A class of simple weight Virasoro modules

Abstract

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 ].

Tensor product weight modules over the Virasoro algebra

“…Now we compare the tensor products V (c, h) ⊗ V α,β with other known simple weight modules. So far known simple weight Virasoro modules with infinite-dimensional weight spaces are from [2,13]. It is proved in [13] that V (c, h) ⊗ V α,β (or its minimal submodule) is not isomorphic to any module in [13].…”

“…Later a four-parameter family of simple weight modules with infinitely dimensional weight spaces were given in [2]. Recently in [13,16] another two huge classes of simple modules with infinitely dimensional weight spaces were obtained using non-weight modules introduced in [20].…”

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]).…”

Tensor Product Weight Representations of the Neveu–Schwarz Algebra

“…For the last few years, the study on irreducible weight and non-weight modules over the Virasoro algebra has been very active, see [9], [15], [16], [17], [18], [19], [20], [21]. The tensor product of highest weight modules with intermediate series modules over the Virasoro algebra were the first such simple Virasoro modules with infinite-dimensional weight spaces [12].…”