20,23]. Our irreducibility result is a certain refinement of these results.…”

Fraction representations and highest-weight-like representations of the Virasoro algebra

“…In particular, a classification of irreducible weight Virasoro modules with finite-dimensional weight spaces was given by Mathieu [14], and a classification of irreducible weight Virasoro modules with one finite-dimensional nonzero weight space was given in [15]. There are some known irreducible weight Virasoro modules with infinitedimensional weight spaces, see [3,4,20], and the very recent papers [9,12]. We remark that the tensor products of intermediate series modules over the Virasoro algebra never give irreducible modules [21].…”

Classification of simple weight modules for super-Virasoro algebra with a finite-dimensional weight space

“…So the irreducible modules with a finite-dimensional weight space over the Virasoro algebra are classified. For the last few years, the study on irreducible weight modules with infinite-dimensional weight spaces and nonweight modules over the Virasoro algebra has been very active (see [2,4,5,[7][8][9]13,17,18,21]). The irreducibility problem for the tensor products over the Virasoro algebra was a long-standing problem (see [18]).…”

“…In [2], the irreducibility for the tensor product of highest weight modules with intermediate series modules over the Virasoro algebra was completely determined by using Feigin-Fuchs' Theorem (see [3] or [1]) on singular vectors of Verma modules and the so called shifting technique. In the process of the study in [2], the main result in [12] plays an important rule. Moreover, the result in [12] is generalized to twisted Schrödinger-Virasoro algebra by Li and Su [6], twisted Heisenberg-Virasoro by Shen and Su [14].…”

“…The tensor product of intermediate series modules over the Virasoro algebra never gives simple modules [21]. In [2], the irreducibility for the tensor product of highest weight modules with intermediate series modules over the Virasoro algebra was completely determined by using Feigin-Fuchs' Theorem (see [3] or [1]) on singular vectors of Verma modules and the so called shifting technique. In the process of the study in [2], the main result in [12] plays an important rule.…”