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

Abstract

We consider the Whittaker modules $M_{1}(\varvec{\lambda },\varvec{\mu })$ for the Weyl vertex algebra M (also called $\beta \gamma$ vertex algebra), constructed in Adamović et al. (J Algebra 539:1–23, 2019), where it was proved that these modules are irreducible for each finite cyclic orbifold $M^{{\mathbb {Z}}_n}$. In this paper, we consider the modules $M_{1}(\varvec{\lambda },\varvec{\mu })$ as modules for the ${{\mathbb {Z}}}$-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 ${\mathcal {M}}(2)$ (cf. Adamović in J Algebra 270:115–132, 2003; Kac and Radul in Transform Groups 1:41–70, 1996; Wang in Commun Math Phys 195:95–111, 1998). Furthermore, these modules are also modules of the Lie algebra $\widehat{\mathfrak {gl}}$ with central charge $c=-1$. We prove that they are reducible as $\widehat{\mathfrak {gl}}$-modules (and therefore also as $M^0$-modules), and we completely describe their irreducible quotients $L(d,\varvec{\lambda },\varvec{\mu })$. We show that $L(d,\varvec{\lambda },\varvec{\mu })$ 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 ${\mathcal {W}}_{1+\infty , c=-1}$.

Appendix A: \(M_1(\varvec{\lambda }, \varvec{\mu })\) as a module for the Heisenberg–Virasoro algebra

In this section, we prove a stronger statement that $M_1(\varvec{\lambda }, \varvec{\mu })$ is a cyclic module for the Heisenberg–Virasoro vertex subalgebra of ${\mathcal {W}}_{1+\infty ,-1}$. Our main argument will be that the action of singlet ${\mathcal {M}}(2)$ on the Whittaker vector $\textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }$ can be replaced by the action of the Virasoro subalgebra of ${\mathcal {M}}(2)$. Next we identify $M_1(\varvec{\lambda }, \varvec{\mu })$ and $L(d,\varvec{\lambda }, \varvec{\mu })$ as restricted modules for the Heisenberg–Virasoro algebra recently classified in [22].

Recall that

$$\begin{aligned} J^k(z) =Y(a^*(-k) a, z) = \sum _{ s \in {{\mathbb {Z}}} } J^k(s) z^{-k-s-1}. \end{aligned}$$

The singlet algebra ${\mathcal {M}}(2)$ (cf. [2, 21]) is generated by the Virasoro vector $L = J^1 + \tfrac{1}{2}:(J^0)^2: + \tfrac{1}{2} D J^0$ and by the primary field H of conformal weight 3. Recall that the ${{\mathbb {Z}}}_2$–orbifold of M is isomorphic to the simple affine vertex algebra $L_{-1/2}(\mathfrak {sl}_2)$ generated by

$$\begin{aligned} e = \tfrac{1}{2}: a^2:, h = -J^0, f =-\tfrac{1}{2}:(a^* )^2:, \end{aligned}$$

the singlet algebra is isomorphic to the parafermion vertex algebra $N_{-1/2}(\mathfrak {sl}_2)$ and H is a scalar multiple of the parafermion generator $W^3$ in [11, Section 2]:

$$\begin{aligned} W^3= & {} k^2 h(-3)\textbf{1} + 3k h(-2) h(-1)\textbf{1} + 2 h(-1) ^3\textbf{1} - 6 k h(-1) e(-1) f(-1)\textbf{1} \\{} & {} + 3 k^2 e(-2) f(-1) \textbf{1} - 3 k^2 f(-2) e(-1) \textbf{1} \quad (k=-1/2), \end{aligned}$$

(see also [19, Section 4]).

We have that ${\mathcal {L}}^{HVir} _{c=-1} = {\mathcal {L}}^{Vir} _{c=-2} \otimes M_1(1)$, where ${\mathcal {L}}^{Vir} _{c=-2}$ denotes the simple Virasoro vertex algebra of central charge $c=-2$ generated by L.

Theorem A.1

We have:

• (1) $M_1(\varvec{\lambda }, \varvec{\mu })$ is an cyclic ${\mathcal {L}}^{HVir} _{c=-1}$-module.

• (2) $L(d,\varvec{\lambda }, \varvec{\mu })$ is an irreducible ${\mathcal {L}}^{HVir}_{c=-1}$-module.

• (3) $M_1(\varvec{\lambda }, \varvec{\mu })$ and $L(d,\varvec{\lambda }, \varvec{\mu })$ are not tensor product ${\mathcal {L}}^{HVir}_{c=-1}$-modules. Moreover, there is a nilpotent subalgebra ${\mathcal {P}}$ of ${\mathcal {H}}$ and one-dimensional ${\mathcal {P}}$-module U such that $M_1(\varvec{\lambda }, \varvec{\mu }) = \text{ Ind } _{{\mathcal {P}} } ^{{\mathcal {H}}} U$.

Proof

The field H can be expressed using $J^0, J^1$ and it contains the non-trivial summand $J^0(-1)^3\textbf{1}$. Set $H(i):= H_{i+2}$, so we have

$$\begin{aligned} H(z) = Y(H, z) = \sum _{i \in {{\mathbb {Z}}} } H(i) z^{-i-3}. \end{aligned}$$

By a direct calculation, we get for $s \in {{\mathbb {Z}}}_{\ge 0}$:

$$\begin{aligned} H(3n + 3m+s ) \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }= \delta _{s,0} q \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }\end{aligned}$$

for certain $q \in {{\mathbb {C}}}$, $q \ne 0$.

Using the following relation in ${\mathcal {M}}(2)$

$$\begin{aligned} \frac{3}{4} H(-6)\textbf{1} - L(-2) H(-4)\textbf{1} +\frac{3}{2} L(-3) H =0, \end{aligned}$$

and using the arguments as in [20, Lemma 3.1], we get that

$$\begin{aligned} {\mathcal {M}}(2). \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }= \langle L \rangle \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }, \end{aligned}$$

so ${\mathcal {M}}(2). \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }$ is isomorphic to the Virasoro submodule obtained by the action of the Virasoro generator L on the Whittaker vector $\textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }$. Since by Theorem 5.2$M_1(\varvec{\lambda }, \varvec{\mu })$ is a cyclic ${\mathcal {M}}(2) \otimes M_1(1)$-module, we conclude that $M_1(\varvec{\lambda }, \varvec{\mu })$ is a cyclic ${\mathcal {L}}^{HVir} _{c=-1}$-module. This proves assertion (1). The assertion (2) is a consequence of (1).

Let us prove assertion (3). For simplicity, we consider the case $n + 1 \ge m$. Other cases can be treat similarly.

By a direct calculation, we get for $s \in {{\mathbb {Z}}}_{>0}$:

$$\begin{aligned}{} & {} J^0(n+m +s) \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }= J^1(n+m +s) \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }= 0, \end{aligned}$$

(A.1)

$$\begin{aligned}{} & {} J^0( n+m) \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }= a_{m} \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }, \dots , J^0(n+1) \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }= a_{1} \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }, \end{aligned}$$

(A.2)

$$\begin{aligned}{} & {} J^1( n+m) \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }= b_{m+1} \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }, \dots , J^1 (n) \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }= b_1 \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }, \end{aligned}$$

(A.3)

where $a_i, b _i \in {{\mathbb {C}}}$, $a_{m}, b_{ m+1} \ne 0$, and $J^0(n) \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }$, $J^1(n-1) \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }$ are not proportional to $\textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }$.

Consider the nilpotent subalgebra ${\mathcal {P}}^{(n)}$ of ${\mathcal {H}}$ spanned by $J^0(r+1), J^1(r)$, $r \ge n$ and $C_1, C_2$. Then ${{\mathbb {C}}} \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }$ is the one-dimensional ${\mathcal {P}}^{(n)}$-module satisfying relations (A.1)-(A.3) and $C_1 \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }= 2 \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }$, $C_2 \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }= - \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }$. Then $M_1(\varvec{\lambda }, \varvec{\mu })$ and $L(d, \varvec{\lambda }, \varvec{\mu })$ are quotients of the universal ${\mathcal {H}}$-module:

$$\begin{aligned} \text{ Ind } _{{\mathcal {P}}^{(n)} }^{{\mathcal {H}}} {{\mathbb {C}}} \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }. \end{aligned}$$

These universal modules and their simple quotients appeared in a slightly general form in [22, Section 7], where it was proved that they are not tensor product modules (see [22, Example 7.5, Remark A.4]). In particular, it was shown that for each $d \in {{\mathbb {C}}}$:

$$\begin{aligned} \frac{ \text{ Ind } _{{\mathcal {P}}^{(n)} }^{{\mathcal {H}}} }{U({\mathcal {H}})(J^0(0)-d ) \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }} \end{aligned}$$

is a simple, restricted ${\mathcal {H}}$-module. This easily implies that as ${\mathcal {H}}$-modules:

$$\begin{aligned} M_1(\varvec{\lambda }, \varvec{\mu }) = \text{ Ind } _{{\mathcal {P}}^{(n)} }^{{\mathcal {H}}} {{\mathbb {C}}} \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }, \quad L(d, \varvec{\lambda }, \varvec{\mu }) = \frac{M_1(\varvec{\lambda }, \varvec{\mu }) }{U({\mathcal {H}})(J^0(0)-d ) \textbf{w}_{ \varvec{ \lambda }, \varvec{ \mu } }}. \end{aligned}$$

The proof follows. $\square$

Corollary A.2

${\mathcal {W}}_{1+\infty , c=-1}$-modules ${{\widetilde{\rho }}}_s (L(d,\varvec{\lambda }, \varvec{\mu }))$ are typical.