Whittaker modules for $\widehat{\mathfrak gl}$ and $\mathcal W_{1+ \infty}$-modules which are not tensor products

TL;DR

This paper studies Whittaker modules for the Weyl vertex algebra and their relation to modules for the Lie algebra fagl, revealing their reducibility, irreducible quotients, and the fact that most are not tensor products.

Contribution

It provides a detailed analysis of the structure and irreducibility of Whittaker modules for fagl and faw_{1+ infinity} at central charge -1, including classification of their irreducible quotients.

Findings

Modules are reducible as fagl-modules

Irreducible quotients are classified and mostly not tensor products

All modules are typical and irreducible over the Heisenberg-Virasoro subalgebra

Abstract

We consider the Whittaker modules $M_{1}(\lambda,\mu)$ for the Weyl vertex algebra $M$, constructed in arXiv:1811.04649, where it was proved that these modules are irreducible for each finite cyclic orbifold $M^{\Bbb Z_n}$. In this paper, we consider the modules $M_{1}(\lambda,\mu)$ as modules for the ${\Bbb Z}$-orbifold of $M$, denoted by $M^0$. $M^0$ is isomorphic to the vertex algebra $\mathcal W_{1+\infty, c=-1} = \mathcal M(2) \otimes M_1(1)$ which is the tensor product of the Heisenberg vertex algebra $M_1(1)$ and the singlet algebra $\mathcal M(2)$. Furthermore, these modules are also modules of the Lie algebra $\widehat{\mathfrak gl}$ with central charge $c=-1$. We prove they are reducible as $\widehat{\mathfrak gl}$-modules (and therefore also as $M^0$-modules), and we completely describe their irreducible quotients $L(d,\lambda,\mu)$. We show that $L(d,\lambda,\mu)$ in most cases are not tensor product modules for the vertex algebra $ \mathcal M(2) \otimes M_1(1)$. Moreover, we show that all constructed modules are typical in the sense that they are irreducible for the Heisenberg-Virasoro vertex subalgebra of $\mathcal W_{1+\infty, c=-1}$.