New irreducible weight modules over Witt algebras with infinite-dimensional weight spaces

Abstract

Let d > 1 be an integer. In 1986, Shen defined a class of weight modules F α b (V ) over the Witt algebra W d for α ∈ C d , b ∈ C, and an irreducible module V over the general linear Lie algebra gl d on which the identity matrix acts as multiplication by b. In 1996, Eswara Rao determined necessary and sufficient conditions for these modules to be irreducible when V is finite-dimensional. In this note, we will determine necessary and sufficient conditions for all these modules F α b (V ) to be irreducible where V is not necessarily finite-dimensional. In this way, we obtain a large new family of irreducible W d -modules with infinite-dimensional weight spaces.

Irreducible modules over the divergence zero algebras and their $q$-analogues

“…Our results for L d and L d generalize the similar result of Liu and Zhao for the Witt algebra W d in [LZ1] and the result of Talboom for the algebra L d for finite-dimensional V ( [T]). Our results for the algebras L d (q) and L d (q) are new except for the case when d = 2 and V is finite-dimensional.…”

“…The representation theory of Witt algebras was studied extensively by many authors, see [B,BF,BMZ,E1,E2,GZ,LZ1,T] and references therein. In 1986, Shen [Sh] defined a class of modules F α b (V ) over the Witt algebra W d for any weight modules over the special linear Lie algebra sl d , where α is any d-dimensional complex vector and b is a complex number.…”

“…These modules as well as the functors F α b , known as the Larson functors now, were also studied by Larson in [L1, L2, L3]. In 1996, Eswara Rao [E1] determined the irreducibility of these modules for finite-dimensional V , and recently G. Liu and K. Zhao studied the case when V is infinite-dimensional (see [LZ1]).…”

Simple Witt Modules that are Finitely Generated over the Cartan Subalgebra

“…Over the last two decades, the weight representation theory of Witt algebras was extensively studied by many algebraists and physicists; see for example [B,E1,E2,BMZ,GLZ,L3,L4,L5,LZ,LLZ,MZ2,Z]. In 1986, Shen defined a class of modules F α b (V ) over the Witt algebra W d for α ∈ C d , b ∈ C, and a simple module V over the special linear Lie algebra sl d , see [Sh], which were also given by Larsson in 1992, see [L3].…”

Classification of irreducible bounded weight modules over the derivation Lie algebras of quantum tori

“…In 1996, Eswara Rao [E1] determined necessary and sufficient conditions for these modules to be irreducible when V is finite dimensional, see [GZ] for a simplified proof. When V is infinite dimensional, F α b (V ) is always irreducible, see [LZ2].…”