Nonzero level Harish-Chandra modules over the Virasoro-like algebra

Abstract

In the present paper, we study the nonzero level Harish-Chandra modules for the Virasoro-like algebra. We prove that a nonzero level Harish-Chandra module of the Virasoro-like algebra is a generalized highest weight (GHW for short) module. Then we prove that a GHW module of the Virasoro-like algebra is induced from an irreducible module of a Heisenberg subalgebra.

Whittaker modules for a Lie algebra of Block type

“…5 We omit the proof of this theorem since it is similar to the proof of Theorem 4.2.2 in [11]. Proof We omit the proof of this corollary since it is similar to the proof of Corollary 4.2.3 in [11].…”

“…It is also a special case of Cartan type S Lie algebra or Cartan type H Lie algebra. Partially due to these facts, the study of this Lie algebra or its analogs has recently attracted some authors' attentions (see, e.g., [3,5,9,10,[12][13][14]16,18]). It is well known that although Cartan type Lie algebras have a long history, their representation theory is far from being well developed.…”

Graded Modules Over the q-Analog Virasoro-Like Algebra

“…We say that the two representations (π, V) and (ρ, V) are isomorphic up to the automorphism σ of G. Now we recall the following result from [12]. (1) If c 1 = 0, then V(0, i, 0) is a trivial L-module.…”

“…The quantum torus is one of the main objects in noncommutative geometry, and plays an important role in the classification of extended affine Lie algebras [3]. Meanwhile, the q analog Virasoro like algebra can be regarded as a q deformation of the Virasoro-like algebra introduced and studied by Arnold, de Wit, etc when they try to generalize the Virasoro algebra ( [1,6,11,[13][14][15][16] and [17]). There are some papers devoted to the study of structure and representations of the q analog Virasoro algebra L. C. Jiang and D. Meng studied its derivation Lie algebra and the automorphism group of its derivation Lie algebra [10].…”

“…Y. Gao constructed a class of principal vertex representations for the extended affine Lie algebras coordinatized by certain quantum tori by using the representation of q analog Virasoro-like algebra [9]. In the present paper, we deal with the classification of irreducible graded L-modules with finite dimensional homogeneous subspaces by using the results on the irreducible modules of Heisenberg algebra obtained by V. Chari (see [4], also [8]), and the results about the Z-graded L modules and the Z 2 -graded L modules given in [7] and [12]. The paper is arranged as follows.…”