Generalized Verma Modules over Lie Algebras of Weyl Type

Abstract

For a field F of characteristic 0 and an additive subgroup Γ of F, there corresponds a Lie algebra W(Γ) of generalized Weyl type. Given a total order of Γ and a weight Λ, a generalized Verma W(Γ)-module M (Λ, ≺) is defined. In this paper, the irreducibility of M (Λ, ≺) is completely determined. It is also proved that an irreducible highest weight module over the W-infinity algebra W1+∞ is quasifinite if and only if it is a proper quotient of a Verma module.

Representations of Generalized Loop Planar Galilean Conformal Algebras W(Γ)

“…Recently, some researchers have investigated the Harish-Chandra modules of many important Lie algebras and Lie super algebras [7][8][9], such as twisted Heisenberg-Virasoro algebra [8], Schrödinger-Virasoro algebra [9] and deformative twisted Schrödinger-Virasoro Lie algebra [9]. In references [10][11][12][13][14][15][16][17], the Verma modules of some Lie algebras and super algebras have been determined. The Verma modules of generalized Virasoro algebras were determined in reference [18].…”

Some properties of generalized reduced Verma modules over $\mathbb{Z}$-graded modular Lie superalgebras

“…The question about the structure of submodules of a Verma module arose in the original paper of Verma [5]. As a natural generalization of Verma modules, the generalized Verma modules are modules induced, starting from arbitrary simple modules (not necessarily finite-dimensional), from a parabolic subalgebra and a complex semisimple Lie algebra (see [6][7][8][9]). One of the main questions about generalized Verma modules is their structure, i.e., reducibility, submodules, equivalence, etc.…”