Graded Modules Over the q-Analog Virasoro-Like Algebra

Abstract

In this paper, we deal with the classification of the irreducible Z-graded and Z 2 -graded modules with finite dimensional homogeneous subspaces for the q analog Virasoro-like algebra L. We first prove that a Z-graded L-module must be a uniformly bounded module or a generalized highest weight module. Then we show that an irreducible generalized highest weight Z-graded module with finite dimensional homogeneous subspaces must be a highest (or lowest) weight module and give a necessary and sufficient condition for such a module with finite dimensional homogeneous subspaces. We use the Z-graded modules to construct a class of Z 2 -graded irreducible generalized highest weight modules with finite dimensional homogeneous subspaces. Finally, we classify the Z 2 -graded L-modules. We first prove that a Z 2 -graded module must be either a uniformly bounded module or a generalized highest weight module. Then we prove that an irreducible nontrivial Z 2 -graded module with finite dimensional homogeneous subspaces must be isomorphic to a module constructed as above. As a consequence, we also classify the irreducible Z-graded modules and the irreducible Z 2 -graded modules with finite dimensional homogeneous subspaces and center acting nontrivial.Presented by Susan Montgomery.

Representations of the Local Area-Preserving Lie Algebra for the Klein Bottle

“…There are some papers devoted to the study of structure and representation of this algebra [8,12,13,19]. Partly motivated by the membrane theory, groups of area-preserving diffeomorphisms of the 2-dimensional surface (including the two-sphere, the two-torus, and the tetrahedron) and their Lie algebras have recently been the focus of much attention in the physics literature [1,2,7,[14][15][16][17][18].…”

Classification of Z-Graded Modules of the Intermediate Series over the q-Analog Virasoro-like Algebra

“…Gao constructed a class of principal vertex representations for the extended affine Lie algebras coordinatized by certain quantum tori by using the representation of the q-analog Virasoro-like algebra( [8]). The classification of the irreducible graded modules with finite dimensional homogeneous subspaces and nontrivial centers over the q-analog Virasoro-like algebra has been completed in [9]. Thus, we only consider the classification of the Z-graded modules of the intermediate series over the centerless q-analog…”

Modules for the core of extended affine Lie algebras of typeA1with coordinates in rank 2 quantum tori

“…For representations of Lie algebras coordinatized by quantum tori, see [Jakobsen and Kac 1989;Berman and Szmigielski 1999;Gao 2000b;2000a;Eswara Rao 2004;Rao 2003] and the references therein. For structure and representations of the q-analogue Virasorolike algebra, see [Zhang and Zhao 1996;Jiang and Meng 1998;Rao and Zhao 2004;Lin and Tan 2006;2008]. This paper is organized as follows.…”