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

Abstract

Communicated by Efim Zelmanov MSC: 17B10 17B20 17B65 17B66 17B68 Keywords: The Virasoro algebra Non-weight representations Fraction representations Highest-weight-like representationsIn this paper two new classes of irreducible modules over the centerless Virasoro algebra V are obtained. These modules are generally not weight modules or Whittaker modules. We first construct a class of modules over V parameterized by any 2n + 2 complex numbers for any nonnegative integer n which we call fraction modules. The necessary and sufficient conditions for fraction modules to be irreducible are determined. Also we determine the necessary and sufficient conditions for two irreducible fraction modules to be isomorphic. Then we define highest-weight-like Verma modules over V . These modules behave like highest weight Verma modules.It is proved that each highest-weight-like Verma module has an irreducible quotient module which is isomorphic to a subquotient of some reducible fraction module.

Tensor product weight modules over the Virasoro algebra

“…At the same time for the last decade, various families of non-weight simple Virasoro modules were studied in [6,9,14,15,18,22,[24][25][26]. These (except the modules in [15,18,24,25]) are basically various versions of Whittaker modules constructed using different tricks.…”

Irreducible Twisted Heisenberg–Virasoro Modules from Tensor Products

“…Other work has focused on non-weight L-modules, which attract attention in the past few years, such as Whittaker modules (see, e.g., [13,18,19,21]), C[L 0 ]-free modules, irreducible modules from Weyl modules and a class of non-weight modules including highest-weight-like modules (see, e.g, [1,5,11,14,23,24]). In the present paper, we shall study non-weight Lmodules.…”

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]).…”