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

DOI10.1007/S11005-023-01663-1arXiv2112.08725MaRDI QIDQ6385863

Dražen Adamović, Veronika Pedić Tomić

Publication date: 16 December 2021

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 $BbbZ$-orbifold of $M$, denoted by $M^0$. $M^0$ is isomorphic to the vertex algebra $mathcal{W}_{1+infty,c=-1}=mathcalM(2)otimes{M}_1(1)$ which is the tensor product of the Heisenberg vertex algebra $M_1(1)$ and the singlet algebra $mathcalM(2)$. Furthermore, these modules are also modules of the Lie algebra $widehatmathfrakgl$ with central charge $c=-1$. We prove they are reducible as $widehatmathfrakgl$-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 $mathcalM(2)otimes{M}_1(1)$. Moreover, we show that all constructed modules are typical in the sense that they are irreducible for the Heisenberg-Virasoro vertex subalgebra of $mathcal{W}_{1+infty,c=-1}$.

Mathematics Subject Classification ID

Vertex operators; vertex operator algebras and related structures (17B69) Infinite-dimensional Lie (super)algebras (17B65) Simple, semisimple, reductive (super)algebras (17B20)

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