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

A Lie suparalgebra $\mathfrak{osp}_{1|2n}$ is a finite-dimensional simple Lie superalgebra whose Dynkin diagram is the same as 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 suparalgebra $\mathfrak{osp}_{1|2n}$ is not a Lie algebra but has similar properties to simple Lie algebras. For example, the category of finite-dimensional $\mathfrak{osp}_{1|2n}$-modules is semisimple and we have the Harish-Chandra isomorphism $Z(\mathfrak{osp}_{1|2n})\simeq\mathbb{C}[\mathfrak{h}]^{W}$, 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 $\mathfrak{osp}_{1|2n}$ doesn’t satisfy the Duflo theorem [Duf77], which says that annihilators of Verma modules in $U(\mathfrak{g})$ is generated by its intersections with the center $Z(\mathfrak{g})$ for simple Lie algebras $\mathfrak{g}$. This problem was founded by Musson [Mus97] and solved by Gorelik and Lantzmann [GL00] by using an extension algebra $\widetilde{Z}(\mathfrak{osp}_{1|2n})$ of $Z(\mathfrak{osp}_{1|2n})$. More precisely, Gorelik and Lantzmann prove that annihilators of Verma modules in $U(\mathfrak{osp}_{1|2n})$ is generated by its intersections with $\widetilde{Z}(\mathfrak{osp}_{1|2n})$. The associative algebra $\widetilde{Z}(\mathfrak{osp}_{1|2n})$ is called the ghost center of $\mathfrak{osp}_{1|2n}$ in [Gor00].

For a Lie superalgebra $\mathfrak{g}$ with $\mathfrak{g}_{\bar{1}}\neq 0$, the ghost center $\widetilde{Z}(\mathfrak{g})$ is introduced by Gorelik in [Gor00] as the direct sum $Z(\mathfrak{g})\oplus\mathcal{A}(\mathfrak{g})$, where $\mathcal{A}(\mathfrak{g})$ is the anticenter defined by $\mathcal{A}(\mathfrak{g})=\{a\in U(\mathfrak{g})\mid ua-(-1)^{p(u)(p(a)+\bar{1})}au=0\ \mathrm{for}\ \mathrm{all}\ u\in\mathfrak{g}\}$. If $\mathfrak{g}$ is a finite-dimensional simple basic classical Lie superalgebra, it is known 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, if $\mathfrak{g}=\mathfrak{osp}_{1|2n}$, there exists $T\in U(\mathfrak{g})_{\bar{0}}$ such that $\mathcal{A}(\mathfrak{osp}_{1|2n})=Z(\mathfrak{osp}_{1|2n})T$ by [ABF97, Mus97, GL00]. The element $T$ is called the Casimir ghost [ABF97] since $T^{2}\in Z(\mathfrak{osp}_{1|2n})$. In case $\mathfrak{g}=\mathfrak{osp}_{1|2}$, [Pin90] also suggested that $T=4Q-4C+\frac{1}{2}$ satisfies $T^{2}=4C+\frac{1}{4}\in Z(\mathfrak{osp}_{1|2})$, where $C$ is the Casimir element in $U(\mathfrak{osp}_{1|2})$ and $Q$ is one in $U(\mathfrak{sl}_{2})$.

The finite $\mathcal{W}$-algebra $U(\mathfrak{g},f)$ is an associative superalgebra over $\mathbb{C}$ defined from a simple basic classical Lie superalgebra $\mathfrak{g}$ and its even nilpotent element $f$ [Pre02, Pre07, Kos78, Lyn79, dBT94, RS99, GG02]. If $\mathfrak{g}$ is a simple Lie algebra and $f$ is a principal nilpotent element $f_{\mathfrak{prin}}$, the corresponding finite $\mathcal{W}$-algebra $U(\mathfrak{g},f_{\mathrm{prin}})$ is isomorphic to the center $Z(\mathfrak{g})$ of $U(\mathfrak{g})$ by Kostant [Kos78].

The $\mathcal{W}$-algebra $\mathcal{W}^{k}(\mathfrak{g},f)$ is a vertex superalgebra defined by the Drinfeld-Sokolov reductions associated to $\mathfrak{g},f$ and level $k\in\mathbb{C}$ [FF92, KRW03]. In general, (Ramond-twisted) simple modules of a $\frac{1}{2}\mathbb{Z}$-graded vertex superalgebras $V$ with a Hamiltonian operator $H$ are classified by the associated superalgebra named as the ($H$-twisted) Zhu algebras of $V$. De Sole and Kac shows that the $H$-twisted Zhu algebra of $\mathcal{W}^{k}(\mathfrak{g},f)$ is isomorphic to the finite $\mathcal{W}$-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 simple positive-energy modules of $\mathcal{W}^{k}(\mathfrak{g},f)$. If $f=f_{\mathrm{prin}}$, the corresponding $\mathcal{W}$-algebra is called the principal $\mathcal{W}$-algebra of $\mathfrak{g}$, which we denoted by $\mathcal{W}^{k}(\mathfrak{g})=\mathcal{W}^{k}(\mathfrak{g},f_{\mathrm{prin}})$.

The finite $\mathcal{W}$-algebra $U(\mathfrak{osp}_{1|2n},f_{\mathrm{prin}})$ associated to $\mathfrak{osp}_{1|2n}$ and its principal nilpotent element $f_{\mathrm{prin}}$ 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. Through the isomorphism in Theorem A, a $\mathbb{Z}_{2}$-grading of $\widetilde{Z}(\mathfrak{osp}_{1|2n})$ is inherited from one of $U(\mathfrak{osp}_{1|2n},f_{\mathrm{prin}})$ so that the even part of $\widetilde{Z}(\mathfrak{osp}_{1|2n})$ is $Z(\mathfrak{osp}_{1|2n})$ and the odd part is $\mathcal{A}(\mathfrak{osp}_{1|2n})$.

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 Section 4 for the definition of $\mu$. The map $\mu$ was originally introduced in [Lyn79]. The injectivity of $\mu$ was only known for non-super cases, but has been recently proved by [Nak20] for super cases. As a corollary of Theorem A, it follows that simple positive-energy Ramond-twisted modules of principal $\mathcal{W}$-algebras $\mathcal{W}^{k}(\mathfrak{osp}_{1|2n})$ are classified by simple modules of the ghost center of $\mathfrak{osp}_{1|2n}$. See also Corollary 6.7. We remark that our definitions of $U(\mathfrak{osp}_{1|2n},f_{\mathrm{prin}})$ is different from those in some literatures [Pol13, PS13, ZS15]. See Remark 4.4.

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

Acknowledgments The author wishes to thank Thomas Creutzig, Tomoyuki Arakawa, Hiroshi Yamauchi and Maria Gorelik for valuable comments and suggestions. Some part of this work was done while he was visiting Instituto de Matemática Pura e Aplicada, Brazil in March and April 2022 and the Centre de Recherches Mathḿatiques, Université de Montréal, Canada in October 2022. He is grateful to those institutes for their hospitality. He is supported by World Premier International Research Center Initiative (WPI), MEXT, Japan and JSPS KAKENHI Grant Number JP21K20317.

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

$\displaystyle[A_{\lambda}B]=\sum_{n=0}^{\infty}\frac{\lambda^{n}}{n!}A_{(n)}B\in\mathbb{C}[\lambda]\otimes V$

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 $[H,Y(A,z)]=z\partial_{z}Y(A,z)+Y(H(A),z)$ for all $A\in V$. The eigenvalue of $H$ is called the conformal weight. If $V$ is conformal and $L(z)=\sum_{n\in\mathbb{Z}}L_{n}z^{-n-2}$ 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 $\circ$-product of $V$ by

$\displaystyle A*B=\sum_{j=0}^{\infty}\binom{\Delta_{A}}{j}A_{(j-1)}B,\quad A\circ B=\sum_{j=0}^{\infty}\binom{\Delta_{A}}{j}A_{(j-2)}B,\quad A,B\in V.$

Then the quotient space

$\displaystyle\operatorname{Zhu}_{H}V=V/V\circ V$

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

$\displaystyle Y_{M}\colon M\ni A\rightarrow Y_{M}(A,z)=\sum_{n\in\mathbb{Z}+\Delta_{A}}A^{M}_{(n)}z^{-n-1}\in(\operatorname{End}M)[\![z^{\frac{1}{2}},z^{-\frac{1}{2}}]\!]$

such that (1) for each $C\in M$, $A^{M}_{(n)}C=0$ if $n\gg 0$, (2) $Y_{M}(|0\rangle,z)=\operatorname{id}_{M}$ and (3) for any $A,B\in V$, $C\in M$, $n\in\mathbb{Z}$, $m\in\mathbb{Z}+\Delta_{A}$ and $\ell\in\mathbb{Z}+\Delta_{B}$,

	$\displaystyle\sum_{j=0}^{\infty}(-1)^{j}\binom{n}{j}\left(A^{M}_{(m+n-j)}B^{M}_{(\ell+j)}-(-1)^{p(A)p(B)}B^{M}_{(\ell+n-j)}A^{M}_{(m+j)}\right)C$
	$\displaystyle=\sum_{j=0}^{\infty}\binom{m}{j}\left(A_{(n+j)}B\right)^{M}_{(m+\ell-j)}C.$

Hence the Ramond-twisted module is a twisted module of $V$ for the automorphism $\mathrm{e}^{2\pi iH}$. 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 $M_{0}\neq 0$ 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 [DSK06, Lemma 2.22], a linear map $V_{\Gamma}\ni A\mapsto A^{M}_{0}|_{M_{0}}\in\operatorname{End}M_{0}$ induces a homomorphism $\operatorname{Zhu}_{H}V\rightarrow\operatorname{End}M_{0}$. 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 $\operatorname{Zhu}_{H}V$-modules. By [DSK06, 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 $\operatorname{Zhu}_{H}V$-modules.

Let $\mathfrak{g}$ be a finite-dimensional simple Lie superalgebra with the normalized even supersymmetric invariant bilinear form $(\cdot|\cdot)$ 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 [KRW03] for the definitions of good gradings and [EK05, Hoy12] 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:

1. (1)

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

2. (2)

   $f\in\mathfrak{g}_{-1}$,

3. (3)

   $\operatorname{ad}(f)\colon\mathfrak{g}_{j}\rightarrow\mathfrak{g}_{j-1}$ is injective for $j\geq\frac{1}{2}$ and surjective for $j\leq\frac{1}{2}$,

4. (4)

   $(\mathfrak{g}_{i}|\mathfrak{g}_{j})=0$ if $i+j\neq 0$,

5. (5)

   $\dim\mathfrak{g}^{f}=\dim\mathfrak{g}_{0}+\dim\mathfrak{g}_{\frac{1}{2}}$, 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 $\mathfrak{g}_{\geq 0}$. Denote by $\Delta_{j}=\{\alpha\in\Delta\mid\mathfrak{g}_{\alpha}\subset\mathfrak{g}_{j}\}$ and $\Pi_{j}=\Pi\cap\Delta_{j}$ for $j\in\frac{1}{2}\mathbb{Z}$. We have $\Pi=\Pi_{0}\sqcup\Pi_{\frac{1}{2}}\sqcup\Pi_{1}$. Let $\chi\colon\mathfrak{g}\rightarrow\mathbb{C}$ be a linear map defined by $\chi(u)=(f|u)$. Since $\operatorname{ad}(f)\colon\mathfrak{g}_{\frac{1}{2}}\rightarrow\mathfrak{g}_{-\frac{1}{2}}$ is an isomorphism of vector spaces, the super skew-symmetric bilinear form $\mathfrak{g}_{\frac{1}{2}}\times\mathfrak{g}_{\frac{1}{2}}\ni(u,v)\mapsto\chi([u,v])\in\mathbb{C}$ 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

$\displaystyle[u_{\lambda}v]=[u,v]+k(u|v)\lambda,\quad u,v\in\mathfrak{g}.$

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

$\displaystyle[{\phi_{\alpha}}_{\lambda}\phi_{\beta}]=\chi(u_{\alpha},u_{\beta}),\quad\alpha,\beta\in\Delta_{\frac{1}{2}}.$

Let $F^{\mathrm{ch}}(\mathfrak{g}_{>0})$ 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

$\displaystyle[{\varphi_{\alpha}}_{\lambda}\varphi^{*}_{\beta}]=\delta_{\alpha,\beta},\quad[{\varphi_{\alpha}}_{\lambda}\varphi_{\beta}]=[{\varphi^{*}_{\alpha}}_{\lambda}\varphi^{*}_{\beta}]=0,\quad\alpha,\beta\in\Delta_{>0}.$

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

	$\displaystyle d=$	$\displaystyle\sum_{\alpha\in\Delta_{>0}}(-1)^{p(\alpha)}u_{\alpha}\varphi^{*}_{\alpha}-\frac{1}{2}\sum_{\alpha,\beta,\gamma\in\Delta_{>0}}(-1)^{p(\alpha)p(\gamma)}c_{\alpha,\beta}^{\gamma}:\!\varphi_{\gamma}\varphi^{*}_{\alpha}\varphi^{*}_{\beta}\!:$
		$\displaystyle+\sum_{\alpha\in\Delta_{\frac{1}{2}}}\phi_{\alpha}\varphi^{*}_{\alpha}+\sum_{\alpha\in\Delta_{>0}}\chi(u_{\alpha})\varphi^{*}_{\alpha}.$

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

$\displaystyle\mathcal{W}^{k}(\mathfrak{g},f)=H(C^{k}(\mathfrak{g},f),d_{(0)}).$

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

$\displaystyle J^{u}(z)=u(z)+\sum_{\alpha,\beta\in\Delta_{>0}}c_{\beta,u}^{\alpha}:\!\varphi^{*}_{\beta}(z)\varphi_{\alpha}(z)\!:,\quad u\in\mathfrak{g}_{\leq 0}.$

Then we have [KW04]

$\displaystyle\mathcal{W}^{k}(\mathfrak{g},f)=H(C^{k}(\mathfrak{g},f),d_{(0)})=H^{0}(C^{k}(\mathfrak{g},f)_{+},d_{(0)}).$

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

	$\displaystyle[{J^{u}}_{\lambda}J^{v}]=J^{[u,v]}+\tau(u|v)\lambda,\quad u,v\in\mathfrak{g}_{\leq 0}$
	$\displaystyle\tau(u|v)=k(u|v)+\frac{1}{2}\kappa_{\mathfrak{g}}(u|v)-\frac{1}{2}\kappa_{\mathfrak{g}_{0}}(u|v),\quad u,v\in\mathfrak{g}_{\leq 0},$

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

$\displaystyle\Upsilon\colon\mathcal{W}^{k}(\mathfrak{g},f)\rightarrow V^{\tau}(\mathfrak{g}_{0})\otimes F(\mathfrak{g}_{\frac{1}{2}})$

by the restriction. The map $\Upsilon$ is called the Miura map and injective thanks to [Fre05, Ara17, Nak20].

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

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

$\displaystyle[\Phi_{\alpha},\Phi_{\beta}]=\chi([u_{\alpha},u_{\beta}]),\quad\alpha,\beta\in\Delta_{\frac{1}{2}}.$

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 $\alpha\in\Delta_{\geq 1}$. 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

$\displaystyle[\psi_{\alpha},\psi^{*}_{\beta}]=\delta_{\alpha,\beta},\quad[\psi_{\alpha},\psi_{\beta}]=[\psi^{*}_{\alpha},\psi^{*}_{\beta}]=0,\quad\alpha,\beta\in\Delta_{>0}.$

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

	$\displaystyle C_{I}=U(\mathfrak{g})\otimes\Phi\otimes\Lambda(\mathfrak{g}_{>0}),\quad d_{I}=\operatorname{ad}(Q),$
	$\displaystyle Q=\sum_{\alpha\in\Delta_{>0}}(-1)^{p(\alpha)}X_{\alpha}\psi_{\alpha}-\frac{1}{2}\sum_{\alpha,\beta,\gamma\in\Delta_{>0}}(-1)^{p(\alpha)p(\gamma)}c_{\alpha,\beta}^{\gamma}\psi_{\gamma}\psi^{*}_{\alpha}\ \psi^{*}_{\beta},$
	$\displaystyle X_{\alpha}=u_{\alpha}+(-1)^{p(\alpha)}(\Phi_{\alpha}+\chi(u_{\alpha})),\quad\alpha\in\Delta_{>0},$

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 $(C_{I},d_{I})$ forms a cochain complex with respect to the charged degree on $\Lambda(\mathfrak{g}_{>0})$ and the cohomology

$\displaystyle U(\mathfrak{g},f)_{I}=H^{\bullet}(C_{I},d_{I})$

(4.1)

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

$\displaystyle j^{u}=u+\sum_{\alpha,\beta\in\Delta_{>0}}c_{\beta,u}^{\alpha}\psi^{*}_{\beta}\ \psi_{\alpha},\quad u\in\mathfrak{g}.$

Then

$\displaystyle\operatorname{ad}(Q)\cdot\psi_{\alpha}=j^{u_{\alpha}}+(-1)^{p(\alpha)}(\Phi_{\alpha}+\chi(u_{\alpha}))=X_{\alpha}+\sum_{\alpha,\beta\Delta_{>0}}c_{\beta,u}^{\alpha}\psi^{*}_{\beta}\ \psi_{\alpha},\quad\alpha\in\Delta_{>0}.$

Let $C_{-}$ be the subalgebra of $C_{I}$ generated by $\psi_{\alpha}$, $\operatorname{ad}(Q)\cdot\psi_{\alpha}$ $(\alpha\in\Delta_{>0})$ and $C_{+}$ be the subalgebra of $C_{I}$ generated by $j^{u}$ $(u\in\mathfrak{g}_{\leq 0})$, $\Phi_{\alpha}$ $(\alpha\in\Delta_{\frac{1}{2}})$ and $\psi^{*}_{\alpha}$ $(\alpha\in\Delta_{>0})$. Then $(C_{\pm},d_{I})$ form subcomplexes and $C_{I}\simeq C_{-}\otimes C_{+}$ as vector superspaces. Since $H(C_{-},d_{I})=\mathbb{C}$, we have

$\displaystyle H(C_{I},d_{I})\simeq\ H(C_{-},d_{I})\otimes H(C_{+},d_{I})=H(C_{+},d_{I}).$

Using the same argument as in [KW04], it follows that $H^{n}(C_{+},d_{I})=0$ for $n\neq 0$. Therefore $U(\mathfrak{g},f)_{I}$ is a subalgebra of $C^{0}_{+}$, which is generated by $j^{u}$ $(u\in\mathfrak{g}_{\leq 0})$ and $\Phi_{\alpha}$ $(\alpha\in\Delta_{\frac{1}{2}})$. Since $[j^{u},j^{v}]=j^{[u,v]}$ for $u,v\in\mathfrak{g}_{\leq 0}$, there exists an isomorphism $C^{0}_{+}\simeq U(\mathfrak{g}_{\leq 0})\otimes\Phi$ as associative $\mathbb{C}$-superalgebras. The projection $\mathfrak{g}_{\leq 0}\twoheadrightarrow\mathfrak{g}_{0}$ induces an associative $\mathbb{C}$-superalgebra surjective homomorphism $U(\mathfrak{g}_{\leq 0})\otimes\Phi\twoheadrightarrow U(\mathfrak{g}_{0})\otimes\Phi$ so that we have

$\displaystyle\mu\colon U(\mathfrak{g},f)_{I}\rightarrow U(\mathfrak{g}_{0})\otimes\Phi$

by the restriction. The map $\mu$ is called the Miura map for the finite $\mathcal{W}$-algebras and injective by [Lyn79, Gen20, Nak20]. Let $\mathbb{C}_{-\chi}$ be the one-dimensional $\mathfrak{g}_{\geq 1}$-module defined by $\mathfrak{g}_{\geq 1}\ni u\mapsto-\chi(u)\in\mathbb{C}$ and $M_{II}$ be the induced left $\mathfrak{g}$-module

$\displaystyle M_{II}=\operatorname{Ind}^{\mathfrak{g}}_{\mathfrak{g}_{\geq 1}}\mathbb{C}_{-\chi}=U(\mathfrak{g})\underset{U(\mathfrak{g}_{\geq 1})}{\otimes}\mathbb{C}_{-\chi}\simeq U(\mathfrak{g})/I_{-\chi},$

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

$\displaystyle U(\mathfrak{g},f)_{II}=(M_{II})^{\operatorname{ad}(\mathfrak{g}_{>0})}.$

(4.2)

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