Quasifinite modules of a Lie algebra related to Block type

Abstract

Let B be the universal central extension of a graded Lie algebra of Block type. In this paper, it is proved that any quasifinite irreducible B-module is either highest weight, lowest weight or uniformly bounded. Furthermore, the quasifinite irreducible highest weight B-modules are classified, and the intermediate series B-modules are classified and constructed.

Automorphisms and Verma modules for generalized Schrödinger–Virasoro algebras

“…Recently, a number of new classes of infinite-dimensional Lie algebras over a field of characteristic 0 related to the Virasoro algebra were studied by several authors (see [6,8,9,11,13,14,16]). Among those algebras, are the generalized Witt algebras, the generalized Virasoro algebras, the Lie algebras of generalized Weyl type and the generalized Heisenberg-Virasoro algebra.…”

Classification of quasifinite representations of a Lie algebra related to Block type

“…The proof of [17,Theorem 4.8] given in [17] involves some heavy computations. In our case here, some new techniques are employed, which render the proof seems to be more elegant, as one may see in Section 3.…”

“…This Lie algebra B, as stated in [17], is interesting in the sense that it contains the following subalgebra Vir = span{L α,0 , C | α ∈ Z}, (1.2) which is isomorphic to the well-known Virasoro algebra (cf. (2.1)), while the Lie algebra B contains no such a subalgebra.…”

“…Thus a classification of irreducible quasifinite B-modules also gives a classification of irreducible quasifiniteB 0,n -modules for all n > 0. The authors in [17] prove that an irreducible quasifinite B-module is either a highest weight module, a lowest weight module or a uniformly bounded module. They also obtain a complete description of irreducible quasifinite highest weight modules and modules of the intermediate series over B.…”

“…However it turns out that these Verma type modules, regarded as Z-graded modules, are all with infinite-dimensional homogenous subspaces (except the subspace spanned by the generator which has dimension 1). Probably because of the above, the authors in[17] turned to study representations of the Lie algebra B related to Block type, which is a subalgebra of B (up to a central element C ), with basis {L α,i , C | α, i ∈ Z, i 0} and relations…”