Finite -algebras of $\mathfrak {osp}_{1|2n}$ and ghost centers

Abstract

We prove that the finite -algebra $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})$ associated to $\mathfrak {osp}_{1|2n}$ and its principal nilpotent element $f_\textrm{prin}$ is isomorphic to Gorelik’s ghost center of $\mathfrak {osp}_{1|2n}$. It is an analogue for $\mathfrak {osp}_{1|2n}$ of a theorem of Kostant (Invent Math 48(2):101–184, 1978).

1 Introduction

The Lie superalgebra $\mathfrak {osp}_{1|2n}$ is the finite-dimensional simple Lie superalgebra whose Dynkin diagram is the same as the one of type $B_n$ except for a unique simple short root, which is replaced by a non-isotropic odd simple root in $\mathfrak {osp}_{1|2n}$. The Lie superalgebra $\mathfrak {osp}_{1|2n}$ is not a Lie algebra but it has properties similar to simple Lie algebras. For example, the category of finite-dimensional $\mathfrak {osp}_{1|2n}$-modules is semisimple and the Harish–Chandra isomorphism $Z(\mathfrak {osp}_{1|2n}) \simeq \mathbb {C}\hspace{0.55542pt}[\mathfrak {h}]^W$ holds, where $Z(\mathfrak {g})$ denotes the center of the universal enveloping algebra $U(\mathfrak {g})$, $\mathfrak {h}$ is a Cartan subalgebra of $\mathfrak {osp}_{1|2n}$ and W is the Weyl group. However, an analogue of Duflo’s theorem [7] does not hold for $\mathfrak {osp}_{1|2n}$, that is the annihilating ideals of Verma modules in $U(\mathfrak {osp}_{1|2n})$ are not generated by their intersections with the center $Z(\mathfrak {osp}_{1|2n})$. This problem was noticed by Musson [25] and solved by Gorelik and Lantzmann [17] by replacing $Z(\mathfrak {osp}_{1|2n})$ with a larger algebra, called the ghost center $\widetilde{Z}(\mathfrak {osp}_{1|2n})$.

For a Lie superalgebra $\mathfrak {g} = \mathfrak {g}_{\bar{0}}\hspace{1.111pt}{\oplus }\hspace{1.111pt}\mathfrak {g}_{\bar{1}}$ with $\mathfrak {g}_{\bar{1}} \ne 0$, the ghost center $\widetilde{Z}(\mathfrak {g})$ was introduced by Gorelik in [14] as the direct sum , where is the anticenter defined by , where $p(\hspace{1.111pt}{\cdot }\hspace{1.111pt})$ denotes the parity. If $\mathfrak {g}$ is a finite-dimensional simple basic classical Lie superalgebra, it is known [14] that $\widetilde{Z}(\mathfrak {g})$ coincides with the center of $U(\mathfrak {g})_{\bar{0}}$ and thus is a purely even subalgebra of $U(\mathfrak {g})$. Moreover, in the case $\mathfrak {g}=\mathfrak {osp}_{1|2n}$, there exists $T \in U(\mathfrak {g})_{\bar{0}}$ such that by [2, 17, 25]. The element T is called the Casimir’s ghost [2] since $T^2\! \in Z(\mathfrak {osp}_{1|2n})$. When $n=1$, in particular, T can be expressed as $4Q - 4C + {1}/{2}$ by using [27] and then $T^2 \!= 4C + {1}/{4}$, where C is the Casimir element in $U(\mathfrak {osp}_{1|2})$ and Q is the one in $U(\mathfrak {sl}_2)$.

The finite -algebra $U(\mathfrak {g}, f)$ is an associative superalgebra over $\mathbb {C}$ defined from a simple finite-dimensional basic classical Lie superalgebra $\mathfrak {g}$ and an even nilpotent element f [3, 11, 23, 24, 30,31,32]. In the case when $\mathfrak {g}$ is a simple Lie algebra and f is a principal nilpotent element $f_\textrm{prin}$, it was proven by Kostant [23] that the corresponding finite -algebra $U(\mathfrak {g}, f_\textrm{prin})$ is isomorphic to the center $Z(\mathfrak {g})$ of $U(\mathfrak {g})$.

The -algebra is a vertex superalgebra defined by the Drinfeld–Sokolov reduction associated to $\mathfrak {g}, f$ and a complex number $k \in \mathbb {C}$, called the level [9, 20]. In general, (Ramond-twisted) positive-energy simple modules of a $\frac{1}{2}\mathbb {Z}$-graded vertex superalgebra V with a Hamiltonian operator H are classified in terms of an associated superalgebra called the (H-twisted) Zhu algebra of V. See Sect. 2 for the definition of Ramond-twisted modules. It was proven by De Sole and Kac [6] that the Zhu algebra of is isomorphic to the finite -algebra $U(\mathfrak {g}, f)$. In particular, there exists a one-to-one correspondence between simple modules of $U(\mathfrak {g}, f)$ and Ramond-twisted positive-energy simple modules of . The -algebra associated to a principal nilpotent element $f = f_\textrm{prin}$ is called the principal -algebra of $\mathfrak {g}$, which we denote by .

Theorem A

(Theorem 6.5) $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})$ is isomorphic to $\widetilde{Z}(\mathfrak {osp}_{1|2n})$ as associative algebras.

The finite -algebra $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})$ associated to $\mathfrak {osp}_{1|2n}$ and its principal nilpotent element $f_\textrm{prin}$ is an associative superalgebra with its non-trivial odd part, while the ghost center $\widetilde{Z}(\mathfrak {osp}_{1|2n})$ is not. However, we prove an isomorphism between them.

To prove Theorem A, we use the Miura map $\mu$ and its injectivity and relationship with the Harish–Chandra homomorphism of $\mathfrak {osp}_{1|2n}$. See Sect. 4 for the definition of $\mu$. The map $\mu$ was originally introduced in [24]. The injectivity of $\mu$ was only known for non-super cases, but has been recently proved by [26] for super cases. As a corollary of Theorem A, it follows that Ramond-twisted positive-energy simple modules of principal -algebras are classified by simple modules of the ghost center $\widetilde{Z}(\mathfrak {osp}_{1|2n})$ (Corollary 6.7). We note that the definition of $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})$ in the paper comes from the theory of vertex superalgebras (Remark 4.4).

We will prove in the next paper that the untwisted Zhu algebra of is isomorphic to the center of $U(\mathfrak {sp}_{2n})$. This is only known in the case $n=1$ due to [22]. Thus, by Theorem A, the untwisted Zhu algebra is isomorphic to the even part of $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})$. It is also proven in (6.1) that the Zhu functor is compatible with the Miura map, and hence with the Harish–Chandra homomorphism. Since simple modules of $Z(\mathfrak {sp}_{2n})$ can be described in terms of the central characters and the Harish–Chandra homomorphism, we may apply this to construct simple modules of inside tensor products of Fock modules and the free fermion F. This allows us to analyze the coset construction of , where $V^{\ell }(\mathfrak {so}_{2n+1})$ is the affine vertex algebra of $\mathfrak {so}_{2n+1}$ at some level $\ell$. We intend to address this issue in sequels to this paper in our joint work with Thomas Creutzig.

Let us remark that a super analog of the Soergel Struktursatz for a suitable Whittaker functor from the integral BGG category of any basic classical simple Lie superalgebra $\mathfrak {g}$ to the category of finite-dimensional modules of $U(\mathfrak {g}, f_\textrm{prin})$ has been established in [4]. We also hope to clarify the relationship between the ghost center of $\mathfrak {g}$ and $U(\mathfrak {g}, f_\textrm{prin})$ in general $\mathfrak {g}$ to apply to the Soergel Struktursatz in our future works.

The paper is organized as follows. In Sect. 2, we introduce H-twisted Zhu algebras. In Sect. 3, we recall the definitions of -algebras . In Sect. 4, we give two definitions $U(\mathfrak {g}, f)_\textrm{I}$ and $U(\mathfrak {g}, f)_\textrm{II}$ of finite -algebras and show the equivalence of the definitions, that is, $U(\mathfrak {g}, f)_\textrm{I} \simeq U(\mathfrak {g}, f)_\textrm{II}$. The proof is similar to [5]. In Sect. 5, we recall the principal -algebra of $\mathfrak {osp}_{1|2n}$. In Sect. 6, we prove Theorem A.

2 H-twisted Zhu algebras

Let V be a vertex superalgebra. Denote by $|0\rangle$ the vacuum vector, by $\partial$ the translation operator, by p(A) the parity of $A \in V$, and by $Y(A, z) = A(z) = \sum _{n \in \mathbb {Z}}A_{(n)}z^{-n-1}$ the field on V corresponding to $A \in V$. Let

$$\begin{aligned}{}[A{}_\lambda B] = \sum _{n=0}^\infty \frac{\lambda ^n}{n!}\,A_{(n)}B \in \mathbb {C}\hspace{0.55542pt}[\lambda ]\hspace{1.111pt}{\otimes }\hspace{1.111pt}V \end{aligned}$$

be the $\lambda$-bracket of A and B for $A, B \in V$. A Hamiltonian operator H on V is a semisimple operator on V satisfying that $[H, Y(A, z)] = z\partial _zY(A, z) + Y(H(A), z)$ for all $A \in V$. The eigenvalue of H is called the conformal weight. If V is conformal and $L(z) = \sum _{n \in \mathbb {Z}}L_nz^{-n-2}$ is the field corresponding to the conformal vector of V, we may choose $H = L_0$ as the Hamiltonian operator.

Suppose that V is a $\frac{1}{2}\mathbb {Z}$-graded vertex superalgebra with respect to a Hamiltonian operator H. Denote by $\Delta _A$ the conformal weight of $A \in V$. Define the $*$-product and $\hspace{0.55542pt}{\circ }\hspace{1.111pt}$-product of V by

$$\begin{aligned} A \hspace{1.111pt}{*}\hspace{1.111pt}B =\sum _{j=0}^\infty \left( {\begin{array}{c}\Delta _A\\ j\end{array}}\right) A_{(j-1)}B,\;\; A \hspace{0.55542pt}{\circ }\hspace{1.111pt}B =\sum _{j=0}^\infty \left( {\begin{array}{c}\Delta _A\\ j\end{array}}\right) A_{(j-2)}B,\quad A, B \in V. \end{aligned}$$

Then the quotient space

$$\begin{aligned} \textrm{Zhu}_H V = V/V\hspace{0.55542pt}{\circ }\hspace{1.111pt}V \end{aligned}$$

has a structure of associative superalgebra with respect to the product induced from $*$, and is called the H-twisted Zhu algebra of V. Here $V \hspace{0.55542pt}{\circ }\hspace{1.111pt}V = {\text {Span}}_\mathbb {C}\{A \hspace{0.55542pt}{\circ }\hspace{0.55542pt}B \,{|}\, A, B \,{\in }\, V\}$. The vacuum vector $|0\rangle$ defines a unit of $\textrm{Zhu}_H V$. A superspace M is called a Ramond-twisted V-module if M is equipped with a parity-preserving linear map

$$\begin{aligned} Y_M :M \ni A \rightarrow Y_M(A, z) =\!\! \sum _{n \in \mathbb {Z}+ \Delta _A}\!\!\!\!A^M_{(n)}z^{-n-1} \in ({\text {End}}M)\bigl [\!\bigl [z^{{1}/{2}}\!, z^{-{1}/{2}}\bigr ]\!\bigr ] \end{aligned}$$

such that (1) for each $C \in M$, $A^M_{(n)}C = 0$ if $n \gg 0$, (2) $Y_M(|0\rangle , z) = {\text {id}}_M$ and (3) for any $A, B \in V$, $C \in M$, $n \in \mathbb {Z}$, $m \in \mathbb {Z}+ \Delta _A$ and $\ell \in \mathbb {Z}+ \Delta _B$,

$$\begin{aligned} \sum _{j=0}^\infty (-1)^j\left( {\begin{array}{c}n\\ j\end{array}}\right) \Bigl ( A^M_{(m+n-j)}B^M_{(\ell +j)}&- (-1)^{p(A)p(B)}B^M_{(\ell +n-j)}A^M_{(m+j)} \Bigr )\hspace{1.111pt}C\\&\qquad = \sum _{j=0}^\infty \left( {\begin{array}{c}m\\ j\end{array}}\right) \bigl (A_{(n+j)}B \bigr )^M_{(m+\ell -j)} \hspace{1.111pt}C. \end{aligned}$$

Hence the Ramond-twisted module is a twisted module of V for the automorphism $\textrm{e}^{2\pi i H}$. In particular, M is just a V-module if V is $\mathbb {Z}$-graded. Define $A^M_n$ by $Y_M(A, z) = \sum _{n \in \mathbb {Z}}A^M_n z^{-n-\Delta _A}$ for $A \in V$. A Ramond-twisted V-module M is called positive-energy if M has an $\mathbb {R}$-grading $M = \bigoplus _{j \in \mathbb {R}}M_j$ with $M_0 \ne 0$ such that $A^M_n M_j\! \subset M_{j+n}$ for all $A \in V$, $n \in \mathbb {Z}$ and $j \in \mathbb {R}$. Then $M_0$ is called the top space. By [6, Lemma 2.22], a linear map $V \ni A \mapsto A^M_0|_{M_0}\! \in {\text {End}}M_0$ induces a homomorphism $\textrm{Zhu}_H V \rightarrow {\text {End}}M_0$. Thus we have a functor $M \mapsto M_0$ from the category of positive-energy Ramond-twisted V-modules to the category of $\mathbb {Z}_2$-graded $\textrm{Zhu}_H V$-modules. By [6, Theorem 2.30], these functors establish a bijection (up to isomorphisms) between simple positive-energy Ramond-twisted V-modules and simple $\mathbb {Z}_2$-graded $\textrm{Zhu}_H V$-modules.

3 -algebras

Let $\mathfrak {g}$ be a finite-dimensional simple Lie superalgebra with the normalized even supersymmetric invariant bilinear form $(\hspace{1.111pt}{\cdot }\hspace{1.111pt}|\hspace{1.111pt}{\cdot }\hspace{1.111pt})$ and f be a nilpotent element in the even part of $\mathfrak {g}$. Then there exists a $\frac{1}{2}\mathbb {Z}$-grading on $\mathfrak {g}$ that is good for f. See [20] for the definitions of good gradings and [8, 18] for the classifications. Let $\mathfrak {g}_j$ be the homogeneous subspace of $\mathfrak {g}$ with degree j. The good grading $\mathfrak {g}=\bigoplus _{j\in \frac{1}{2}\mathbb {Z}}\mathfrak {g}_j$ for f on $\mathfrak {g}$ satisfies the following properties:

• $[\mathfrak {g}_i, \mathfrak {g}_j]\subset \mathfrak {g}_{i+j}$,

• $f\in \mathfrak {g}_{-1}$,

• ${\text {ad}}\hspace{0.55542pt}(f):\mathfrak {g}_j\rightarrow \mathfrak {g}_{j-1}$ is injective for $j\geqslant {1}/{2}$ and surjective for $j\leqslant {1}/{2}$,

• $(\mathfrak {g}_i\hspace{0.55542pt}{|}\hspace{1.111pt}\mathfrak {g}_j)=0$ if $i+j\ne 0$,

• $\dim \mathfrak {g}^f=\dim \mathfrak {g}_0+\dim \mathfrak {g}_{{1}/{2}}$, where $\mathfrak {g}^f$ is the centralizer of f in $\mathfrak {g}$.

Then we can choose a set of simple roots $\Pi$ of $\mathfrak {g}$ for a Cartan subalgebra $\mathfrak {h}\subset \mathfrak {g}_0$ such that all positive root vectors lie in $\mathfrak {g}_{\geqslant 0}$. Denote $\Delta _j = \{ \alpha \,{\in }\,\Delta \,{|}\, \mathfrak {g}_\alpha \,{\subset }\, \mathfrak {g}_j\}$ and $\Pi _j = \Pi \cap \Delta _j$ for $j \in \frac{1}{2}\mathbb {Z}$. We have $\Pi =\Pi _0\sqcup \Pi _{ {1}/{2}}\sqcup \Pi _1$. Let $\chi :\mathfrak {g}\rightarrow \mathbb {C}$ be a linear map defined by $\chi (u)=(f\hspace{0.55542pt}{|}\hspace{1.111pt}u)$. Since ${\text {ad}}(f):\mathfrak {g}_{ {1}/{2}}\rightarrow \mathfrak {g}_{- {1}/{2}}$ is an isomorphism of vector spaces, the super skew-symmetric bilinear form $\mathfrak {g}_{{1}/{2}}\times \mathfrak {g}_{{1}/{2}}\ni (u, v)\mapsto \chi ([u, v])\in \mathbb {C}$ is non-degenerate. We fix a root vector $u_\alpha$ and denote by $p(\alpha )$ the parity of $u_\alpha$ for $\alpha \in \Delta$.

Let $V^k(\mathfrak {g})$ be the affine vertex superalgebra associated to $\mathfrak {g}$ at level $k \in \mathbb {C}$, which is generated by u(z) ($u \in \mathfrak {g}$) whose parity is the same as u, satisfying that

$$\begin{aligned}{}[u_\lambda v] = [u, v] + k(u\hspace{1.111pt}{|}\hspace{1.111pt}v)\hspace{1.111pt}\lambda ,\quad u, v \in \mathfrak {g}. \end{aligned}$$

Let $F(\mathfrak {g}_{{1}/{2}})$ be the neutral vertex superalgebra associated to $\mathfrak {g}_{{1}/{2}}$, which is strongly generated by $\phi _{\alpha }(z)$ ($\alpha \in \Delta _{{1}/{2}}$) whose parity is equal to $p(\alpha )$, satisfying that

$$\begin{aligned}{}[{\phi _\alpha }_\lambda \phi _\beta ] = \chi (u_\alpha , u_\beta ),\quad \alpha , \beta \in \Delta _{{1}/{2}}. \end{aligned}$$

Let $F^\textrm{ch}(\mathfrak {g}_{>0})$ be the charged fermion vertex superalgebra associated to $\mathfrak {g}_{>0}$, which is strongly generated by $\varphi _\alpha (z), \varphi ^*_\alpha (z)$ ($\alpha \in \Delta _{>0}$) whose parities are equal to $p(\alpha ) + \bar{1}$, satisfying that

$$\begin{aligned}{}[{\varphi _\alpha }_\lambda \varphi ^*_\beta ] = \delta _{\alpha , \beta },\;\; [{\varphi _\alpha }_\lambda \varphi _\beta ] = [{\varphi ^*_\alpha }_\lambda \varphi ^*_\beta ] = 0,\quad \alpha , \beta \in \Delta _{>0}. \end{aligned}$$

Let $C^k(\mathfrak {g},f) = V^k(\mathfrak {g}) \hspace{1.111pt}{\otimes }\hspace{1.111pt}F(\mathfrak {g}_{{1}/{2}}) \hspace{1.111pt}{\otimes }\hspace{1.111pt}F^\textrm{ch}(\mathfrak {g}_{>0})$ and d be an odd element in $C^k(\mathfrak {g},f)$ defined by

$$\begin{aligned} d \,=&\sum _{\alpha \in \Delta _{>0}}\!(-1)^{p(\alpha )}u_\alpha \varphi ^*_\alpha - \frac{1}{2}\sum _{\alpha , \beta , \gamma \in \Delta _{>0}}\!\!\!\!(-1)^{p(\alpha )p(\gamma )}c_{\alpha , \beta }^\gamma :\!\varphi _\gamma \varphi ^*_\alpha \varphi ^*_\beta \!:\\&\qquad \qquad \qquad \qquad \qquad \quad \,+\sum _{\alpha \in \Delta _{{1}/{2}}}\!\!\phi _\alpha \varphi ^*_\alpha + \!\sum _{\alpha \in \Delta _{>0}}\!\!\chi (u_\alpha )\hspace{1.111pt}\varphi ^*_\alpha . \end{aligned}$$

Then $(C^k(\mathfrak {g},f), d_{(0)})$ defines a cochain complex with respect to the charged degree: ${\text {charge}}\varphi _\alpha = -{\text {charge}}\varphi ^*_\alpha = 1$ ($\alpha \in \Delta _{>0}$) and ${\text {charge}}A=0$ for all $A \in V^k(\mathfrak {g})\hspace{1.111pt}{\otimes }\hspace{1.111pt}F(\mathfrak {g}_{ {1}/{2}})$. The (affine) -algebra associated to $\mathfrak {g}$, f at level k is defined by

Let $C^k(\mathfrak {g},f)_+$ be a subcomplex generated by $\phi _\alpha (z)$ ($\alpha \in \Delta _{ {1}/{2}}$), $\varphi ^*_\alpha (z)$ ($\alpha \in \Delta _{>0}$) and

$$\begin{aligned} J^u(z)\, =\, u(z) + \!\sum _{\alpha , \beta \in \Delta _{>0}}\!\!\!c_{\beta , u}^\alpha :\!\varphi ^*_\beta (z)\varphi _\alpha (z)\!:,\quad u \in \mathfrak {g}_{\leqslant 0}. \end{aligned}$$

Then we have [21]

Thus, is a vertex subalgebra of $C^k(\mathfrak {g},f)_+$. Using the fact that

$$\begin{aligned}{}[{J^u}_\lambda J^v]&= J^{[u, v]} + \tau (u\hspace{1.111pt}{|}\hspace{1.111pt}v)\hspace{1.111pt}\lambda ,\quad u, v \in \mathfrak {g}_{\leqslant 0},\\ \tau (u\hspace{1.111pt}{|}\hspace{1.111pt}v)&= k(u\hspace{1.111pt}{|}\hspace{1.111pt}v) + \frac{1}{2}\,\kappa _{\mathfrak {g}}(u\hspace{1.111pt}{|}\hspace{1.111pt}v) - \frac{1}{2}\,\kappa _{\mathfrak {g}_0}(u\hspace{1.111pt}{|}\hspace{1.111pt}v),\quad u, v \in \mathfrak {g}_{\leqslant 0}, \end{aligned}$$

where $\kappa _{\mathfrak {g}}$ denotes the Killing form on $\mathfrak {g}$, it follows that the vertex algebra generated by $J^u(z)$ $(u \in \mathfrak {g}_{\leqslant 0})$ is isomorphic to the affine vertex superalgebra associated to $\mathfrak {g}_{\leqslant 0}$ and $\tau$, which we denote by $V^{\tau }(\mathfrak {g}_{\leqslant 0})$. Therefore the homogeneous subspace of $C^k(\mathfrak {g},f)_+$ with charged degree 0 is isomorphic to $V^{\tau }(\mathfrak {g}_{\leqslant 0}) \hspace{1.111pt}{\otimes }\hspace{1.111pt}F(\mathfrak {g}_{ {1}/{2}})$. The projection $\mathfrak {g}_{\leqslant 0} \twoheadrightarrow \mathfrak {g}_0$ induces a vertex superalgebra surjective homomorphism $V^{\tau }(\mathfrak {g}_{\leqslant 0}) \hspace{1.111pt}{\otimes }\hspace{1.111pt}F(\mathfrak {g}_{{1}/{2}}) \twoheadrightarrow V^{\tau }(\mathfrak {g}_{0}) \hspace{1.111pt}{\otimes }\hspace{1.111pt}F(\mathfrak {g}_{ {1}/{2}})$ so that we have

by the restriction. The map $\Upsilon$ is called the Miura map and is injective thanks to [1, 10, 26].

4 Finite -algebras

Recall the definitions of finite -algebras $U(\mathfrak {g}, f)$, following [5]. We give two definitions in (4.1), (4.2) denoted by $U(\mathfrak {g}, f)_\textrm{I}$, $U(\mathfrak {g}, f)_\textrm{II}$, respectively, and prove the isomorphism $U(\mathfrak {g}, f)_\textrm{I} \simeq U(\mathfrak {g}, f)_\textrm{II}$ in Theorem 4.2.

Let $\Phi$ be an associative $\mathbb {C}$-superalgebra generated by $\Phi _{\alpha }$ $(\alpha \in \Delta _{ {1}/{2}})$ that has the same parity as $u_\alpha$, satisfying that

$$\begin{aligned}{}[\Phi _{\alpha }, \Phi _{\beta }] = \chi ([u_\alpha , u_\beta ]),\quad \alpha , \beta \in \Delta _{ {1}/{2}}. \end{aligned}$$

Here [A, B] denotes $AB - (-1)^{p(A)\,p(B)}BA$. We extend the definition of $\Phi _\alpha$ for all $\alpha \in \Delta _{>0}$ by $\Phi _{\alpha } = 0$ for $\alpha \in \Delta _{\geqslant 1}$. Let $\Lambda (\mathfrak {g}_{>0})$ be the Clifford superalgebra associated to $\mathfrak {g}_{>0}$, which is an associative $\mathbb {C}$-superalgebra generated by $\psi _\alpha , \psi ^*_\alpha$ $(\alpha \in \Delta _{>0})$ with the opposite parity to that of $u_\alpha$, satisfying that

$$\begin{aligned}{}[\psi _\alpha , \psi ^*_\beta ] = \delta _{\alpha , \beta },\quad [\psi _\alpha , \psi _\beta ] = [\psi ^*_\alpha , \psi ^*_\beta ] = 0,\quad \alpha , \beta \in \Delta _{>0}. \end{aligned}$$

The Clifford superalgebra $\Lambda (\mathfrak {g}_{>0})$ has the charged degree defined by $\deg \hspace{0.55542pt}(\psi _\alpha ) = 1 = -\deg \hspace{0.55542pt}(\psi ^*_\alpha )$ for all $\alpha \in \Delta _{>0}$. Set

$$\begin{aligned} C_{\hspace{1.111pt}\mathrm I}&= U(\mathfrak {g})\hspace{1.111pt}{\otimes }\hspace{1.111pt}\Phi \hspace{1.111pt}{\otimes }\hspace{1.111pt}\Lambda (\mathfrak {g}_{>0}),\quad d_{\hspace{1.111pt}\mathrm I}={\text {ad}}\hspace{0.55542pt}(Q),\\ Q&= \!\! \sum _{\alpha \in \Delta _{>0}}\!\!(-1)^{p(\alpha )}X_\alpha \psi _\alpha -\frac{1}{2}\sum _{\alpha ,\beta ,\gamma \in \Delta _{>0}}\!\!\!\!(-1)^{p(\alpha )p(\gamma )}c_{\alpha ,\beta }^\gamma \psi _\gamma \psi ^*_\alpha \ \psi ^*_\beta ,\\ X_\alpha&= u_\alpha + (-1)^{p(\alpha )}(\Phi _{\alpha } + \chi (u_\alpha )),\quad \alpha \in \Delta _{>0}, \end{aligned}$$

where $c_{\alpha , \beta }^\gamma$ is the structure constant defined by $[u_\alpha , u_\beta ] = \sum _{\gamma \in \Delta _{>0}} c_{\alpha , \beta }^\gamma u_\gamma$. Then a pair $(C_{\hspace{1.111pt}\mathrm I}, d_{\hspace{1.111pt}\mathrm I})$ forms a cochain complex with respect to the charged degree on $\Lambda (\mathfrak {g}_{>0})$ and the cohomology

$$\begin{aligned} U(\mathfrak {g}, f)_\textrm{I} = H^\bullet (C_{\hspace{1.111pt}\mathrm I}, d_{\hspace{1.111pt}\mathrm I}) \end{aligned}$$

(4.1)

has a structure of an associative $\mathbb {C}$-superalgebra inherited from that of $C_{\hspace{1.111pt}\mathrm I}$. Let

$$\begin{aligned} j^u = u + \!\!\sum _{\alpha , \beta \in \Delta _{>0}}\!\!\! c_{\beta , u}^\alpha \psi ^*_\beta \ \psi _\alpha ,\quad u \in \mathfrak {g}. \end{aligned}$$

Then

$$\begin{aligned} {\text {ad}}\hspace{0.55542pt}(Q) \hspace{1.111pt}{\cdot }\hspace{1.111pt}\psi _{\alpha } = j^{u_\alpha }\! + (-1)^{p(\alpha )}(\Phi _{\alpha } + \chi (u_\alpha )) = X_\alpha + \!\!\sum _{\alpha , \beta \in \Delta _{>0}}\!\!\!c_{\beta , u}^\alpha \psi ^*_\beta \ \psi _\alpha ,\;\; \alpha \in \Delta _{>0}. \end{aligned}$$

Let $C_-$ be the subalgebra of $C_\textrm{I}$ generated by $\psi _\alpha$, ${\text {ad}}\hspace{0.55542pt}(Q) \hspace{1.111pt}{\cdot }\hspace{1.111pt}\psi _{\alpha }$ $(\alpha \in \Delta _{>0})$ and $C_+$ be the subalgebra of $C_{\hspace{1.111pt}\mathrm I}$ generated by $j^u$ $(u \in \mathfrak {g}_{\leqslant 0})$, $\Phi _\alpha$ $(\alpha \in \Delta _{ {1}/{2}})$ and $\psi ^*_\alpha$ $(\alpha \in \Delta _{>0})$. Then $(C_\pm , d_{\hspace{1.111pt}\mathrm I})$ form subcomplexes and $C_{\hspace{1.111pt}\mathrm I} \simeq C_- \hspace{1.111pt}{\otimes }\hspace{1.111pt}C_+$ as vector superspaces. Since $H(C_-, d_{\hspace{1.111pt}\mathrm I})=\mathbb {C}$, we have

$$\begin{aligned} H(C_\textrm{I}, d_{\hspace{1.111pt}\mathrm I}) \simeq \ H(C_-, d_{\hspace{1.111pt}\mathrm I}) \hspace{1.111pt}{\otimes }\hspace{1.111pt}H(C_+, d_{\hspace{1.111pt}\mathrm I}) = H(C_+, d_{\hspace{1.111pt}\mathrm I}). \end{aligned}$$

Using the same argument as in [21], it follows that $H^n(C_+, d_{\hspace{1.111pt}\mathrm I})=0$ for $n\ne 0$. Therefore $U(\mathfrak {g}, f)_\textrm{I}$ is a subalgebra of $C^0_+$, which is generated by $j^u$ $(u \in \mathfrak {g}_{\leqslant 0})$ and $\Phi _\alpha$ $(\alpha \in \Delta _{{1}/{2}})$. Since $[j^u\!, j^v] = j^{[u, v]}$ for $u, v \in \mathfrak {g}_{\leqslant 0}$, there exists an isomorphism $C^0_+ \simeq U(\mathfrak {g}_{\leqslant 0}) \hspace{1.111pt}{\otimes }\hspace{1.111pt}\Phi$ as associative $\mathbb {C}$-superalgebras. The projection $\mathfrak {g}_{\leqslant 0} \twoheadrightarrow \mathfrak {g}_0$ induces an associative $\mathbb {C}$-superalgebra surjective homomorphism $U(\mathfrak {g}_{\leqslant 0}) \hspace{1.111pt}{\otimes }\hspace{1.111pt}\Phi \twoheadrightarrow U(\mathfrak {g}_{0}) \hspace{1.111pt}{\otimes }\hspace{1.111pt}\Phi$ so that we have

$$\begin{aligned} \mu :U(\mathfrak {g}, f)_\textrm{I} \rightarrow U(\mathfrak {g}_{0}) \hspace{1.111pt}{\otimes }\hspace{1.111pt}\Phi \end{aligned}$$

by the restriction. The map $\mu$ is called the Miura map for the finite -algebras and it is injective by [13, 24, 26]. Let $\mathbb {C}_{-\chi }$ be the one-dimensional $\mathfrak {g}_{\geqslant 1}$-module defined by $\mathfrak {g}_{\geqslant 1} \ni u \mapsto -\chi (u) \in \mathbb {C}$ and $M_\textrm{II}$ be the induced left $\mathfrak {g}$-module

$$\begin{aligned} M_\textrm{II} = {\text {Ind}}^\mathfrak {g}_{\mathfrak {g}_{\geqslant 1}}\mathbb {C}_{-\chi } = U(\mathfrak {g})\underset{U(\mathfrak {g}_{\geqslant 1})}{\hspace{1.111pt}{\otimes }\hspace{1.111pt}}\mathbb {C}_{-\chi } \simeq U(\mathfrak {g})/I_{-\chi }, \end{aligned}$$

where $I_{-\chi }$ is a left $U(\mathfrak {g})$-module generated by $u + \chi (u)$ for all $u \in \mathfrak {g}_{\geqslant 1}$. Then $M_\textrm{II}$ has a structure of the ${\text {ad}}\hspace{0.55542pt}(\mathfrak {g}_{>0})$-module inherited from that of $U(\mathfrak {g})$. Set the ${\text {ad}}\hspace{0.55542pt}(\mathfrak {g}_{>0})$-invariant subspace

$$\begin{aligned} U(\mathfrak {g}, f)_\textrm{II} = (M_\textrm{II})^{{\text {ad}}(\mathfrak {g}_{>0})}. \end{aligned}$$

(4.2)

Then $U(\mathfrak {g}, f)_\textrm{II}$ also has a structure of an associative $\mathbb {C}$-superalgebra inherited from that of $U(\mathfrak {g})$. We may also define $U(\mathfrak {g}, f)_\textrm{II}$ as the Chevalley cohomology $H(\mathfrak {g}_{>0}, M_\textrm{II})$ of the left $\mathfrak {g}_{>0}$-module $M_\textrm{II}$:

Lemma 4.1

([11, 26])

$$\begin{aligned} H(\mathfrak {g}_{>0}, M_\textrm{II}) = H^0(\mathfrak {g}_{>0}, M_\textrm{II}) = (M_\textrm{II})^{{\text {ad}}(\mathfrak {g}_{>0})}. \end{aligned}$$

Proof

Though the assertion is proved in [11] for Lie algebras $\mathfrak {g}$, the same proof together with [26, Corollary 2.6] applies. $\square$

Theorem 4.2

([5, Theorem A.6]) There exists an isomorphism $U(\mathfrak {g}, f)_\textrm{I} \simeq U(\mathfrak {g}, f)_\textrm{II}$ as associative $\mathbb {C}$-superalgebras.

Proof

Though the assertion is proved in [5] for Lie algebras $\mathfrak {g}$, the same proof applies as follows. Let $C_\textrm{II} = \Lambda (\mathfrak {g}_{>0})_c \hspace{1.111pt}{\otimes }\hspace{1.111pt}M_\textrm{II}$ be the Chevalley cohomology complex of the left $\mathfrak {g}_{>0}$-module $M_\textrm{II}$, where $\Lambda (\mathfrak {g}_{>0})_c$ is the subalgebra of $\Lambda (\mathfrak {g}_{>0})$ generated by $\psi ^*_\alpha$ for all $\alpha \in \Delta _{>0}$, and $d_{\hspace{1.111pt}\mathrm II}$ be the derivation of the cochain complex $C_\textrm{II}$. Let $U(\mathfrak {g}_{>0})_{-\chi } = U(\mathfrak {g}_{>0})\hspace{1.111pt}{\otimes }\hspace{1.111pt}\mathbb {C}_{-\chi }$ be a left $\mathfrak {g}_{\geqslant 1}$-module defined by the diagonal action, where $U(\mathfrak {g}_{>0})$ is considered as a left $\mathfrak {g}_{\geqslant 1}$-module by the left multiplication, and $M_\textrm{III}$ be the induced left $\mathfrak {g}$-module

$$\begin{aligned} M_\textrm{III} = {\text {Ind}}^\mathfrak {g}_{\mathfrak {g}_{\geqslant 1}}U(\mathfrak {g}_{>0})_{-\chi } = U(\mathfrak {g})\underset{U(\mathfrak {g}_{\geqslant 1})}{\hspace{1.111pt}{\otimes }\hspace{1.111pt}}U(\mathfrak {g}_{>0})_{-\chi }. \end{aligned}$$

Let $\mathbb {C}_\chi$ be the one-dimensional $\mathfrak {g}_{\geqslant 1}$-module defined by $\mathfrak {g}_{\geqslant 1} \ni u \mapsto \chi (u) \in \mathbb {C}$ and $U(\mathfrak {g})_\chi = U(\mathfrak {g}) \hspace{1.111pt}{\otimes }\hspace{1.111pt}\mathbb {C}_\chi$ be a right $\mathfrak {g}_{\geqslant 1}$-module defined by the diagonal action, where $U(\mathfrak {g})$ is considered as a right $\mathfrak {g}_{\geqslant 1}$-module by the right multiplication. Then we have

$$\begin{aligned} M_\textrm{III} \simeq U(\mathfrak {g})_\chi \underset{U(\mathfrak {g}_{\geqslant 1})}{\hspace{1.111pt}{\otimes }\hspace{1.111pt}}U(\mathfrak {g}_{>0}) \end{aligned}$$

so that $M_\textrm{III}$ is a left $\mathfrak {g}$- right $\mathfrak {g}_{>0}$-bimodule. Note that there is an isomorphism$\Lambda (\mathfrak {g}_{>0}) \simeq \Lambda (\mathfrak {g}_{>0})_h \hspace{1.111pt}{\otimes }\hspace{1.111pt}\Lambda (\mathfrak {g}_{>0})_c$ of vector superspaces, where $\Lambda (\mathfrak {g}_{>0})_h$ is the subalgebra of $\Lambda (\mathfrak {g}_{>0})$ generated by $\psi _\alpha$ for all $\alpha \in \Delta _{>0}$. Let $d_h$ be the derivation of the Chevalley homology complex $M_\textrm{III}\hspace{1.111pt}{\otimes }\hspace{1.111pt}\Lambda (\mathfrak {g}_{>0})_h$ of the right $\mathfrak {g}_{>0}$-module $M_\textrm{III}$. Then $M_\textrm{III}\hspace{1.111pt}{\otimes }\hspace{1.111pt}\Lambda (\mathfrak {g}_{>0})_h$ is clearly a left $\mathfrak {g}_{>0}$-module with respect to the adjoint $\mathfrak {g}_{>0}$-action. Now, let $\overline{d}_c$ be the derivation of the Chevalley cohomology complex $\Lambda (\mathfrak {g}_{>0})_c \hspace{1.111pt}{\otimes }\hspace{1.111pt}M_\textrm{III}{\otimes }\hspace{1.111pt}\Lambda (\mathfrak {g}_{>0})_h$ of the left $\mathfrak {g}_{>0}$-module $M_\textrm{III}\hspace{1.111pt}{\otimes }\hspace{1.111pt}\Lambda (\mathfrak {g}_{>0})_h$. Then, as in [5], we get a new cochain complex $(C_\textrm{III}, d_{\hspace{1.111pt}\mathrm III})$ defined by

$$\begin{aligned} C_{\textrm{III}} = \Lambda (\mathfrak {g}_{>0})_c \hspace{1.111pt}{\otimes }\hspace{1.111pt}M_\textrm{III}\hspace{1.111pt}{\otimes }\hspace{1.111pt}\Lambda (\mathfrak {g}_{>0})_h,\quad d_{\hspace{1.111pt}\mathrm III} = d_c +(-1)^{\delta -1}\hspace{1.111pt}{\otimes }\hspace{1.111pt}d_h, \end{aligned}$$

where $\delta$ denotes the parity of the part of elements in $\Lambda (\mathfrak {g}_{>0})_c$. Then it is easy to check that the following linear map

$$\begin{aligned} i_{\hspace{1.111pt}\mathrm III} :C_\textrm{III} \ni \ \psi ^*_{\beta _1}\cdots \psi ^*_{\beta _i} \hspace{1.111pt}{\otimes }\hspace{1.111pt}\bigl (v_1 \cdots v_s \underset{U(\mathfrak {g}_{\geqslant 1} )}{\hspace{1.111pt}{\otimes }\hspace{1.111pt}}u_{\alpha _1} \cdots u_{\alpha _t}\bigr ) \hspace{1.111pt}{\otimes }\hspace{1.111pt}\psi _{\gamma _1}\cdots \psi _{\gamma _j}\\ \mapsto \ \psi ^*_{\beta _1}\cdots \psi ^*_{\beta _i} \cdot v_1 \cdots v_s \cdot X_{\alpha _1} \cdots X_{\alpha _t} \cdot \psi _{\gamma _1}\cdots \psi _{\gamma _j} \in \overline{C}_\textrm{I} \end{aligned}$$

with $v_1, \ldots , v_s \in \mathfrak {g}$, $\alpha _1,\ldots ,\alpha _t, \beta _1, \ldots ,\beta _i, \gamma _1, \ldots , \gamma _j \in \Delta _{>0}$ is well defined and induces an isomorphism of complexes $(C_\textrm{III}, d_{\hspace{1.111pt}\mathrm III}) \rightarrow (C_{\hspace{1.111pt}\mathrm I}, d_{\hspace{1.111pt}\mathrm I} )$ since $i_{\hspace{1.111pt}\mathrm III \rightarrow I} \hspace{0.55542pt}{\circ }\hspace{1.111pt}d_{\hspace{1.111pt}\mathrm III} = d_{\hspace{1.111pt}\mathrm I} \hspace{0.55542pt}{\circ }\hspace{1.111pt}i_{\hspace{1.111pt}\mathrm III \rightarrow I}$. Now

$$\begin{aligned} H_n(C_\textrm{III}, d_h)&= \Lambda (\mathfrak {g}_{>0})_c \hspace{1.111pt}{\otimes }\hspace{1.111pt}H_n\bigl (M_\textrm{III}\hspace{1.111pt}{\otimes }\hspace{1.111pt}\Lambda (\mathfrak {g}_{>0})_h, d_h\bigr )\\&= \Lambda (\mathfrak {g}_{>0})_c \hspace{1.111pt}{\otimes }\hspace{1.111pt}U(\mathfrak {g})_\chi \underset{U(\mathfrak {g}_{\geqslant 1})}{\hspace{1.111pt}{\otimes }\hspace{1.111pt}} H_n(\mathfrak {g}_{>0}, U(\mathfrak {g}_{>0}))\\&= \delta _{n, 0}\hspace{1.111pt}\Lambda (\mathfrak {g}_{>0})_c \hspace{1.111pt}{\otimes }\hspace{1.111pt}U(\mathfrak {g})_\chi \underset{U(\mathfrak {g}_{\geqslant 1})}{\hspace{1.111pt}{\otimes }\hspace{1.111pt}}\mathbb {C}\simeq \delta _{n,0}\ C_\textrm{II}. \end{aligned}$$

Thus, since $d_c$ and $(-1)^{\delta -1}\hspace{1.111pt}{\otimes }\hspace{1.111pt}d_h$ commute, we have

$$\begin{aligned} H(C_\textrm{III}, d_{\hspace{1.111pt}\mathrm III}) \simeq H(H(C_\textrm{III},d_h), d_c) \simeq H(C_\textrm{II}, d_{\hspace{1.111pt}\mathrm II}). \end{aligned}$$

The above argument together with the isomorphism $i_\mathrm{III \rightarrow I}$ of complexes shows that $(C_\textrm{I}, d_{\hspace{1.111pt}\mathrm I} )$ and $(C_\textrm{II}, d_{\hspace{1.111pt}\mathrm II})$ are quasi-isomorphic via the following quasi-isomorphism

$$\begin{aligned} {\begin{matrix} i_{\hspace{1.111pt}\mathrm I \rightarrow II}:C_{\hspace{1.111pt}\mathrm I} \ni \ \psi ^*_{\beta _1}\cdots \psi ^*_{\beta _i} &{}\cdot v_1 \cdots v_s \cdot X_{\alpha _1} \cdots X_{\alpha _t} \cdot \psi _{\gamma _1}\cdots \psi _{\gamma _j}\\ &{} \mapsto \ \delta _{t,0}\hspace{1.111pt}\delta _{j,0}\ \psi ^*_{\beta _1}\cdots \psi ^*_{\beta _i} \cdot v_1 \cdots v_s \in C_\textrm{II}, \end{matrix}} \end{aligned}$$

(4.3)

which preserves the associative superalgebra structures on the cohomologies. $\square$

Definition 4.3

The finite -algebra $U(\mathfrak {g}, f)$ associated to $\mathfrak {g}, f$ is defined to be the superalgebra $U(\mathfrak {g}, f)_\textrm{I}$, which is isomorphic to $U(\mathfrak {g}, f)_\textrm{II}$ due to Theorem 4.2.

Remark 4.4

The same result as Theorem 4.2 for Poisson superalgebra versions has been studied in [33]. Also remark that our definitions of the finite -algebra $U(\mathfrak {g}, f)$ are not necessarily equivalent to the definitions in some literature [28, 29, 34]. In fact, in case that $\mathfrak {g}=\mathfrak {osp}_{1|2n}$ and $f = f_\textrm{prin}$ its principal nilpotent element, we have $\dim \mathfrak {g}_{{1}/{2}} = \dim \mathfrak {g}_{ {1}/{2}, \bar{1}} = 1$ and thus $\mathfrak {g}_{\geqslant 1} \subsetneq \mathfrak {g}_{>0}$. Then $U(\mathfrak {g}, f) \simeq U(\mathfrak {g}, f)_\textrm{II} = (U(\mathfrak {g})/I_{-\chi })^{{\text {ad}}(\mathfrak {g}_{>0})}$ is a proper subalgebra of $(U(\mathfrak {g})/I_{-\chi })^{{\text {ad}}(\mathfrak {g}_{\geqslant 1})} = {\text {End}}_{U(\mathfrak {g})}U(\mathfrak {g})/I_{-\chi }$.

The vertex superalgebra $C^k(\mathfrak {g}, f)$ has a conformal vector $\omega$ if $k\ne -h^\vee$, which defines the conformal weights on $C^k(\mathfrak {g}, f)$ by $L_0$, where $\omega (z) = \sum _{n \in \mathbb {Z}}L_n z^{-n-2}$. See [20] for the details. Then $H=L_0$ defines a Hamiltonian operator on $C^k(\mathfrak {g}, f)$, the vertex subalgebra $C^k(\mathfrak {g},f)_+$, and the corresponding -algebra . Moreover the Hamiltonian operator $L_0$ is well defined for all $k \in \mathbb {C}$. Recall that $\textrm{Zhu}_H V$ is the H-twisted Zhu algebra of V, see Sect. 2. Let $x \in \mathfrak {h}$ be such that $[x,u] = j u$ for $u \in \mathfrak {g}_j$. Then by [1, 6],

$$\begin{aligned} \textrm{Zhu}_H C^k(\mathfrak {g},f)_+ \simeq C_+,\;\; J^u \mapsto j^u + \tau (x\hspace{1.111pt}{|}\hspace{1.111pt}u),\;\; \phi _\alpha \mapsto \Phi _\alpha ,\;\; \varphi _\alpha ^* \mapsto \psi ^*_\alpha \end{aligned}$$

(4.4)

for $u \in \mathfrak {g}_{\leqslant 0}$, $\alpha \in \Delta _{>0}$ and $\textrm{Zhu}_H H^0(C^k(\mathfrak {g},f)_+,d_{(0)}) \simeq H^0(C_+, d_{\hspace{1.111pt}\mathrm I})$ so that

(4.5)

Let $V_1, V_2$ be any $\frac{1}{2}\mathbb {Z}_{\geqslant 0}$-graded vertex superalgebras with the Hamiltonian operators and $g :V_1\rightarrow V_2$ any vertex superalgebra homomorphism preserving the conformal weights. Since $g(V_1\hspace{0.55542pt}{\circ }\hspace{1.111pt}V_1)=g(V_1)\hspace{0.55542pt}{\circ }\hspace{1.111pt}g(V_1)\subset V_2\hspace{0.55542pt}{\circ }\hspace{1.111pt}V_2$, the map g induces an algebra homomorphism

$$\begin{aligned} {\textrm{Zhu}}_H (g):\textrm{Zhu}_H V_1\rightarrow \textrm{Zhu}_H V_2. \end{aligned}$$

Apply for $g = \Upsilon$. Then we get

$$\begin{aligned} {\textrm{Zhu}}_H(\Upsilon ) = \mu \end{aligned}$$

by construction.

5 Principal -algebras of $\mathfrak {osp}_{1|2n}$

Consider the case that

$$\begin{aligned} \mathfrak {g} = \mathfrak {osp}_{1|2n} = \left\{ u = \left( \begin{array}{c|cc} 0 &{} {}^t\hspace{-1.111pt}y &{} -\hspace{1.111pt}{}^t\hspace{-1.111pt}x \\ \hline x &{} a &{} b \\ y &{} c &{} -\hspace{1.111pt}{}^t\hspace{-1.111pt}a \end{array} \right) \in \mathfrak {gl}_{1|2n} \mathrel {} \Bigg | \mathrel {} \begin{array}{l} a, b, c \in {\text {Mat}}_\mathbb {C}(n \hspace{1.111pt}{\times }\hspace{1.111pt}n),\\ x, y \in {\text {Mat}}_\mathbb {C}(n \hspace{1.111pt}{\times }\hspace{1.111pt}1),\\ b = {}^t\hspace{-1.111pt}b,\ c = {}^t\hspace{-1.111pt}c \end{array} \right\} , \end{aligned}$$

where ${}^t\hspace{-1.111pt}A$ denotes the transpose of A. Let $\{e_{i, j}\}_{i, j \in I}$ be the standard basis of $\mathfrak {gl}_{1|2n}$ with the index set $I = \{ 0, 1, \ldots , n, -1, \ldots , -n\}$ and $h_i = e_{i, i} - e_{-i, -i}$ $(i = 1, \ldots , n)$. Then $\mathfrak {h} = {\text {Span}}_\mathbb {C}\{h_i\}_{i=1}^n$ is a Cartan subalgebra of $\mathfrak {osp}_{1|2n}$. Define $\epsilon _i \in \mathfrak {h}^*$ by $\epsilon _i(h_j) = \delta _{i, j}$. Then $\Delta _+ = \{\epsilon _i, 2\epsilon _i\}_{i=1}^n \sqcup \{\epsilon _i-\epsilon _j,\epsilon _i+\epsilon _j\}_{1 \leqslant i<j \leqslant n}$ forms a set of positive roots with simple roots $\Pi = \{ \alpha _i\}_{i=1}$, $\alpha _i = \epsilon _i - \epsilon _{i+1}$ $(i=1,\ldots ,n-1)$ and $\alpha _n = \epsilon _n$, and $\epsilon _1, \ldots , \epsilon _n$ are the (non-isotropic) odd roots in $\Delta _+$. Set $\Delta _- = -\Delta _+$ and $(u\hspace{1.111pt}{|}\hspace{1.111pt}v) = -{\text {str}}\hspace{0.55542pt}(uv)$ for $u, v \in \mathfrak {osp}_{1|2n}$. We may identify $\mathfrak {h}^*$ with $\mathfrak {h}$ through $\nu :\mathfrak {h}^* \ni \lambda \mapsto \nu (\lambda ) \in \mathfrak {h}$ defined by $\lambda (h) = (h\hspace{1.111pt}{|}\hspace{1.111pt}\nu (\lambda ))$ for $h \in \mathfrak {h}$, which induces a non-degenerate bilinear form on $\mathfrak {h}^*$ by $(\lambda \hspace{1.111pt}{|}\hspace{1.111pt}\mu ) = (\nu (\lambda )\hspace{1.111pt}{|}\hspace{1.111pt}\nu (\mu ))$ so that $(\epsilon _i\hspace{0.55542pt}{|}\hspace{1.111pt}\epsilon _j) = \delta _{i, j}/2$. Then $h_i$ corresponds to $2\epsilon _i = 2 \sum _{j=i}^n \alpha _j$ by $\nu$. We have

$$\begin{aligned} (\alpha _i\hspace{0.55542pt}{|}\hspace{1.111pt}\alpha _i) = 1,\quad (\alpha _i\hspace{0.55542pt}{|}\hspace{1.111pt}\alpha _{i+1}) = -\frac{1}{2}\hspace{0.55542pt},\;\; i = 1, \ldots , n-1;\quad (\alpha _n\,{|}\,\alpha _n) = \frac{1}{2}\hspace{0.55542pt}. \end{aligned}$$

Note that the dual Coxeter number of $\mathfrak {osp}_{1|2n}$ is equal to $n+{1}/{2}$. Let

$$\begin{aligned} f_\textrm{prin} = \sum _{i=1}^{n-1} u_{-\alpha _i} + u_{-2\alpha _n} \end{aligned}$$

be a principal nilpotent element in the even part of $\mathfrak {osp}_{1|2n}$, where $u_\alpha$ denotes some root vector for $\alpha \in \Delta$. Then there exists a unique good grading on $\mathfrak {osp}_{1|2n}$ such that $\Pi _1 = \{\alpha _i\}_{i=1}^{n-1}$ and $\Pi _{{1}/{2}} = \{\alpha _n\}$. Thus

$$\begin{aligned} \mathfrak {g}_0 = \mathfrak {h},\quad \mathfrak {g}_{>0} = \mathfrak {n} :=\!\bigoplus _{\alpha \in \Delta _+}\!\mathfrak {g}_\alpha ,\quad \mathfrak {g}_{<0} = \mathfrak {n}_- :=\!\bigoplus _{\alpha \in \Delta _-}\!\mathfrak {g}_\alpha . \end{aligned}$$

Let

be the principal -algebra of $\mathfrak {osp}_{1|2n}$ at level k. The Miura map for is

where $\pi$ is the Heisenberg vertex algebra generated by even fields $\alpha _i(z)$, $i = 1, \ldots , n$, satisfying that

$$\begin{aligned}{}[{\alpha _i}_\lambda \alpha _j] = \biggl (k+n+\frac{1}{2}\biggr )(\alpha _i\hspace{0.55542pt}{|}\hspace{1.111pt}\alpha _j)\hspace{1.111pt}\lambda ,\quad i, j = 1,\ldots ,n, \end{aligned}$$

and F is the free fermion vertex superalgebra generated by an odd field $\phi (z)$ satisfying that

$$\begin{aligned}{}[\phi _\lambda \phi ] = 1. \end{aligned}$$

By [12, Theorem 6.4], is strongly generated by $G, W_2, W_4, \ldots , W_{2n}$ for odd G and even $W_2, W_4, \ldots , W_{2n}$ elements of conformal weights $n + {1}/{2}$ and $2, 4, \ldots , 2n$ such that

$$\begin{aligned} \begin{aligned} \Upsilon (G)(z)&=\ :\!(2(k+n)\hspace{1.111pt}\partial + h_1(z)) \cdots (2(k+n)\hspace{1.111pt}\partial + h_n(z))\hspace{1.111pt}\phi (z)\!:,\\ \Upsilon (W_{2i})(z)&\equiv \!\sum _{1\leqslant j_1<\cdots <j_{i}\leqslant n}\!\!\!:\!h_{j_1}^2(z)\cdots h_{j_{i}}^2(z)\!:\quad ({\text {mod}}\ C_2(\pi \hspace{1.111pt}{\otimes }\hspace{1.111pt}F) ),\\ C_2(\pi \hspace{1.111pt}{\otimes }\hspace{1.111pt}F)&=\{ A_{(-2)}B\mid A, B\in \pi \hspace{1.111pt}{\otimes }\hspace{1.111pt}F\}, \end{aligned} \end{aligned}$$

(5.1)

and

$$\begin{aligned}{}[G_\lambda G] = W_{2n} + \sum _{i=1}^{n-1}\gamma _i\biggl (\frac{\lambda ^{2i-1}}{(2i-1)!}\,W_{2n-2i+1}+\frac{\lambda ^{2i}}{(2i)!}\,W_{2n-2i}\biggr ) + \gamma _n \hspace{1.111pt}\frac{\lambda ^{2n}}{(2n)!} \end{aligned}$$

(5.2)

for some , where

$$\begin{aligned} h_i(z) = 2\sum _{j=i}^n \alpha _j(z),\quad \gamma _i = (-1)^i\prod _{j=1}^i \hspace{1.111pt}( 2(2j-1)(k+n)-1 ) ( 4j(k+n)+1 ), \end{aligned}$$

which satisfy that

$$\begin{aligned}{}[{h_i}_\lambda h_j] = (2k+2n+1)\hspace{1.111pt}\delta _{i, j}\lambda ,\quad i, j=1,\ldots , n. \end{aligned}$$

If $k + n + {1}/{2} \ne 0$,

$$\begin{aligned} L = \frac{W_2}{2(2k+2n+1)} \end{aligned}$$

is a unique conformal vector of with the central charge

$$\begin{aligned} c(k) ={} -\frac{(2n+1)(2(2n-1)(k+n)-1)(4n(k+n)+1)}{2(2k+2n+1)}\hspace{0.55542pt}. \end{aligned}$$

6 Zhu algebras of

By (4.5), we have an isomorphism

Then $\iota _1$ is induced by (4.4):

$$\begin{aligned}&{\textrm{Zhu}}_H C^k(\mathfrak {osp}_{1|2n},f_\textrm{prin}) \xrightarrow { \ \simeq \ } C_+,\\&J^u \mapsto j^u + (2k+2n+1)(\rho _{\mathfrak {osp}}\hspace{1.111pt}{|}\hspace{1.111pt}u),\;\; \phi _\alpha \mapsto \Phi _\alpha ,\;\; \varphi ^*_\alpha \mapsto \psi ^*_\alpha , \end{aligned}$$

where

$$\begin{aligned} \rho _{\mathfrak {osp}} = \frac{1}{2}\sum _{\alpha \in \Delta _+}(-1)^{p(\alpha )}\alpha . \end{aligned}$$

Let $\mathbb {C}\hspace{0.55542pt}[\mathfrak {h}^*] = U(\mathfrak {h})$ and set an isomorphism

$$\begin{aligned}&\iota _2 :\textrm{Zhu}_H \pi \hspace{1.111pt}{\otimes }\hspace{1.111pt}\textrm{Zhu}_H F \xrightarrow { \ \simeq \ } \mathbb {C}\hspace{0.55542pt}[\mathfrak {h}^*]\hspace{1.111pt}{\otimes }\hspace{1.111pt}\Phi ,\\&h_i \mapsto h_i + (2n-2i+1)\biggl (k+n+\frac{1}{2}\biggr ),\quad \phi _{\alpha _n} \mapsto \Phi _{\alpha _n}. \end{aligned}$$

Then we have a commutative diagram of Miura maps

(6.1)

By [6], has a PBW basis generated by $G, W_2, W_4, \ldots , W_{2n}$. By abuse of notation, we shall use the same notation for the generators of $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})$ corresponding to $G, W_2, W_4, \ldots , W_{2n}$ by $\iota _1$.

Lemma 6.1

$\mu (G) = (h_1 + \rho _{\mathfrak {osp}}(h_1))(h_2 + \rho _{\mathfrak {osp}}(h_2)) \cdots (h_n + \rho _{\mathfrak {osp}}(h_n))\hspace{1.111pt}{\otimes }\hspace{1.111pt}\Phi _{\alpha _n}$.

Proof

We have

Thus

$$\begin{aligned} \mu (G)&= \iota _2\bigl (({-}\hspace{1.66656pt}(2n-1)(k+n) + h_1) \hspace{1.111pt}{*}\hspace{1.111pt}({-}\hspace{1.66656pt}(2n-3)(k+n) + h_2) \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \cdots * (-(k+n) + h_n)\hspace{1.111pt}{*}\hspace{1.111pt}\phi \bigr )\\&=\biggl (h_1 + n-1 + \frac{1}{2}\biggr )\biggl (h_2 + n-2+\frac{1}{2}\biggr ) \cdots \biggl (h_n + \frac{1}{2}\biggr )\hspace{1.111pt}{\otimes }\hspace{1.111pt}\Phi _{\alpha _n}. \end{aligned}$$

Therefore the assertion follows from the fact that $\rho _{\mathfrak {osp}}(h_i) = n-i+ {1}/{2}$. $\square$

For a basic classical Lie superalgebra $\mathfrak {g}$ such that $\mathfrak {g}_{\bar{1}} \ne 0$, denote by

called the center, the anticenter and the ghost center of $U(\mathfrak {g})$, respectively due to [14]. Then the ghost center $\widetilde{Z}(\mathfrak {g})$ coincides with the center of $U(\mathfrak {g})_{\bar{0}}$ by [14, Corollary 4.4.4]. In case that $\mathfrak {g}=\mathfrak {osp}_{1|2n}$, there exists $T \in U(\mathfrak {g})_{\bar{0}}$ [2, 17, 25] such that

where

$$\begin{aligned} \eta :U(\mathfrak {osp}_{1|2n}) \twoheadrightarrow U(\mathfrak {h}) = \mathbb {C}\hspace{0.55542pt}[\mathfrak {h}^*] \end{aligned}$$

is the projection induced by the decomposition

$$\begin{aligned} U(\mathfrak {osp}_{1|2n}) \simeq \mathfrak {n}_- U(\mathfrak {osp}_{1|2n}) \hspace{1.111pt}{\oplus }\hspace{1.111pt}U(\mathfrak {h}) \hspace{1.111pt}{\oplus }\hspace{1.111pt}U(\mathfrak {osp}_{1|2n})\hspace{1.111pt}\mathfrak {n} \end{aligned}$$

and $\sigma$ is an isomorphism defined by

$$\begin{aligned} \sigma :\mathbb {C}\hspace{0.55542pt}[\mathfrak {h}^*] \rightarrow \mathbb {C}\hspace{0.55542pt}[\mathfrak {h}^*],\quad f \mapsto (\sigma (f) :\lambda \mapsto f(\lambda -\rho _{\mathfrak {osp}})). \end{aligned}$$

The element T is called the Casimir’s ghost [2] since $T^2 \!\in Z(\mathfrak {osp}_{1|2n})$ is such that $(\sigma \hspace{0.55542pt}{\circ }\hspace{1.111pt}\eta )(T^2) = h_1^2\cdots h_n^2$, and is studied for general $\mathfrak {g}$ in [14]. It is well known [15, 19] that the restriction of $\sigma \hspace{0.55542pt}{\circ }\hspace{1.111pt}\eta$ to $Z(\mathfrak {g})$ is injective and maps onto $\mathbb {C}\hspace{0.55542pt}[\mathfrak {h}^*]^W$, where W is the Weyl group of $\mathfrak {sp}_{2n}$, called the Harish–Chandra homomorphism of $\mathfrak {osp}_{1|2n}$. Recall that

$$\begin{aligned} U(\mathfrak {osp}_{1|2n}, f_\textrm{prin}) \simeq U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})_\textrm{II} = (U(\mathfrak {osp}_{1|2n})/I_{-\chi })^{{\text {ad}}\mathfrak {n}}, \end{aligned}$$

where $I_{-\chi }$ is a left $U(\mathfrak {osp}_{1|2n})$-module generated by $u_\alpha + (f_\textrm{prin}\hspace{0.55542pt}{|}\hspace{1.111pt}u_\alpha )$ for all $\alpha \in \Delta _+\hspace{1.111pt}{\setminus }\hspace{1.111pt}\{\alpha _n\}$. Define the projections $q_1, q_2$ by

$$\begin{aligned}&q_1 :U(\mathfrak {osp}_{1|2n}) \twoheadrightarrow U(\mathfrak {osp}_{1|2n})/I_{-\chi },\\&q_2 :U(\mathfrak {osp}_{1|2n})/I_{-\chi } \simeq \mathfrak {n}_-U(\mathfrak {osp}_{1|2n})/I_{-\chi } \hspace{1.111pt}{\oplus }\hspace{1.111pt}U(\mathfrak {h}) \hspace{1.111pt}{\oplus }\hspace{1.111pt}U(\mathfrak {h})\hspace{1.111pt}u_{\alpha _n} \!\twoheadrightarrow U(\mathfrak {h}) \hspace{1.111pt}{\oplus }\hspace{1.111pt}U(\mathfrak {h})\hspace{1.111pt}u_{\alpha _n} \end{aligned}$$

and a linear map $q_3$ by

$$\begin{aligned} q_3 :U(\mathfrak {h}) \hspace{1.111pt}{\oplus }\hspace{1.111pt}U(\mathfrak {h})\hspace{1.111pt}u_{\alpha _n}\! \rightarrow \mathbb {C}\hspace{0.55542pt}[\mathfrak {h}^*] \hspace{1.111pt}{\otimes }\hspace{1.111pt}\Phi ,\quad (f_1,\mathrel {} f_2 \cdot u_{\alpha _n}) \mapsto f_1\hspace{1.111pt}{\otimes }\hspace{1.111pt}1+ f_2\hspace{1.111pt}{\otimes }\hspace{1.111pt}\Phi _{\alpha _n}. \end{aligned}$$

Then, using the quasi-isomorphism $i_{\hspace{1.111pt}\mathrm I \rightarrow II}$ in (4.3), the Miura map $\mu$ can be identified with the restriction of the composition map $q_3 \hspace{0.55542pt}{\circ }\hspace{1.111pt}q_2$ to $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})_\textrm{II}$ since $u_{\alpha _n} = X_{\alpha _n} + \Phi _{\alpha _n}$.

Lemma 6.2

$q_1(Tu_{\alpha _n})$ is the element of $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})_\textrm{II}$ corresponding to G.

Proof

First of all, we show that $q_1(Tu_{\alpha _n}) \in U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})_\textrm{II}$. It is enough to show that $[u_\alpha , Tu_{\alpha _n}] \equiv 0$ $({\text {mod}} I_{-\chi })$ for all $\alpha \in \Delta _+$. Let $\Delta _{+, \hspace{1.111pt}\bar{i}} = \{ \alpha \in \Delta _+ \,{|}\, p(u_\alpha ) = \bar{i}\}$. Since $[u_\alpha , T]=0$ for $\alpha \in \Delta _{+, \bar{0}}$, we have

$$\begin{aligned}{}[u_\alpha , Tu_{\alpha _n}] = T[u_\alpha , u_{\alpha _n}] \equiv 0\quad ({\text {mod}} I_{-\chi }),\quad \alpha \in \Delta _{+, \bar{0}}. \end{aligned}$$

Next, for $\alpha \in \Delta _{+, \bar{1}}\hspace{1.111pt}{\setminus }\hspace{1.111pt}\{\alpha _n\}$, since $u_\alpha T + Tu_\alpha = 0$, we also have

$$\begin{aligned}{}[u_\alpha , Tu_{\alpha _n}] = {}-T[u_\alpha , u_{\alpha _n}] + 2Tu_{\alpha _n}u_\alpha \equiv 0\quad ({\text {mod}} I_{-\chi }),\quad \alpha \in \Delta _{+, \bar{1}}\hspace{1.111pt}{\setminus }\hspace{1.111pt}\{\alpha _n\}. \end{aligned}$$

Finally, in case that $\alpha = \alpha _n$,

$$\begin{aligned}{}[u_{\alpha _n}, Tu_{\alpha _n}] = (u_{\alpha _n}T + Tu_{\alpha _n})\hspace{1.111pt}u_{\alpha _n} = 0. \end{aligned}$$

Therefore, $q_1(Tu_{\alpha _n})$ belongs to $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})_\textrm{II}$. Now $\mu = q_3 \hspace{0.55542pt}{\circ }\hspace{1.111pt}q_2|_{U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})_\textrm{II}}$ and by definition,

$$\begin{aligned} ((\sigma \hspace{1.111pt}{\otimes }\hspace{1.111pt}1) \hspace{0.55542pt}{\circ }\hspace{1.111pt}\mu )(q_1(Tu_{\alpha _n}))&= ((\sigma \hspace{1.111pt}{\otimes }\hspace{1.111pt}1) \hspace{0.55542pt}{\circ }\hspace{1.111pt}q_3 \hspace{0.55542pt}{\circ }\hspace{1.111pt}q_2 \hspace{0.55542pt}{\circ }\hspace{1.111pt}q_1)(Tu_{\alpha _n})\\&= (\sigma \hspace{0.55542pt}{\circ }\hspace{1.111pt}\eta )(T) \hspace{1.111pt}{\otimes }\hspace{1.111pt}\Phi _{\alpha _n} = h_1\cdots h_n \hspace{1.111pt}{\otimes }\hspace{1.111pt}\Phi _{\alpha _n}. \end{aligned}$$

By Lemma 6.1, $((\sigma \hspace{1.111pt}{\otimes }\hspace{1.111pt}1) \hspace{0.55542pt}{\circ }\hspace{1.111pt}\mu )(G) = h_1\cdots h_n \hspace{1.111pt}{\otimes }\hspace{1.111pt}\Phi _{\alpha _n}$. Since $(\sigma \hspace{1.111pt}{\otimes }\hspace{1.111pt}1) \hspace{0.55542pt}{\circ }\hspace{1.111pt}\mu$ is injective, we have $q_1(Tu_{\alpha _n}) = G$. $\square$

Theorem 6.3

$U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})_{\bar{0}} \simeq Z(\mathfrak {osp}_{1|2n})$.

Proof

Since $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})$ has a PBW basis generated by $G,W_2,W_4, \ldots , W_{2n}$ and G is a unique odd generator, $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})_{\bar{0}}$ has a PBW basis generated by $W_2,W_4, \ldots , W_{2n}$. Now $\Phi$ is a superalgebra generated by $\Phi _{\alpha _n}$ with the relation $2 \Phi _{\alpha _n}^2 = \chi (u_{\alpha _n}, u_{\alpha _n})$. Thus $\mu$ maps $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})_{\bar{0}}$ to $\mathbb {C}\hspace{0.55542pt}[\mathfrak {h}^*]$. By (5.1), $\mu (W_{2i})$ for $i = 1, \ldots , n$ are algebraically independent in $\mathbb {C}\hspace{0.55542pt}[\mathfrak {h}^*]$ with degree 2i (but not necessary homogeneous). Now, by definition, $q_2 \hspace{0.55542pt}{\circ }\hspace{1.111pt}q_1 =\eta$ on $Z(\mathfrak {osp}_{1|2n})$. Hence $q_2 \hspace{0.55542pt}{\circ }\hspace{1.111pt}q_1 |_{Z(\mathfrak {osp}_{1|2n})}$ is injective. In particular, $q_1 |_{Z(\mathfrak {osp}_{1|2n})}$ is injective. Clearly, $q_1(Z(\mathfrak {osp}_{1|2n}))$ is ${\text {ad}}\mathfrak {n}$-invariant. Thus, $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})\simeq U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})_\textrm{II}$ contains $Z(\mathfrak {osp}_{1|2n})$ through $q_1$. Moreover

$$\begin{aligned} \mu (Z(\mathfrak {osp}_{1|2n})) = (q_3 \hspace{0.55542pt}{\circ }\hspace{1.111pt}q_2 \hspace{0.55542pt}{\circ }\hspace{1.111pt}q_1 )(Z(\mathfrak {osp}_{1|2n})) = \eta (Z(\mathfrak {osp}_{1|2n})) = \sigma ^{-1}(\mathbb {C}\hspace{0.55542pt}[\mathfrak {h}^*]^W). \end{aligned}$$

Since $\mathbb {C}[ \mathfrak {h}^*]^W$ is a symmetric algebra of $h_1^2, \ldots , h_n^2$, $\mu (Z(\mathfrak {osp}_{1|2n}))$ must contain all $\mu (W_{2i})$ for $i = 1, \ldots , n$. Therefore

$$\begin{aligned} U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})_{\bar{0}} \simeq Z(\mathfrak {osp}_{1|2n}). \end{aligned}$$

This completes the proof. $\square$

Corollary 6.4

.

Proof

The assertion is immediate from Theorem 6.3 and the fact that

$\square$

Consider a linear isomorphism

defined by $\xi (z, a) = (z, a\,u_{\alpha _n})$. Then by Lemma 6.2 and the fact that , we have $(q_1 \hspace{0.55542pt}{\circ }\hspace{1.111pt}\xi )(\widetilde{Z}(\mathfrak {osp}_{1|2n})) \subset U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})_\textrm{II}$.

Theorem 6.5

The map $q_1 \hspace{0.55542pt}{\circ }\hspace{1.111pt}\xi :\widetilde{Z}(\mathfrak {osp}_{1|2n}) \rightarrow U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})$ is an isomorphism of associative algebras.

Proof

By definition and Lemma 6.2, $(q_3 \hspace{0.55542pt}{\circ }\hspace{1.111pt}q_2 \hspace{0.55542pt}{\circ }\hspace{1.111pt}q_1 \hspace{0.55542pt}{\circ }\hspace{1.111pt}\xi )(zT) = (q_3 \hspace{0.55542pt}{\circ }\hspace{1.111pt}q_2 \hspace{0.55542pt}{\circ }\hspace{1.111pt}q_1)(zTu_{\alpha _n}) = \eta (z)\hspace{0.55542pt}G$ for all $z \in Z(\mathfrak {osp}_{1|2n})$. Thus, is injective. In particular, is injective. Using the fact that $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})$ has a PBW basis generated by $G,W_2,W_4, \ldots , W_{2n}$ and Theorem 6.3, it follows that $q_1 \hspace{0.55542pt}{\circ }\hspace{1.111pt}\xi$ is a linear isomorphism. Now, we may suppose that $\chi (u_{\alpha _n}, u_{\alpha _n}) = 2$. Then $\Phi _{\alpha _n}^2 = 1$ so that $\mu (T^2) = \sigma ^{-1}(h_1^2\cdots h_n^2) = \mu (G^2)$. Therefore $q_1 \hspace{0.55542pt}{\circ }\hspace{1.111pt}\xi$ defines an isomorphism of associative algebras. $\square$

Let $L(\lambda )$ be the simple highest weight $\mathfrak {osp}_{1|2n}$-module with the highest weight $\lambda$. Then there exists $\chi _\lambda :Z(\mathfrak {osp}_{1|2n}) \rightarrow \mathbb {C}$ such that z acts on $\chi _\lambda (z)$ on $L(\lambda )$ for all $z \in Z(\mathfrak {osp}_{1|2N})$. The map $\chi _\lambda$ is called a central character of $\mathfrak {osp}_{1|2n}$ and is induced by $\eta$ and one-dimensional $\mathbb {C}\hspace{0.55542pt}[\mathfrak {h}^*]$-module $\mathbb {C}_\lambda$ defined by $f \mapsto f(\lambda )$. Using the Harish–Chandra homomorphism, it follows that $\chi _{\lambda _1} = \chi _{\lambda _2}$ if and only if $\lambda _2 = w(\lambda _1 + \rho _{\mathfrak {osp}})- \rho _{\mathfrak {osp}}$ for some $w \in W$. Let

$$\begin{aligned} D =\Bigl \{\hspace{1.111pt}\lambda \in \mathfrak {h}^* \ \Big | \,\prod _{\alpha \in \Delta _{\bar{1}}}\!(\lambda +\rho _{\mathfrak {osp}}\hspace{1.111pt}{|}\hspace{1.111pt}\alpha ) = 0\Bigr \}. \end{aligned}$$

Denote by $\chi _\lambda \in D$ if $\lambda \in D$. Since $w(\Delta _{\bar{1}}) \subset \Delta _{\bar{1}}$ for all $w \in W$, we have $\lambda \in D \Rightarrow w(\lambda + \rho _{\mathfrak {osp}}) - \rho _{\mathfrak {osp}}\in D$ for any $w \in W$ so that $\chi _\lambda \in D$ is well defined.

From now on, we will identify $\widetilde{Z}(\mathfrak {osp}_{1|2n})$ with $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})$ by Theorem 6.5. Then $\widetilde{Z}(\mathfrak {osp}_{1|2n})$ is a superalgebra such that . Let E be a finite-dimensional $\mathbb {Z}_2$-graded simple $\widetilde{Z}(\mathfrak {osp}_{1|2n})$-module. Then $Z(\mathfrak {osp}_{1|2n})$ acts on E as $\chi _\lambda$ for some $\lambda \in \mathfrak {h}^*$. For a non-zero parity-homogeneous element $v \in E$, Tv has an opposite parity to v such that $T^2 v = \chi _\lambda (T^2)\hspace{1.111pt}v$. Recall that the set $\{h_1, \ldots , h_n\}$ is identified with $2\Delta _{+, \bar{1}}$ by $\mathfrak {h} \simeq \mathfrak {h}^*$. Then, using the fact that $\eta (T^2) = \sigma ^{-1}(h_1^2\cdots h_n^2)$, it follows that

$$\begin{aligned} \chi _\lambda (T^2) = \prod _{i=1}^n\hspace{1.111pt}\bigl ((\lambda +\rho _{\mathfrak {osp}})(h_i)\bigr )^2 = \!\!\prod _{\alpha \in \Delta _{+, \bar{1}}}\!\!(\lambda + \rho _{\mathfrak {osp}}\hspace{1.111pt}{|}\hspace{1.111pt}2\alpha )^2. \end{aligned}$$

Hence $\chi _\lambda (T^2) = 0$ if and only if $\chi _\lambda \in D$. Since E is simple, $E = \mathbb {C}v$ if $\chi _\lambda \in D$ and $E = \mathbb {C}v \hspace{1.111pt}{\oplus }\hspace{1.111pt}\mathbb {C}Tv$ if $\chi _\lambda \notin D$, which we denote by $E_{\chi _\lambda }$. Here we identify $E_{\chi _\lambda }$ with the parity change of $E_{\chi _\lambda }$ if $\chi _\lambda (T^2) = 0$. Therefore we obtain the following results:

Proposition 6.6

A finite-dimensional $\mathbb {Z}_2$-graded simple $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})$-module is isomorphic to $E_{\chi _\lambda }$ for some $\lambda \in \mathfrak {h}^*$. In particular, there exists one-to-one correspondence between isomorphism classes (up to the parity change) of finite-dimensional $\mathbb {Z}_2$-graded simple $U(\mathfrak {osp}_{1|2n}, f_\textrm{prin})$-modules and central characters of $\mathfrak {osp}_{1|2n}$.

Corollary 6.7

There exists a bijective correspondence between central characters of $\mathfrak {osp}_{1|2n}$ and isomorphism classes (up to the parity change) of simple positive-energy Ramond-twisted -modules with finite-dimensional top spaces.

Proof

The assertion is immediate from , Proposition 6.6 and [6, Theorem 2.30]. $\square$

Corollary 6.7 implies that dimensions of the top spaces $E_{\chi _\lambda }$ of simple positive-energy Ramond-twisted -modules are equal to 2 if and only if $(\lambda +\rho _{\mathfrak {osp}}\,{|}\,\alpha ) \ne 0$ for all $\alpha \in \Delta _{\bar{1}}$. We remark that this condition is equivalent to one that the annihilator of the Verma module $M(\lambda )$ is generated by its intersection with the center $Z(\mathfrak {osp}_{1|2n})$ by [16].