Irreducible weight modules with a finite-dimensional weight space over the twisted N=1 Schr\"{o}dinger-Neveu-Schwarz algebra

TL;DR

This paper proves that all irreducible weight modules with finite-dimensional weight spaces over the twisted N=1 Schrödinger-Neveu-Schwarz algebra are Harish-Chandra modules, showing a classification result in representation theory.

Contribution

It establishes that no simple mixed modules exist over this algebra, confirming all such modules are Harish-Chandra modules, thus advancing understanding of its representation structure.

Findings

No simple mixed modules over the algebra

All irreducible weight modules with finite-dimensional weight spaces are Harish-Chandra modules

Provides a classification of modules over the twisted N=1 Schrödinger-Neveu-Schwarz algebra

Abstract

It is shown that there are no simple mixed modules over the twisted N=1 Schr\"{o}dinger-Neveu-Schwarz algebra, which implies that every irreducible weight module over it with a nontrivial finite-dimensional weight space, is a Harish-Chandra module.

Equations

$$[L_n,L_m]=(m-n)L_{m+n}+\frac{n^3-n}{12}{\delta }_{m,-n}\mathfrak{c},$$

$$[L_n,G_r]=(r-\frac{n}{2})G_{r+n},[G_r,G_s]=2L_{r+s}+\frac{1-4r^2}{12}{\delta }_{r,-s}\mathfrak{c},$$

$$\displaystyle[L_{n},Y_{p}]=\left\{\begin{array}[]{cc}(p-\frac{n}{2})Y_{p+n},&{\rm if}\ p\in\mathbb{Z},\\[6.0pt] pY_{p+n},&{\rm if}\ p\in\frac{1}{2}+\mathbb{Z},\end{array}\right.\ \ \,[L_{n},M_{p}]=\left\{\begin{array}[]{cc}pM_{p+n},&{\rm if}\ p\in\mathbb{Z},\\[6.0pt] (p+\frac{n}{2})M_{p+n},&{\rm if}\ p\in\frac{1}{2}+\mathbb{Z},\end{array}\right.$$

$$\displaystyle[G_{r},Y_{p}]=\left\{\begin{array}[]{cc}\frac{1}{2}(p-r)Y_{p+r},&{\rm if}\ p\in\mathbb{Z},\\[6.0pt] 2Y_{p+r},&{\rm if}\ p\in\frac{1}{2}+\mathbb{Z},\end{array}\right.\ \,[G_{r},M_{p}]=\left\{\begin{array}[]{cc}\frac{p}{2}M_{p+r},&{\rm if}\ p\in\mathbb{Z},\\[6.0pt] 2M_{p+r},&{\rm if}\ p\in\frac{1}{2}+\mathbb{Z},\end{array}\right.$$

$$\displaystyle[Y_{p},Y_{q}]=\left\{\begin{array}[]{ccc}\frac{1}{2}(q-p)M_{q+p},&{\rm if}\ p,q\in\mathbb{Z},\\[6.0pt] \frac{q}{2}M_{q+p},&{\rm if}\ p\in\mathbb{Z},q\in\frac{1}{2}+\mathbb{Z},\\[6.0pt] 2M_{q+p},&{\rm if}\ p,q\in\frac{1}{2}+\mathbb{Z}.\end{array}\right.$$

$${\widetilde{\mathfrak{tsns}}}_{\bar{0}}$$

$${\widetilde{\mathfrak{tsns}}}_{\bar{1}}$$

$${\widetilde{\mathfrak{tsns}}}_{+}$$

$${\widetilde{\mathfrak{tsns}}}_{-}$$

$${\widetilde{\mathfrak{tsns}}}_0$$

$$\widetilde{\mathfrak{tsns}}$$

$$G_{\frac{3}{2}}v=G_{\frac{1}{2}}v=Y_{\frac{1}{2}}v=M_{\frac{1}{2}}v=0\text{mbox}or{G}_{-\frac{3}{2}}v=G_{-\frac{1}{2}}v=Y_{-\frac{1}{2}}v=M_{-\frac{1}{2}}v=0.$$

$$\displaystyle\big{(}\frac{1}{2}L_{1}G_{\frac{1}{2}}-G_{\frac{3}{2}}\big{)}G_{\frac{3}{2}}v=0=Y_{p}v=M_{p}v\ \ {\rm for\ all}\ p\in\frac{1}{2}\mathbb{Z}^{+}.$$

$$\displaystyle\big{(}\frac{1}{2}L_{-1}G_{-\frac{1}{2}}+G_{-\frac{3}{2}}\big{)}G_{-\frac{3}{2}}v=0=Y_{-p}v=M_{-p}v\ \ {\rm for\ all}\ p\in\frac{1}{2}\mathbb{Z}^{-}.$$

$$\displaystyle 0=L_{-1}G_{-\frac{1}{2}}(\frac{1}{2}L_{1}G_{\frac{1}{2}}-G_{\frac{3}{2}})w=(2L^{2}_{0}-3L_{0})w=\big{(}2(\mu+1)^{2}-3(\mu+1)\big{)}.$$

$$[G_{\frac{1}{2}},G_{\frac{1}{2}}]=2L_1,[L_1,Y_r]=r{Y}_{1+r},[L_1,M_r]=(r+\frac{1}{2})M_{r+1},\forall r\in \frac{1}{2}+{\mathbb{Z}}_{+},$$

$$L_{1}W=Y_{r}W=M_{r}W=0,\forall r\in \frac{1}{2}+{\mathbb{Z}}_{+}.$$

$$[G_{\frac{1}{2}},Y_p]=2Y_{p+\frac{1}{2}},[Y_p,Y_q]=2M_{p+q},\forall p,q\in \frac{1}{2}+\mathbb{Z},$$

$$Y_{n}W=M_{n}W=0,\forall n\in {\mathbb{Z}}_{+}.$$

$$Y_{p}W=M_{p}W=0,\forall p\in (\frac{1}{2}{\mathbb{Z}}_{+}{)}^{\ast }.$$

$$Y_p{G}_{\frac{3}{2}}W=M_p{G}_{\frac{3}{2}}W=0,\forall p\in (\frac{1}{2}{\mathbb{Z}}_{+}{)}^{\ast }.$$

$$[L_{-1},Y_p]=p{Y}_{p-1},$$

$$Y_{p}v=M_{p}v=0,\forall p,q\in (\frac{1}{2}{\mathbb{Z}}_{-}{)}^{\ast }.$$

$$Y_{p}v=M_{p}v=0,\forall p,q\in (\frac{1}{2}{\mathbb{Z}}_{+}{)}^{\ast },$$

$$Y_{0}v=M_{0}v=0,$$

$$Y_{p}v=M_{p}v=0,\forall p\in \frac{1}{2}\mathbb{Z}.$$

$$\dim G_{\frac{3}{2}}W=\infty ,$$

$$Y_p{G}_{\frac{3}{2}}W=M_p{G}_{\frac{3}{2}}W=0,\forall p\in (\frac{1}{2}{\mathbb{Z}}_{+}{)}^{\ast }.$$

$$L_{-1}v=Y_{-1}v=M_{-1}v=0.$$

$$Y_{p}v=M_{p}v=0,\forall p,q\in (\frac{1}{2}{\mathbb{Z}}_{+}{)}^{\ast }.$$

$$[L_{-1},Y_{1-p}]$$

$$Y_{p}v=0,\forall p\in \frac{1}{2}{\mathbb{Z}}_{-}.$$

$$Y_{p}v=0,\forall p\in \frac{1}{2}\mathbb{Z}.$$

$$[Y_p,Y_q]$$

Full text

An irreducible weight module $V$ is called a mixed module if there exist $\lambda\in\mathbb{C}$ and $k\in\mathbb{Z}^{*}$ such that ${\rm dim\,}V_{\lambda}=\infty$ and ${\rm dim\,}V_{\lambda+k}<\infty$. The following conjecture was given in [6]:

The positive answers to the above question and conjecture were given in [7]. Such question and conjecture have already been solved on the truncated Virasoro algebras, the $W$-algebra $W(2,2)$, the twisted Schrödinger-Virasoro algebra, the twisted Heisenberg-Virasoro algebra and the Neveu-Schwarz algebra in [1, 4, 5, 9, 11]. In this note, we also give the positive answers to the above question and conjecture for the twisted $N\!=\!1$ Schrödinger-Neveu-Schwarz algebra. Our main result is the following:

Throughout this paper, we respectively denote $\mathbb{Z}^{*}$, $\mathbb{Z}^{+}$, $\mathbb{Z}^{-}$, $\mathbb{Z}_{+}$ and $\mathbb{Z}_{-}$ the sets of the nonzero, positive, negative, nonnegative and non-positive integers.

2  Proof of Theorem 1.3

We first recall a main result about the irreducible weight Neveu-Schwarz-modules given in [11], which we cited here as the following lemma:

Since $\mathfrak{I}={\rm{Span}}_{\mathbb{C}}\{Y_{p},M_{p}\,|\,p\in\frac{1}{2}\mathbb{Z}\}$ is a nontrivial ideal of $\widetilde{\mathfrak{tsns}}$, $\mathfrak{I}V$ is a submodule of $V$ for any $\widetilde{\mathfrak{tsns}}$-module $V$, which gives the following lemma (such lemma was not used in [11]).

For any irreducible weight $\widetilde{\mathfrak{tsns}}$-module $V$, if we can prove $\mathfrak{I}V=0$, then $V$ will degenerate to be an irreducible weight $\widetilde{\mathfrak{ns}}$-module. Then Theorem 1.3 follows from Lemma 2.1 in this case. Denote the universal enveloping algebra of $\widetilde{\mathfrak{tsns}}$ by $U(\widetilde{\mathfrak{tsns}})$. For any irreducible weight $\widetilde{\mathfrak{tsns}}$-module $V$, if there exists some $0\neq v\in V$ such that $\mathfrak{I}v=0$, then $U(\widetilde{\mathfrak{tsns}})v=V$. And the following lemma follows:

It is easily to see that $\{G_{\frac{3}{2}},\,G_{\frac{1}{2}},\,Y_{\frac{1}{2}},\,M_{\frac{1}{2}}\}$ and $\{G_{-\frac{3}{2}},\,G_{-\frac{1}{2}},\,Y_{-\frac{1}{2}},\,M_{-\frac{1}{2}}\}$ respectively generate $\widetilde{\mathfrak{tsns}}_{+}$ and $\widetilde{\mathfrak{tsns}}_{-}$. Then the following lemma follows from the fact that both highest and lowest weight $\widetilde{\mathfrak{tsns}}$-modules are Harish-Chandra modules (similar to the Virasoro algebra case investigated in [7]).

[TABLE]

Then $V$ is a Harish-Chandra module.

Proof of Theorem 1.3   We shall prove this theorem step by step by several lemmas.

Assume now that $V$ is an irreducible weight $\widetilde{\mathfrak{tsns}}$-module such that there exists $\lambda\in\mathbb{C}$ satisfying ${\rm dim\,}V_{\lambda}=\infty$. Denote the set $S_{\lambda}\!=\!\{p\,|\,{\rm dim\,}V_{\lambda+p}<\infty,\,p\in\frac{1}{2}\mathbb{Z}^{*}\}$.

Proof   Suppose there are two different elements $p$ and $q$ in $S_{\lambda}$. Without loss of generality, we can assume $p=\frac{1}{2}$, $q>\frac{1}{2}$.

Case 1  $p=\frac{1}{2}$, $q>\frac{1}{2}$ and $q\in\frac{1}{2}+\mathbb{Z}$.

Let $W$ be the intersection of the kernels of the linear maps $X_{\frac{1}{2}}:\,V_{\lambda}\rightarrow V_{\lambda+\frac{1}{2}}$ and $X_{q}:\,V_{\lambda}\rightarrow V_{\lambda+q}$ for $X=G,\,Y,\,M$. Then $X_{\frac{1}{2}}W=X_{q}W=0$ for all $X\in\{G,\,Y,\,M\}$. The assumptions ${\rm dim\,}V_{\lambda}=\infty$, ${\rm dim\,}V_{\lambda+\frac{1}{2}}<\infty$ and ${\rm dim\,}V_{\lambda+q}<\infty$ force ${\rm dim\,}W=\infty$. Using $G_{\frac{1}{2}}W=Y_{\frac{1}{2}}W=M_{\frac{1}{2}}W=0$, $G_{q}W=Y_{q}W=M_{q}W=0$ and the given brackets, we obtain $L_{n}W=G_{r}W=Y_{p}W=M_{p}W=0$ for all $n=1,2q,2q+1,2q+2,\cdots$, $r=\frac{1}{2},q,q+1,q+2,\cdots$, $p\in\frac{1}{2}\mathbb{Z}_{>0}$. We claim that $q>\frac{3}{2}$ and $G_{\frac{3}{2}}w\neq 0$ for all $0\neq w\in W$. Otherwise, $W$ would be a Harish-Chandra module by Lemma 2.4. Thus ${\rm dim\,}G_{\frac{3}{2}}W=\infty$. Since ${\rm dim\,}V_{\lambda+\frac{1}{2}}<\infty$, there exists some $w\in W$ such that $v=G_{\frac{3}{2}}w$ and $L_{-1}v=Y_{-1}v=M_{-1}v=0$. It is easily to verify that $\widetilde{\mathfrak{tsns}}_{+}v=0$, which forces $V$ to be a Harish-Chandra module according to Lemma 2.4. This contracts with our assumptions. Hence this case is impossible.

Case 2  $p=\frac{1}{2}$, $q>1$ and $q\in\mathbb{Z}$.

Let $W$ be the intersection of the kernels of the linear maps $X_{\frac{1}{2}}:\,V_{\lambda}\rightarrow V_{\lambda+\frac{1}{2}}$ and $Z_{q}:\,V_{\lambda}\rightarrow V_{\lambda+q}$ for $Z=L,\,Y,\,M$. Then $X_{\frac{1}{2}}W=Z_{q}W=0$ for all $X\in\{G,\,Y,\,M\}$ and $Z\in\{L,\,Y,\,M\}$. The assumptions ${\rm dim\,}V_{\lambda}=\infty$, ${\rm dim\,}V_{\lambda+\frac{1}{2}}<\infty$ and ${\rm dim\,}V_{\lambda+q}<\infty$ force ${\rm dim\,}W=\infty$. Using $G_{\frac{1}{2}}W=Y_{\frac{1}{2}}W=M_{\frac{1}{2}}W=0$, $L_{q}W=Y_{q}W=M_{q}W=0$ and the given brackets, we obtain $L_{n}W=G_{r}W=Y_{p}W=M_{p}W=0$ for all $n=1,q,q+1,q+2,\cdots$, $r=\frac{1}{2},q+\frac{1}{2},q+\frac{3}{2},\cdots$, $p\in\frac{1}{2}\mathbb{Z}^{+}$. We claim that $G_{\frac{3}{2}}w\neq 0$ for all $0\neq w\in W$. Otherwise, $W$ would be a Harish-Chandra module by Lemma 2.4. Thus ${\rm dim\,}G_{\frac{3}{2}}W=\infty$. Similar to the proof of the first case, one can prove that this case is also impossible. Then this lemma follows. $\Box$

The following lemma can be easily verified, partially given in Lemma 2 of [11].

[TABLE]

(ii)  Let $0\neq v\in V$ be such that $G_{-\frac{1}{2}}v=Y_{-\frac{1}{2}}v=M_{-\frac{1}{2}}v=0$. Then

[TABLE]

According to Lemma 2.5, we can assume $V$ is an irreducible weight $\widetilde{\mathfrak{tsns}}$-module such that there exists some $\mu\in\mathbb{C}$ satisfying ${\rm dim\,}V_{\mu}<\infty$, ${\rm dim\,}V_{\mu+\frac{1}{2}}<\infty$ and ${\rm dim\,}V_{\mu+p}=\infty$ for all $p\in\mathbb{Z}^{*}\cup\{\frac{1}{2}+\mathbb{Z}^{*}\}$.

Proof   Let $U$ be the intersection of the kernels of the linear maps $X_{\frac{1}{2}}:\,V_{\mu-\frac{1}{2}}\rightarrow V_{\mu}$ for $X=G,\,Y,\,M$. Then $G_{\frac{1}{2}}U=Y_{\frac{1}{2}}U=M_{\frac{1}{2}}U=0$. The assumptions ${\rm dim\,}V_{\mu}<\infty$ and ${\rm dim\,}V_{\mu-{\frac{1}{2}}}=\infty$ force ${\rm dim\,}U=\infty$. We claim that $G_{\frac{3}{2}}u\neq 0$ for all $0\neq u\in U$. Otherwise, $M$ would be a Harish-Chandra module by Lemma 2.4. Thus ${\rm dim\,}G_{\frac{3}{2}}U=\infty$. Since ${\rm dim\,}M_{\mu}<\infty$, there exists some $u\in U$ such that $v=G_{\frac{3}{2}}u$ and $G_{-\frac{1}{2}}v=Y_{-\frac{1}{2}}v=M_{-\frac{1}{2}}v=0$. Using Lemma 2.6, we obtain $(\frac{1}{2}L_{1}G_{\frac{1}{2}}-G_{\frac{3}{2}})v=0$ and $Y_{p}u=M_{p}u=0$ for all $p\in\frac{1}{2}\mathbb{Z}^{+}$. Then $U$ is simply a $\widetilde{\mathfrak{ns}}$-module. Then we can use the same discussions as those given in Lemma 3 of [11], and have the following observations:

[TABLE]

Then the lemma follows. $\Box$

According to Lemma 2.7, any irreducible weight $\widetilde{\mathfrak{tsns}}$-module $V$ possessing only two different finite-dimensional weight spaces can be assumed to be of the following type: ${\rm dim\,}V_{\frac{1}{2}}<\infty$, ${\rm dim\,}V_{1}<\infty$ and ${\rm dim\,}V_{p+\frac{1}{2}}=\infty$ for all $p\in\mathbb{Z}^{*}\cup\{\frac{1}{2}+\mathbb{Z}^{*}\}$. Without loss of generality, we will just prove that this case can not happen, which can be formulated as the following lemma:

Proof   Let $W$ be the intersection of the kernels of the linear maps $X_{\frac{1}{2}}:\,V_{0}\rightarrow V_{\frac{1}{2}}$ for all $X\in\{G,\,Y,\,M\}$. Then $G_{\frac{1}{2}}W=Y_{\frac{1}{2}}W=M_{\frac{1}{2}}W=L_{0}W=0$ and ${\rm dim\,}W=\infty$. According to the following facts:

[TABLE]

we can prove the following identities:

[TABLE]

Recalling the following identities:

[TABLE]

we can obtain the following identities:

[TABLE]

Then combining (2.1) and (2.2), we arrive at the following identities:

[TABLE]

Using the above obtained results, we can prove the following identities:

[TABLE]

We claim that $G_{\frac{3}{2}}w\neq 0$ for all $0\neq w\in W$. Otherwise, $V$ would become a Harish-Chandra module by Lemma 2.4, which forces ${\rm dim\,}G_{\frac{3}{2}}W=\infty$. Recalling our suppose ${\rm dim\,}V_{1}<\infty$, we can deduce that there exists $0\neq v\in G_{\frac{3}{2}}W$ such that $G_{-\frac{1}{2}}v=Y_{-\frac{1}{2}}v=M_{-\frac{1}{2}}v=0$. Then $L_{-1}v=G_{-\frac{1}{2}}G_{-\frac{1}{2}}v=0$. According to the following identities:

[TABLE]

we can deduce the following results:

[TABLE]

The identities given in (2.4), imply

[TABLE]

Combining $Y_{0}=\frac{1}{2}[G_{-\frac{1}{2}},Y_{\frac{1}{2}}]$, $M_{0}=\frac{1}{2}[G_{-\frac{1}{2}},M_{\frac{1}{2}}]$, (2.5) and (2.6), we can deduce

[TABLE]

which together with (2.5) and (2.6), gives

[TABLE]

Then recalling Lemma 2.3, we can finally deduce that $Y_{p},M_{p}$ act trivially on the whole $V$ for all $p\in\frac{1}{2}\mathbb{Z}$ and $V$ is simply an irreducible weight module over $\widetilde{\mathfrak{ns}}$. Then this lemma follows from Lemma 2.1. $\Box$

By Lemmas 2.5, 2.7 and 2.8, we get the following lemma immediately.

According to Lemma 2.9, we can suppose there exists some $p\in\frac{1}{2}\mathbb{Z}^{*}$, such that ${\rm dim\,}V_{\lambda+p}<\infty$. For convenience, we denote $\lambda+p$ by $\mu$ in the following lemma.

Proof   According to our suppose, we know ${\rm dim\,}V_{\mu}<\infty$ with $\mu\in S_{\lambda}$. Let $W$ be the intersection of the kernels of the linear maps $X_{\frac{1}{2}}:\,V_{\mu-\frac{1}{2}}\rightarrow V_{\mu}$ for $X\in\{G,\,Y,\,M\}$. Then $G_{\frac{1}{2}}W=Y_{\frac{1}{2}}W=M_{\frac{1}{2}}W=L_{0}W=0$ and ${\rm dim\,}W=\infty$. Similar to the corresponding proof of Lemma 2.8, we can prove the following identities:

[TABLE]

Recalling our suppose that ${\rm dim\,}V_{\mu}<\infty$, we claim that there exists some nonzero element $v\in G_{\frac{3}{2}}W$ such that

[TABLE]

According to (2.8), we also know that

[TABLE]

The following identities hold:

[TABLE]

from which we can deduce

[TABLE]

Combining (2.10) and (2.13), we arrive at the following result:

[TABLE]

The identity $M_{p}v=0$ for all $p\in\frac{1}{2}\mathbb{Z}$ follows from 2.13 and the follows identities:

[TABLE]

Then again recalling Lemma 2.3, we can finally deduce that $Y_{p},M_{p}$ act trivially on the whole $V$ for all $p\in\frac{1}{2}\mathbb{Z}$ and $V$ is simply an irreducible weight module over $\widetilde{\mathfrak{ns}}$. Then this lemma follows from Lemma 2.1. $\Box$

Then Theorem 1.3 follows immediately from Lemma 2.10. $\blacksquare$

Acknowledgements  Supported by a NSF grant BK20160403 of Jiangsu Province and NSF grants 11501417, 11671056, 11271056, 11101056 of China, Innovation Program of Shanghai Municipal Education Commission, Program for Young Excellent Talents in Tongji University and the Fundamental Research Funds for the Central Universities.