Whittaker modules for a Lie algebra of Block type

Abstract

In this paper, we study Whittaker modules for a Lie algebra of Block type. We define Whittaker modules and under some conditions, obtain a bijective correspondence between the set of isomorphism classes of Whittaker modules over this algebra and the set of ideals of a polynomial ring, parallel to a result from the classical setting and the case of the Virasoro algebra.

Whittaker pairs for the Virasoro algebra and the Gaiotto-Bonelli-Maruyoshi-Tanzini states

“…Proof. Let f ψr ∈ V * c,∆ be a form satisfying conditions (14), (15). Equations (14) imply that f ψr vanishes on all basis vectors L −i 1 .…”

“…The construction was based on the triangular decomposition g = n − ⊕ h ⊕ n + and a regular Lie algebra homomorphisms φ : n + → C. For non-regular homomorphisms the Kostant construction was analyzed in [4,5,6]. More recently the Whittaker modules have been intensively investigated for various infinite-dimensional algebras with a triangular decomposition: the Heisenberg and the affine Lie algebras [7], the generalized Weyl algebras [8], the Virasoro algebra [9,10], the twisted Heisenberg-Virasoro algebra [11], the Schrödinger-Witt algebra [12], the graded Lie algebras [13], the W -algebra W (2, 2) [14] and the Lie algebras of Block type [15].…”

Blocks and modules for Whittaker pairs

“…Now we are getting closer to the general situation in which one can consider Whittaker modules (see [14,26,30,16,32,15,33]). We define the general Whittaker set-up as follows: consider a Lie algebra g and a quasi-nilpotent Lie subalgebra n of g such that the action of n on the adjoint n-module g/n is locally nilpotent (in particular, g/n ∈ W n n ).…”

“…The original motivation for this paper stems from the recent activities on Whittaker modules for some infinitedimensional (Lie) algebras, which resulted in the papers [24,25,29,4,7,26,30,16,32,15,33]. Whittaker modules for the Lie algebra sl 2 appear first in the work of Arnal and Pinczon [1].…”