2.1 Queer superalgebra

For a positive integer n, we set $I_{n|n}:=\{-1,\ldots ,-n,1,2, \ldots ,n\}$ , on which we define the parity of $i \in I_{n|n}$ to be $|i|=0$ if $i>0$ and $|i|=1$ if $i<0$ .

Let $V:=\mathbb C^n \oplus \mathbb C^n$ be the superspace with standard basis $v_i$ of parity i for $i \in I_{n|n}$ . Its endomorphism ring $\operatorname {End}_{\mathbb C}(V)$ is an associative superalgebra with standard basis $E_{ij}$ of parity $|i|+|j|$ for $i,j \in I_{n|n}$ . It is also a Lie superalgebra under the standard supercommutator

$$ \begin{align*}[x,y]=xy-(-1)^{|x||y|}yx,\end{align*} $$

for all homogeneous $x,y \in \operatorname {End}_{\mathbb C}(V)$ . It is denoted by $\mathfrak{gl}(n|n)$ .

The queer superalgebra $\operatorname {\mathfrak{q}}(n)$ can be defined in two ways. The map $E_{ij} \mapsto E_{-i,-j}$ is an involutive automorphism of $\mathfrak{gl}(n|n)$ and we define $\operatorname {\mathfrak{q}}(n)$ to be the fixed point sub-superalgebra of this automorphism.

Alternatively, we define $P:V\to V$ by $Pv_i=(-1)^{|v_i|}v_{-i}$ and $\mathfrak{q}(n):=\{f \in \operatorname {End}_{\mathbb C}(V): [f,P]=0\}$ . It is closed under the superbracket and we call it the queer superalgebra.

With respect to the ordered basis $B:=B_1 \cup B_0=\{e_{-1},\dots e_{-n},e_1,\dots ,e_n\}$ , the operator P can be written in matrix form as

$$ \begin{align*}P=\begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix}\end{align*} $$

and $\mathfrak{q}(n)$ can be expressed in the matrix form as

$$ \begin{align*}\mathfrak{q}(n)=\left\{\begin{pmatrix} A & B \\ B & A \end{pmatrix}: A, B \,\, \text{are arbitrary } n \times n \text{ matrices}\right\}.\end{align*} $$

In terms of basis elements of $\mathfrak{gl}(n|n)$ we have $P=E_{-n,n}+E_{-(n-1),n-1}+\cdots +E_{-1,1}-E_{1,-1}-\cdots -E_{n,-n}$ . Note that $\mathfrak{q}(n)$ as the Lie sub-superalgebra of $\mathfrak{gl}(n|n)$ is spanned by $F_{ij}:=E_{ij}+E_{-i,-j}$ for $i>0$ . Note that we have $F_{ij}=F_{-i,-j}=F_{-j,-i}$ for all $i,j$ .

The superalgebra $\mathfrak{gl}(n|n)$ and hence $\mathfrak{q}(n)$ acts on V by matrix multiplication and on the k-fold tensor product $V^{\otimes k}$ by

(2.1) $$ \begin{align} x\cdot(v_1 \otimes v_2 \otimes \cdots \otimes v_k)=\sum_{j=1}^k(-1)^{(|v_1|+\cdots +|v_{j-1}|)|v|}v_1 \otimes v_2 \otimes \cdots \otimes xv_j \otimes v_{j+1} \otimes \cdots \otimes v_k, \end{align} $$

where the elements $x \in \mathfrak{q}(n)$ and $v_i \in V$ are all homogeneous. We then extend to all of $\mathfrak{q}(n)$ acting on all of $V^{\otimes k}$ by linearity.

2.2 The Sergeev superalgebras

Let $S_k$ be the symmetric group on k letters. It is generated by the transpositions $s_1,\dots ,s_{k-1}$ , where $s_i=(i,i+1)$ for all i.

The Sergeev superalgebra $Ser_k$ is the associative superalgebra generated by $s_1,\dots ,s_{k-1}$ and $c_1,\dots ,c_{k-1},c_k$ with the following defining relations:

• • $s_i^2=1,\ s_is_{i+1}s_i=s_{i+1}s_is_{i+1},\ s_is_j=s_js_i \quad (|i-j|>1),$

• • $c_i^2=-1,\ c_ic_j=-c_jc_i,\quad (i \neq j)$

• • $s_ic_is_i=c_{i+1},\ s_ic_j=c_js_i,\quad j \neq i,i+1$ .

The generators $s_1,\dots ,s_{k-1}$ are regarded as even and the subalgebra generated by them is isomorphic to the group algebra $\mathbb C[S_k]$ of $S_k$ ; the generators $c_1,\dots ,c_{k-1},c_k$ are called odd and the $\mathbb C$ -subalgebra generated by them is isomorphic to the Clifford superalgebra $\mathbf {C}l_k$ .

The Sergeev superalgebra $Ser_k$ is isomorphic to the superalgebra $\mathbb C[S_k] \ltimes \mathbf {C}l_k$ , which is $\mathbb C[S_k] \otimes \mathbf {C}l_k$ as a superspace with the multiplication given by

$$ \begin{align*}(\sigma \otimes c_{i_1}\ldots c_{i_l})(\tau \otimes c_{j_1}\ldots c_{j_m})=\sigma \tau \otimes c_{\tau^{-1}(i_1)}\ldots c_{\tau^{-1}(i_l)} c_{j_1}\ldots c_{j_m},\end{align*} $$

where $1 \leq i_s, j_t \leq k$ .

The superalgebra $Ser_k$ has as basis the elements which can be written in the form $\sigma \otimes c_1^{\epsilon _1}\ldots c_k^{\epsilon _k}$ where $\sigma \in S_k$ and $\epsilon _i \in \mathbb Z_2$ for all i.

2.3 Action of the Sergeev algebra and Schur–Weyl–Sergeev duality

The symmetric group $S_k$ acts on the k-fold tensor product $V^{\otimes k}$ by

(2.2) $$ \begin{align} s_i\kern1.5pt{\cdot}\kern1.5pt (v_1 \kern1.5pt{\otimes}\kern1.5pt v_2 \otimes \cdots \otimes v_i \otimes v_{i+1}\otimes \cdots \otimes v_k)\kern1.5pt{=}\kern1.5pt(-1)^{|v_i||v_{i+1}|}v_1 \otimes v_2 \otimes \cdots \otimes v_{i+1} \otimes v_i\otimes \cdots \otimes v_k, \end{align} $$

where $v_i$ ’s are $\mathbb Z_2$ -homogeneous. We then extend the action to $V^{\otimes k}$ by linearity.

More generally the action of $S_k$ on $V^{\otimes k}$ is defined as follows: for $\sigma \in S_k$ and homogeneous $\pmb {v}=v_1 \otimes v_2 \otimes \cdots \otimes v_k$ , the action of $\sigma $ on $\pmb {v}$ is given by:

$$ \begin{align*} \sigma\cdot({v_1}\otimes\dots\otimes {v_k})= (-1)^{\gamma(\pmb{v},\sigma^{-1})}({v_{\sigma^{-1}(1)}}\otimes\dots\otimes {v_{\sigma^{-1}(k)}}), \end{align*} $$

where $\gamma (\pmb {v},\sigma ^{-1})=\prod _{(i,j)\in \text {Inv}(\sigma )}{|v_i||v_j|}$ , with $\text {Inv}(\sigma )=\{(i,j)\colon i<j\text { and }\sigma (i)>\sigma (j)\}$ . It can be easily verified that $\gamma (\pmb {v},\sigma \tau )=\gamma (\sigma ^{-1}\pmb {v},\tau )+\gamma (\pmb {v},\sigma )$ for two permutations $\sigma $ and $\tau $ .

The generators $c_j$ acts on $V^{\otimes k}$ by

$$ \begin{align*}c_j\cdot(v_1 \otimes \cdots \otimes v_k)=(-1)^{|v_1|+\cdots +|v_{j-1}|}v_1 \otimes \cdots \otimes v_{j-1} \otimes P(v_{j}) \otimes v_{j+1} \otimes v_{j+2} \otimes \cdots \otimes v_k,\end{align*} $$

where $v_1, \dots ,v_k \in V$ are homogeneous elements and we then extend the action to $V^{\otimes k}$ by linearity.

More generally, for $\varepsilon _i\in \mathbb Z_2$ , $i=1,\ldots , k$ , we have

$$ \begin{align*}c_1^{\varepsilon_1}\ldots c_k^{\varepsilon_k}\cdot(v_1\otimes \cdots \otimes v_k)= (-1)^{\sum_{i>j}\varepsilon_i|v_j|}P^{\varepsilon_1}v_1\otimes\cdots\otimes P^{\varepsilon_k}v_k.\end{align*} $$

The actions of $S_k$ and $\mathbf {C}l_k$ give rise to a left action of the Sergeev superalgebra $\operatorname {\mathrm{Ser}}_k$ on $V^{\otimes k}$ . So we get an algebra homomorphism

$$ \begin{align*}\Psi_k: \operatorname{\mathrm{Ser}}_k \rightarrow \operatorname{End}_{\mathbb C} (V^{\otimes k}).\end{align*} $$

We record that for a basis element $\sigma ^{-1}\otimes c_1^{\varepsilon _1}\ldots c_k^{\varepsilon _k} \in Ser_k$ , we have

$$ \begin{align*} (\sigma^{-1}\otimes c_1^{\epsilon_1}\ldots c_k^{\epsilon_k})\cdot ({v_1}\otimes\dots\otimes {v_k})= (-1)^{\gamma(\pmb{v},\sigma)+\sum_{i>j}\epsilon_i|v_j|} {P^{\epsilon_{\sigma(1)}}v_{\sigma(1)}}\otimes\dots\otimes {P^{\epsilon_{\sigma(k)}}v_{\sigma(k)}}. \end{align*} $$

Let $\operatorname {End}_{\operatorname {\mathfrak{q}}(n)} (V^{\otimes k})$ be the centralizer algebra of the $\operatorname {\mathfrak{q}}(n)$ -action on $V^{\otimes k}$ . That is

$$ \begin{align*}\operatorname{End}_{\operatorname{\mathfrak{q}}(n)}(V^{\otimes k}):=\{f \in \operatorname{End}_{\mathbb C} (V^{\otimes k})\colon xf=fx,\quad\text{for all } x \in \operatorname{\mathfrak{q}}(n)\}.\end{align*} $$

Let $\rho : \operatorname {\mathfrak{q}}(n) \rightarrow \operatorname {End}_{\mathbb C}(V^{\otimes k})$ be the superalgebra homomorphism induced from the action of $\operatorname {\mathfrak{q}}(n)$ on $V^{\otimes k}$ . In [Reference Savage16], Sergeev proved the following double centralizer theorem along the lines of the Schur–Weyl duality for the queer superalgebra.

3.1

Let $\operatorname {\mathfrak{g}}$ be either a simple Lie superalgebra or $\mathfrak{gl}(m|n)$ , or any Lie superalgebra among $\mathfrak{q}(n)$ or $\mathfrak{p}(n)$ . Let A be an associative, commutative and finitely generated $\mathbb C$ -algebra with unity. Then $\mathfrak{g} \otimes A$ has a natural structure of Lie superalgebra:

$$\begin{align*}[x \otimes a, y \otimes b]=[xy] \otimes ab, \,\, \,\, \text{for} \,\, x,y \in \mathfrak{g}, a,b \in A.\end{align*}$$

With this the even part of $\mathfrak{g} \otimes A$ is ${(\mathfrak{g})}_0 \otimes A$ and the odd part is ${(\mathfrak{g})}_1 \otimes A$ . Let ${\mathfrak{g} = \mathfrak{n}^- \oplus \mathfrak h \oplus \mathfrak{n}^+}$ be the standard triangular decomposition of $\mathfrak{g}$ . Then $\mathfrak{g} \otimes A = (\mathfrak{n}^- \otimes A) \oplus (\mathfrak h \otimes A) \oplus (\mathfrak{n}^- \otimes A)$ is a triangular decomposition of $\mathfrak{g} \otimes A$ .

3.2

Let $T(\mathfrak{g} \otimes A)=\bigoplus _{k =0}^{\infty } T^k(\mathfrak{g} \otimes A)$ , where $T^k(\mathfrak{g} \otimes A)=(\mathfrak{g} \otimes A)^{\otimes k}$ , be the tensor algebra of $\mathfrak{g} \otimes A$ and let $U(\mathfrak{g} \otimes A)$ be the enveloping algebra of $\mathfrak{g} \otimes A$ . The universal enveloping algebra of $\mathfrak{g} \otimes A$ is a quotient of the tensor algebra $T(\mathfrak{g} \otimes A)$ modulo the two-sided ideal I generated in $T(\mathfrak{g} \otimes A)$ by all elements of the form $x \otimes y-(-1)^{|x||y|} y \otimes x-[x,y]$ , where $x,y \in \mathfrak{g} \otimes A$ . Note that $T(\mathfrak{g} \otimes A)$ has a $\mathbb Z_2$ -grading extending that on $\mathfrak{g} \otimes A$ . Since the ideal I is homogeneous, $U(\mathfrak{g} \otimes A)$ is $\mathbb Z_2$ -graded.

We have a natural surjective algebra homomorphism:

$$\begin{align*}\pi: T(\mathfrak{g} \otimes A) \rightarrow U(\mathfrak{g} \otimes A).\end{align*}$$

Let

$$\begin{align*}U(\mathfrak{g} \otimes A)^{\mathfrak{g}}=\{\pmb{y}\in U(\operatorname{\mathfrak{g}}\otimes A)\colon [x,\pmb{y}]=0,\quad\text{for all }x\in\operatorname{\mathfrak{g}}\}\end{align*}$$

be the subalgebra of $\mathfrak{g}$ -invariants. Note that it is naturally $\mathbb Z_2$ -graded.

If we take $A=\mathbb C$ , then $U(\mathfrak{g} \otimes A)^{\mathfrak{g}}$ reduces to $U(\mathfrak{g})^{\mathfrak{g}}$ . Using a PBW basis, we see that if an element of $U(\operatorname {\mathfrak{g}})$ supercommutes with all elements of $\operatorname {\mathfrak{g}}$ , it supercommutes with all elements of the PBW basis. So $U(\operatorname {\mathfrak{g}})^{\operatorname {\mathfrak{g}}}$ is the center of $U(\mathfrak{g})$ , which we denote by $Z(\mathfrak{g})$ .

Let $\mathfrak h=\mathfrak h_0+\mathfrak h_1$ be a Cartan subalgebra of $\mathfrak{g}$ and $\mathfrak{g}=\mathfrak n^- \oplus \mathfrak h \oplus \mathfrak n^+$ be a triangular decomposition of $\mathfrak{g}$ . We note that for basic Lie superalgebras $\mathfrak h=\mathfrak h_0$ , whereas for $\operatorname {\mathfrak{g}}=\operatorname {\mathfrak{q}}(n)$ and $\operatorname {\mathfrak{p}}(n)$ , $\mathfrak h_1 \neq 0$ . Let $\{x_1,x_2,\dots ,x_m\}$ , $\{h_1,h_2,\dots ,h_n\}$ , and $\{y_1,y_2,\dots ,y_p\}$ be homogeneous bases for $\mathfrak{n}^-$ , $\mathfrak{h}$ and $\mathfrak{n}^+$ respectively. Then the monomials $x_1^{r_1}\dots x_m^{r_m}h_1^{s_1}\dots h_n^{s_n}y_1^{t_1}\dots y_p^{t_p}$ , where the exponents of the even elements runs through the set of all positive integers, and the exponents of odd elements ranges over $\{0,1\}$ , form a PBW basis of $U(\operatorname {\mathfrak{g}})$ . The monomials where $r_1=\dots =r_m=t_1=\dots =t_p=0$ form a basis of $U(\mathfrak{h})$ , while the set of all remaining monomials is a basis of $(\mathfrak n^- U(\mathfrak{g}) + U(\mathfrak{g})\mathfrak n^+)$ . This yields a decomposition of $U(\mathfrak{g})$ as ${U(\mathfrak{g})=U(\mathfrak h) \oplus (\mathfrak n^- U(\mathfrak{g}) + U(\mathfrak{g})\mathfrak n^+)}$ . The Harish Chandra homomorphism $HC$ is the restriction to the center $Z(\operatorname {\mathfrak{g}})$ of the projection map $\pi \colon U(\operatorname {\mathfrak{g}})\to U(\mathfrak{h})$ with kernel $(\mathfrak n^- U(\mathfrak{g}) + U(\mathfrak{g})\mathfrak n^+)$ . With the identification of $U(\mathfrak{h} )$ to the symmetric algebra $S(\mathfrak{h} )$ , it is well known that for all basic Lie superalgebras and $\mathfrak{q}(n) $ , the image of the Harish Chandra homomorphism is contained in the commutative algebra $S(\mathfrak{h}_0)$ (see [Reference Cheng and Wang3, Sections 2.2 and 2.3]). Since the Harish Chandra homomorphism is injective, it follows that the center $Z(\operatorname {\mathfrak{g}})=U(\operatorname {\mathfrak{g}})^{\operatorname {\mathfrak{g}}}$ comprises only even elements, hence $U(\mathfrak{g} \otimes A)^{\mathfrak{g}}$ also contains only even elements. For an alternative proof of this fact, we refer to [Reference Wei, Zheng and Shu20, 3.2, Proposition]. It is known that the center of $U(\mathfrak{p}(n))=\mathbb C$ (see [Reference Gorelik7]). We give an alternate proof of this fact in Section 6. Therefore, it now follows that the center of $U(\operatorname {\mathfrak{g}}\otimes A)$ contains only even elements for all basic Lie superalgebras, $\operatorname {\mathfrak{q}}(n)$ and $\operatorname {\mathfrak{p}}(n)$ .

3.3

For any $\theta \in [\mathfrak{g} ^{\otimes k}]^{\mathfrak{g}}$ , $\theta =\sum _i x_1^i \otimes x_2^i \otimes \cdots \otimes x_k^i$ and for any $a_1, a_2, \dots ,a_k \in A$ , we define

$$\begin{align*}\theta(a_1,a_2, \dots, a_k)=\sum_i (x_1^i \otimes a_1) (x_2^i \otimes a_2) \ldots (x_k^i \otimes a_k) \in U(\mathfrak{g} \otimes A).\end{align*}$$

The element $\hat {\theta }(a_1,a_2, \dots ,a_k) := \sum _i (x_1^i \otimes a_1) \otimes (x_2^i \otimes a_2) \otimes \cdots \otimes (x_k^i \otimes a_k) \in T(\mathfrak{g} \otimes A)$ maps to $\theta (a_1,a_2, \dots ,a_k)$ under the surjective map $\pi $ and since $[x,\hat {\theta }(a_1,a_2, \ldots ,a_k)]=0$ for all $x \in \mathfrak{g}$ , we get that $[x,\theta (a_1,a_2, \ldots ,a_k)]=0$ for all $x \in \mathfrak{g}$ . Hence, we have

$$ \begin{align*}\theta(a_1,a_2, \dots ,a_k) \in U(\mathfrak{g} \otimes A)^{\mathfrak{g}}.\end{align*} $$

We then have the following proposition.

Proof The map

$$\begin{align*}\pi: T(\operatorname{\mathfrak{g}} \otimes A) \rightarrow U(\operatorname{\mathfrak{g}} \otimes A).\end{align*}$$

is surjective. By using the PBW basis for $U(\operatorname {\mathfrak{g}} \otimes A)$ , we see that $\pi $ splits. So we have a surjective degree preserving algebra homomorphism

$$\begin{align*}T(\operatorname{\mathfrak{g}} \otimes A)^{\operatorname{\mathfrak{g}}} \rightarrow U(\operatorname{\mathfrak{g}} \otimes A)^{\operatorname{\mathfrak{g}}}.\end{align*}$$

Note that $T(\operatorname {\mathfrak{g}} \otimes A)^{\operatorname {\mathfrak{g}}}$ is a graded algebra and the grading comes from the grading of $T(\operatorname {\mathfrak{g}})^{\operatorname {\mathfrak{g}}}$ . Since $\operatorname {\mathfrak{g}} \otimes \mathbb Ca_1 \cong \operatorname {\mathfrak{g}}$ as $\operatorname {\mathfrak{g}}$ -modules, $(\operatorname {\mathfrak{g}} \otimes \mathbb Ca_1) \otimes (\operatorname {\mathfrak{g}} \otimes \mathbb Ca_2) \otimes \cdots \otimes (\operatorname {\mathfrak{g}} \otimes \mathbb Ca_k)$ as a $\operatorname {\mathfrak{g}}$ -module is isomorphic to $\operatorname {\mathfrak{g}} \otimes \operatorname {\mathfrak{g}} \otimes \cdots \otimes \operatorname {\mathfrak{g}}$ . Then the spanning (resp. algebra generating) set of $T(\operatorname {\mathfrak{g}})^{\operatorname {\mathfrak{g}}}$ maps to a spanning (resp. algebra generating) set of $U(\operatorname {\mathfrak{g}} \otimes A)^{\operatorname {\mathfrak{g}}}$ .

4.1 Isomorphisms

Let V be a super vector space over $\mathbb C$ .

1. We have $V \otimes V^* \cong \operatorname {End}_{\mathbb C}(V)$ defined by $(v,\phi ) \mapsto T_{(v,\phi )}$ where $T_{(v,\phi )}(w)=\phi (w)v$ for $v,w \in V$ .

This implies that $\operatorname {End}_{\mathbb C}(V)^{\otimes k} \cong (V \otimes V^*)^{\otimes k}$ for $k \geq 1$ .

2. For two super vector spaces V and W, we have an isomorphism $V \otimes W \rightarrow W \otimes V$ defined by $v\otimes w \mapsto (-1)^{|v||w|}w \otimes v$ , where v and w are homogeneous elements of V and W, respectively.

3. Let V and W be two super vector spaces. For simple homogeneous tensors $\pmb {v}=v_1\otimes v_2\otimes \dots \otimes v_k \in V^{\otimes k}$ and $\pmb {w}=w_1\otimes w_2\otimes \dots \otimes w_k\in W^{\otimes k}$ , we define where , for $1\leq j\leq k-1$ . We extend this definition by linearity to arbitrary tensors.

We have an evaluation map $(V^*)^{\otimes k} \times V^{\otimes k} \rightarrow \mathbb C$ defined by:

$$\begin{align*}(f_1\otimes f_2\otimes \cdots \otimes f_k, v_1\otimes v_2\otimes \cdots \otimes v_k) \mapsto (-1)^{p(\pmb{f},\pmb{v})}f_1(v_1)\ldots f_k(v_k),\end{align*}$$

where $\pmb {f}=f_1\otimes f_2\otimes \cdots \otimes f_k$ , and $\pmb {v}=v_1\otimes v_2\otimes \cdots \otimes v_k$ for homogeneous elements $f_i \in V^*$ and $v_i \in V$ . This evaluation map extends to a non-degenerate bilinear form and hence we get an isomorphism between $(V^*)^{\otimes k}$ and $(V^{\otimes k})^*$ .

4. Let $\tau $ be the permutation which takes $(1,2,\dots , k,k+1, \dots ,2k)$ to $(\tau (1), \tau (2), \dots ,\tau (k),\tau (k+1), \dots ,\tau (2k))=(1,3,5,\ldots ,2k-1, 2,4,6, \dots ,2k)$ . By applying the permutation $\tau $ to $(V \otimes V^*)^{\otimes k}=(V \otimes V^*) \otimes (V \otimes V^*) \otimes \cdots \otimes (V \otimes V^*)$ we get a $\operatorname {End}_{\mathbb C}(V)$ -module isomorphism between $(V \otimes V^*)^{\otimes k}$ and $V^{\otimes k} \otimes (V^*)^{\otimes k}$ . We then have

$$\begin{align*}\operatorname{End}_{\mathbb C}(V)^{\otimes k} \cong (V \otimes V^*)^{\otimes k} \cong V^{\otimes k} \otimes V^{*\otimes k}.\end{align*}$$

Let $\pmb {v}=v_1 \otimes v_2 \otimes \cdots \otimes v_k$ and $\pmb {f}=f_1\otimes f_2\otimes \cdots \otimes f_k$ . Then the map in the reverse direction is defined by $\pmb {v} \otimes \pmb {f} \mapsto (-1)^{p(\pmb {v},\pmb {f})}(v_1 \otimes f_1) \otimes (v_2 \otimes f_2) \otimes \cdots \otimes (v_k \otimes f_k)$ .

Remark: All the isomorphisms mentioned above are in particular $\operatorname {\mathfrak{q}}(n)$ -equivariant.

4.2 Tensor FFT

The tensor version of the first fundamental theorem (FFT) of invariant theory (see [Reference Goodman and Wallach6]) for $\mathfrak{q}(n) $ describes a spanning set for $(V^{\otimes k} \otimes V^{*{\otimes k}})^{\mathfrak{q}(n) }$ which can be derived from the Shur–Weyl–Sergeev duality described above.

For $n\geq 1$ , let $V=\mathbb {C}^n\oplus \mathbb {C}^n$ . Then $\operatorname {End}_{\mathbb C}(V)=\mathfrak{gl}(n|n)$ . We take the standard basis of V as $\{e_{-1},\dots ,e_{-n},e_1,\dots ,e_n\}$ and we denote the index set for this basis by B. Put $|i|=0$ if $i>0$ and 1 otherwise. The $\mathbb {Z}_2$ -gradation of V is defined by setting $|e_i|=|i|$ . The standard basis of $\operatorname {End}_{\mathbb C}(V)$ consists of the matrix units $E_{ij}$ where $E_{ij}e_k=\delta _{jk}e_i$ . Note that $\operatorname {End}_{\mathbb C}(V)$ is also $\mathbb Z_2$ -graded by $|E_{ij}|=|i|+|j|$ .

To every basis element of the Sergeev algebra $\operatorname {\mathrm{Ser}}_k$ , we wish to associate an element of $V^{\otimes k}\otimes V^{\ast \otimes k} $ . In order to avoid cumbersome notations, we let $\pmb {c}$ denote the element $c_1^{\epsilon _1}\ldots c_k^{\epsilon _k}$ . Let $\{i_1,\dots ,i_k\}$ be a multi-subset of B of cardinality k. The basis element of $V^{\otimes k}$ which corresponds to I is . We simply write $\gamma (I,\sigma )$ for $\gamma (\pmb {e_I},\sigma )$ . For any two length-k multi-subsets of indices $I=\{i_1, \ldots ,i_k\}$ and $J=\{j_1, \ldots ,j_k\}$ of B, we define .

Now we assign the element $\sigma ^{-1}\otimes \pmb {c} \in \operatorname {\mathrm{Ser}}_k$ to the following element:

(4.1) $$ \begin{align} \xi^{(k)}_{\sigma^{-1},\pmb{c}} = \sum_{I} \varsigma(\pmb{c},I,\sigma)({P^{\epsilon_{{\sigma(1)}}}e_{i_{\sigma(1)}}}\otimes\dots\otimes {P^{\epsilon_{{\sigma(k)}}}e_{\sigma(i_k)}}) \otimes({e_{i_1}^*}\otimes\dots\otimes {e_{i_k}^*}), \end{align} $$

where $I=\{i_1,\dots ,i_k\}$ runs over all multi-subsets of B of cardinality k and

$$ \begin{align*}\varsigma(\pmb{c},I,\sigma)=(-1)^{\gamma(I,\sigma)+\sum_{s>t}\epsilon_s|e_{i_t|}+p(I,I)}.\end{align*} $$

With the notations just introduced, the first fundamental theorem for $\mathfrak{q}(n) $ is stated below.

Proof By the isomorphisms (1) and (3) in the previous subsection, we have an isomorphism $\Theta \colon V^{\otimes k}\otimes V^{\ast \otimes k} \to \operatorname {End}_{\mathbb C}(V^{\otimes k})$ .

Since the tensor product action of $\mathfrak{q}(n) $ on $V\otimes V^*$ corresponds to the adjoint action on $\mathfrak{q}(n) $ , $\Theta $ induces an isomorphism $\tilde {\Theta }$ to the subspace $\mathfrak{q}(n) $ -invariants:

(4.2) $$ \begin{align} \tilde{\Theta} \colon (V^{\otimes k}\otimes V^{\ast\otimes k})^{\mathfrak{q}(n)}\to \operatorname{End}_{\mathbb C}(V^{\otimes k})^{\mathfrak{q}(n)}= \operatorname{End}_{\mathfrak{q}(n)}(V^{\otimes k}). \end{align} $$

By Schur–Weyl–Sergeev duality (Theorem 2.1), we have $\Psi _k(\operatorname {\mathrm{Ser}}_k)=\operatorname {End}_{\mathfrak{q}(n) }(V^{\otimes k})$ . We claim that $\Theta $ maps $\xi ^{(k)}_{\sigma ^{-1},\pmb {c}}$ to $\Psi _k(\sigma ^{-1}\otimes \pmb {c})$ . To show this, we take $J=\{j_1, j_2, \ldots ,j_k\}$ and then we have

$$ \begin{align*} \Theta(\xi^{(k)}_{\sigma^{-1},\pmb{c}})({e_{j_1}}\otimes\dots\otimes {e_{j_k}}) &= \begin{aligned}[t] \sum_{I} \varsigma(\pmb{c},I,\sigma) &({P^{\epsilon_{\sigma(1)}}e_{i_{\sigma(1)}}}\otimes\dots\otimes {P^{\epsilon_{\sigma(k)}}e_{i_{\sigma(k)}}})\\ \otimes &({e_{i_1}^*}\otimes\dots\otimes {e_{i_k}^*}) ({e_{j_1}}\otimes\dots\otimes {e_{j_k}}) \end{aligned}\\ &=\sum_{I} (-1)^{p(I,J)} \varsigma(\pmb{c},I,\sigma) [e_{i_1}^*(e_{j_1})\ldots e_{i_k}^*(e_{j_k})]\\ &=(-1)^{\gamma(J,\sigma)+\sum_{s>t}\epsilon_s|e_{j_t}|} ({P^{\epsilon_{\sigma(1)}}e_{j_{\sigma(1)}}}\otimes\dots\otimes {P^{\epsilon_{\sigma(k)}}e_{j_{\sigma(k)}}})\\&=\Psi_k(\sigma^{-1}\otimes\pmb{c}). \end{align*} $$

Since the isomorphism $\Theta $ is $\mathfrak{q}(n) $ -equivariant, for all $x\in \mathfrak{q}(n) $ , it follows that

$$\begin{align*}\Theta(x\cdot\xi^{(k)}_{\sigma^{-1},\pmb{c}})=x\cdot\Theta(\xi^{(k)}_{\sigma^{-1},\pmb{c}})=[x,\Psi_k(\sigma^{-1}\otimes\pmb{c})]=0.\end{align*}$$

The last equality is valid as $\Psi _k(\sigma ^{-1}\otimes \pmb {c})\in \operatorname {End}_{\mathfrak{q}(n) }(V^{\otimes k})$ . Since $\Theta $ is injective, we conclude that $\xi ^{(k)}_{\sigma ^{-1},\pmb {c}} \in (V^{\otimes k} \otimes V^{*\otimes k})^{\mathfrak{q}(n)}$ , for all $\sigma \in S_k$ , $\pmb {c} \in \mathbf {C}l_k$ . Since $\tilde {\Theta }$ is an isomorphism and the elements $\sigma ^{-1}\otimes \pmb {c}$ span the Sergeev superalgebra $\operatorname {\mathrm{Ser}}_k$ , it follows that the set $\{\xi ^{(k)}_{\sigma ^{-1},\pmb {c}} \colon \pmb {c}\in \mathbf {C}l_k,\ \sigma \in S_k\}$ spans $(V^{\otimes k}\otimes V^{\ast \otimes k} )^{\mathfrak{q}(n) }$ .

5.1 Center of $\operatorname {\mathfrak{q}}(n)$

In this subsection, we describe a set of algebra generators of $Z(\mathfrak{q}(n) )=U(\operatorname {\mathfrak{q}}(n))^{\operatorname {\mathfrak{q}}(n)}$ . In what follows we denote the k-cycle $(12\ldots k)$ by $\sigma _k$ and $|e_i|$ by just $|i|$ .

Theorem 5.3 The centre $Z(\mathfrak{q}(n) )$ is generated as an algebra by the following elements:

where k runs over the set of all odd positive integers and I over all multi-subsets of B of cardinality k.

Proof Choose an arbitrary basis element $\pmb {c}=c_1^{\epsilon _1}\ldots c_k^{\epsilon _k}$ of $\mathbf {C}l_k$ . Then formula (5.3) shows that the invariant associated with $\sigma _k^{-1}\otimes \pmb {c}$ in $(\operatorname {\mathfrak{q}}(n)^{\otimes k})^{\operatorname {\mathfrak{q}}(n)}$ is given by

(5.5) $$ \begin{align} \eta^{(k)}_{\sigma_k^{-1},\pmb{c}}=\sum_{I} \nu(\pmb{c},I,\sigma_k) F_{(-1)^{\epsilon_1}i_1,i_2}\otimes\dots\otimes F_{(-1)^{\epsilon_k}i_k,i_1} \end{align} $$

We begin by calculating explicitly the element in Equation (5.6). In this case, the sign $\nu (\pmb {c},I,\sigma _k)$ is given by

(5.6) $$ \begin{align} \nu(\pmb{c},I,\sigma_k)= (-1)^{\gamma(I,\sigma_k)+p(I,I)+p(J_I^{\sigma\otimes\pmb{c}},J_I)+\sum_t \epsilon_t |e_{i_t}|+\sum_{s>t}\epsilon_s|e_{i_t|}}. \end{align} $$

Since $\text {Inv}(\sigma _k)=\{(1,k),(2,k),\dots ,(k-1,k)\}$ , it follows that $ \gamma (I,\sigma _k) = |i_1|(|i_2|+\cdots |i_k|).$

By definition

$$ \begin{align*}p(I,I)=|i_1|(|i_2|+\cdots |i_k|)+|i_2|(|i_3|+\cdots |i_k|)+\cdots +|i_{k-1}||i_k|.\end{align*} $$

We also have

$$ \begin{align*}\sum_{s>t}\epsilon_s|e_{i_t|}+\sum_t\epsilon_t|e_{i_t}|=|i_1|(\epsilon_1+\dots+\epsilon_k)+|i_2|(\epsilon_2+\dots+\epsilon_k)+\dots+|i_k|\epsilon_k.\end{align*} $$

Recall that ${J_I}^{\sigma \otimes \pmb {c}}=((-1)^{\epsilon _{\sigma (k)}}i_{\sigma (k)},(-1)^{\epsilon _{\sigma (1)}}i_{\sigma (1)},\dots ,(-1)^{\epsilon _{\sigma (k-1)}}i_{\sigma (k-1)})$ for $\sigma \in S_k$ where $J_I=(i_2,\dots ,i_k,i_1)$ . So for $\sigma =\sigma _k$ , we get that

$$ \begin{align*} p({J_I}^{\sigma_k\otimes\pmb{c}},J_I) = {|i_2|}(|{(-1)^{\epsilon_2}i_2}|+\dots+|{(-1)^{\epsilon_k}i_k}|) +{|i_3|}(|{(-1)^{\epsilon_3}i_3}|+\dots+|{(-1)^{\epsilon_k}i_k}|) \\ +\dots+ |i_k||(-1)^{\epsilon_k}i_k|. \end{align*} $$

Since $|(-1)^\epsilon i|=\epsilon +|i|$ and $|i|^2=|i|$ , we get that

(5.7) $$ \begin{align} p({J_I}^{\sigma_k\otimes\pmb{c}},J_I)\kern1.5pt{=}\kern1.5pt (|i_2|+|i_3|+\cdots +|i_k|)\kern1.5pt{+}\kern1.5pt|i_2|(\epsilon_2+\dots+\epsilon_k)+\dots+ & |i_{k-1}|(\epsilon_{k-1}\kern1.5pt{+}\kern1.5pt\epsilon_k)\kern1.5pt{+}\kern1.5pt|i_k|\epsilon_k \\ & +p(I,I)-\gamma(I,\sigma_k).\nonumber \end{align} $$

$$ \begin{align*}=(|i_2|+|i_3|+\cdots +|i_k|)+\sum_{s>t}\epsilon_s|e_{i_t|}+\sum_t\epsilon_t|e_{i_t}|-(\epsilon_1+\dots+\epsilon_k)|i_1|+p(I,I)-\gamma(I,\sigma_k).\end{align*} $$

So $\nu (\pmb {c},I,\sigma _k)= (-1)^{(\epsilon _1+\dots +\epsilon _k)|i_1|+(|i_2|+\dots +|i_k|)}.$

In particular, if $\pmb {c}=c_{t_1}\ldots c_{t_m}$ with $0\ne t_1>\dots >t_m$ , then $\nu (\pmb {c},I,\sigma _k)= (-1)^{m|i_1|+|i_2|+\dots +|i_k|}$ . Using this information we can rewrite Equation (5.5) corresponding to $\sigma =\sigma _k$ , and $\pmb {c}=c_{t_1}\ldots c_{t_m}$ as

(5.5a) $$ \begin{align} \sum_{I}(-1)^{m|i_1|+|i_2|+\dots+|i_k|} F_{i_1,i_2}&\otimes\cdots \otimes F_{i_{{t_1}-1},i_{t_1}}\otimes F_{-i_{t_1,i_{{t_1}+1}}} \\ &\otimes\dots\otimes F_{i_{{t_m}-1},i_{t_m}}\otimes{F_{-i_{t_m,i_{{t_m}+1}}}}\otimes\dots\otimes {F_{i_k,i_1}}.\nonumber \end{align} $$

In the sum (5.5a) we first replace $i_{t_1}$ by $-i_{t_1}$ , and then using the relation $F_{i,j}=F_{-i,-j}$ , we replace the factor $F_{i_{{t_1}-1},i_{t_1}}\otimes F_{-i_{t_1,i_{{t_1}+1}}}$ in (5.5a) by $F_{-i_{{t_1}-1},i_{t_1}}\otimes F_{i_{t_1,i_{{t_1}+1}}}$ . Since for any $1\leq q\leq k$ , we have $(-1)^{|i_q|}=-(-1)^{|-i_q|}$ , this replacement will have the sole effect of multiplication by $-1$ . Performing the same operation inductively on $i_{{t_1}-1}$ and then up to on $i_2$ , we can bring the negative sign to $i_1$ while keeping the signs of all intermediate subscripts between $i_2$ and $i_{t_1}$ positive. This implies that the sum (5.5a) is same as

(5.5b) $$ \begin{align} (-1)^{t_1-1} \sum_I (-1)^{m|i_1|+|i_2|+\dots+|i_k|} F_{-i_1,i_2}&\otimes\cdots \otimes F_{i_{{t_1}-1},i_{t_1}}\otimes F_{i_{t_1,i_{{t_1}+1}}} \\&\otimes\dots\otimes F_{i_{{t_m}-1},i_{t_m}}\otimes{F_{-i_{t_m,i_{{t_m}+1}}}}\otimes\dots\otimes {F_{i_k,i_1}}.\nonumber \end{align} $$

Now, we repeat the same procedure in the sum (5.5b) with $i_{t_2}$ and then inductively up to $i_{t_m}$ to obtain that the sum (5.5a) is same as:

(5.5c) $$ \begin{align} (-1)^{t_1+\dots+t_m-m} \sum_I (-1)^{m|i_1|+|i_2|+\dots+|i_k|} F_{(-1)^mi_1,i_2}\otimes{F_{i_2,i_3}}\otimes\dots\otimes {F_{i_k,i_1}}. \end{align} $$

We denote by f the sum in (5.5c) without the sign $(-1)^{t_1+\dots +t_m-m}$ . When m is an even integer, we have $f=C_n^{(k)}$ . If m is odd we show that $f=0$ .

When m is odd, we have

$$ \begin{align*} f=\sum_I (-1)^{|i_1|+|i_2|+\dots+|i_k|} F_{-i_1,i_2}\otimes{F_{i_2,i_3}}\otimes\dots\otimes {F_{i_k,i_1}}. \end{align*} $$

By changing the signs of all the subscripts and using $F_{i,j}=F_{-i,-j}$ , we get that ${f=(-1)^k f}$ . Therefore, $f=0$ when k is odd. We have the following identity:

$$\begin{align*}|F_{-i_1,i_2}|=|-i_1|+|i_2|=1+|i_1|+|i_2|=|F_{i_2,i_3}\otimes\dots\otimes F_{i_k,i_1}|+1.\end{align*}$$

So, f can also be written as

$$\begin{align*}f=\sum_I (-1)^{|i_1|+|i_2|+\dots+|i_k|} {F_{i_2,i_3}}\otimes\dots\otimes {F_{i_k,i_1}}\otimes F_{-i_1,i_2}.\end{align*}$$

Now we replace $i_2,\dots ,i_k$ by their negatives to obtain $f=(-1)^{k-1}f$ . This implies that $f=0$ when k is even too.

To sum up, we have obtained that for any positive integer k, the element in (5.5) corresponding to $(\sigma _k^{-1}\otimes c_{t_1}\ldots c_{t_m})$ is $0$ when m is odd; and when m is even, it equals either $C_n^{(k)}$ or $-C_n^{(k)}$ . Therefore, for each $k\geq 1$ , we associate to it the element $C_n^{(k)}$ .

Since the symmetric group $S_k$ is generated by the cycles $(12)=\sigma _2$ , and $(12\ldots k)=\sigma _k$ , it follows that any element of $\cup _{k\geq 1}S_k$ can be written as a product of various $\sigma _k$ ’s where $k\in \mathbb {N}$ . The multiplication for the Sergeev algebra enables us to write $(\sigma _1\sigma _2\otimes \pmb {c})=(\sigma _1\otimes 1)\cdot (\sigma _2\otimes \pmb {c})$ , for any $\sigma _1,\sigma _2\in S_k$ . As $\Psi _k$ is an algebra homomorphism, we get that $\eta ^{(k)}_{\sigma ,\pmb {c}}=\eta ^{(k)}_{\sigma _1,\pmb {1}} \eta ^{(k)}_{\sigma _2,\pmb {c}}$ . From here we conclude that the elements $\{C_n^{k}\colon k\in \mathbb N\}$ generate $Z(\mathfrak{q}(n) )$ as an algebra.

Corollary 5.4 Let A be a finitely generated associative commutative $\mathbb C$ -algebra with identity. Then $U(\mathfrak{q}(n) \otimes A)^{\mathfrak{q}(n) }$ is generated as an algebra by the following elements:

$$ \begin{align*} C_n^{(k)}(a_1,\dots,a_k)=\sum_{I\colon|I|=k} (-1)^{|i_2|+\dots+|i_k|}{(F_{i_1,i_2}\otimes a_1)}\otimes\dots\otimes {(F_{i_k,i_1}\otimes a_k)}, \end{align*} $$

where k runs over the set of all odd positive integers, $a_1,\dots ,a_k\in A$ are arbitrary, and I runs over all multi-subsets of B of cardinality k.

Remark: Let $V(\lambda _1),\dots , V(\lambda _m)$ be finite-dimensional irreducible modules for $\operatorname {\mathfrak{q}}(n)$ . Let $A=\mathbb C[t,t^{-1}]$ be the algebra of Laurent polynomials in one variable, and $d_1,\dots ,d_m$ be m distinct nonzero complex numbers. We denote the m-tuples $(\lambda _1,\dots ,\lambda _m)$ , and $(d_1,\dots ,d_m)$ , respectively by $\pmb {\lambda }$ and $\pmb {d}$ . Then is a $\mathfrak{q}(n) \otimes A$ -module, where the action is defined by

$$ \begin{align*} & (x\otimes f(t))\cdot ({v_1}\otimes\dots\otimes {v_m})=f(d_1) ({x\cdot v_1}\otimes\dots\otimes {v_m}) +(-1)^{|x||v_1|}f(d_2)( v_1\otimes{x\cdot v_2}\otimes\dots\otimes {v_m})\\ & +\dots+(-1)^{|x|(|v_1|+\dots+|v_{m-1}|}f(d_m)({v_1}\otimes\dots\otimes {x\cdot v_m}) \,\ \text{for} \,\, x\otimes f(t) \in \mathfrak{q}(n) \otimes A \,\, \text{and} \,\, v_i \in V(\lambda_i). \end{align*} $$

Then $V(\pmb {\lambda },\pmb {d})$ with the above action is called an evaluation module. A proof similar to the proof given in [Reference Rao14, Section IV, Part C] shows that the $\mathfrak{q}(n) \otimes A$ -module $V(\pmb {\lambda },\pmb {d})$ is irreducible. Moreover, if $V(\pmb {\lambda },\pmb {d})=\bigoplus _{\mu \in \mathfrak{h} ^*}V(\pmb {\lambda },\pmb {d})_\mu $ is the weight space decomposition, where $V(\pmb {\lambda },\pmb {d})_\mu =\{v\in V(\pmb {\lambda },\pmb {d})\colon h\cdot v=\mu (h)v, \,\, \forall \,\, h \in \mathfrak{h} \}$ , then we have that

$$\begin{align*}C_n^{(k)}(p_{i_1}(t),\dots,p_{i_k}(t))\cdot V(\pmb{\lambda},\pmb{d})_\mu^+\subseteq V(\pmb{\lambda},\pmb{d})_\mu^+,\end{align*}$$

where $V(\pmb {\lambda },\pmb {d})_\mu ^+=\{v\in V(\pmb {\lambda },\pmb {d})_\mu \colon \mathfrak{q}(n) ^+ v=0\}$ , $\mathfrak{q}(n) ^+$ is the sum of all root spaces, $p_i$ is the Lagrange’s polynomial defined by $p_i(t)=\prod _{k\ne i}(t-d_k)/(d_i-d_k)$ , for $1\leq i\leq m$ , $i_1,\dots ,i_k\in \{1,\dots ,m\}$ , and $C_n^{(k)}(p_1,\dots ,p_k)=\sum _I (-1)^{|i_2|+\dots +|i_k|}F_{i_1,i_2}(p_1)\otimes \dots \otimes F_{i_k,i_1}(p_k),$ with $F_{p,q}(p_i)=F_{p,q}\otimes p_i$ . This means that the central operators $C_n^{(k)}(p_{i_1}(t),\dots ,p_{i_k}(t))$ send a highest weight vector to another highest weight vector. This is an effective method of producing new highest weight vectors from a given one. For details, we refer to [Reference Rao14, Section IV, Part D].

6.1 Periplectic Lie superalgebra

The periplectic Lie superalgebra is the Lie superalgebra preserving an odd non-degenerate symmetric or skew-symmetric bilinear form. It is thus a superanalog of the orthogonal or symplectic Lie algebra.

For $n \in \mathbb N$ , let $V=\mathbb C^n \oplus \mathbb C^n$ equipped with a non-degenerate odd symmetric bilinear form

$$ \begin{align*}\beta: V \otimes V \rightarrow \mathbb C, \beta(v,w)=\beta(w,v) \,\, \text{and} \,\, \beta(v,w)=0 \,\, \text{for} \,\, |v|=|w|.\end{align*} $$

Then $\mathfrak{p}(V)$ is the Lie sub-superalgebra of $\operatorname {End}_{\mathbb C}(V)$ preserving $\beta $ , i.e., satisfying

$$ \begin{align*}\beta(Xv,w)+(-1)^{|X||v|}\beta(v,Xw)=0.\end{align*} $$

It can also be defined as the fixed point Lie sub-superalgebra of the involution $\sigma : \mathfrak{gl}(n|n) \rightarrow \mathfrak{gl}(n|n)$ by $E_{i,j} \mapsto -(-1)^{|i||j|+|i|}E_{n+j,n+i}$ .

We note that $\mathfrak{p}(V)$ acts on V by matrix multiplication and on the r-fold tensor product $V^{\otimes r}$ by the rule given in Equation (2.1).

Since the bilinear form is nondegenerate on V, we can choose bases $v_1, v_2, \ldots ,v_n$ for $V_{0}$ and $v_{n+1},v_{n+2}, \ldots ,v_{2n}$ for $V_1$ such that $\beta (v_{n+i},v_j)=\beta (v_j, v_{n+i})=\delta _{ij}$ and $\beta (v_i,v_j)=\beta (v_{n+i},v_{n+j})=0$ for $i,j=1,2, \ldots ,n$ . The basis elements $v_i$ and $v_{n+i}$ are dual to each other with respect to the bilinear form. This form enables us to identify V and $V^*$ . An element $f\in V^*$ is identified with $v_f\in V$ such that $f(u)=\beta (u,v_f)$ for every $u\in V$ . The matrix of $\beta $ with respect to this basis is

$$ \begin{align*}\begin{pmatrix} 0 & I_n \\ I_n & 0 \end{pmatrix}\end{align*} $$

We use the notation $v_{i+n}^*:=v_i$ and $v_i^*:=v_{i+n}$ .

With respect to the above basis an element $X \in \mathfrak{p}(V)$ can be represented in matrix form as

$$ \begin{align*}X=\begin{pmatrix} A & B \\ C & -A^T \end{pmatrix},\end{align*} $$

where $A, B,C$ are $n \times n$ matrices such that $B=B^T$ and $C=-C^T$ . We identify $\mathfrak{p}(V)$ with the space $\mathfrak{p}(n)$ of matrices in the above form. The elements $E_{i,j}-E_{j+n, i+n}, E_{i,j+n}+E_{j,i+n}, E_{i+n,j}-E_{j+n,i}, \text { for } 1 \leq i,j \leq n$ form a basis for $\mathfrak{p}(n)$ .

There is a grading $\mathfrak{p}(n)=\mathfrak{p}(n)_{-1} \oplus \mathfrak{p}(n)_0\oplus \mathfrak{p}(n)_1$ , where $\mathfrak{p}(n)_0 \cong \mathfrak{gl}_n$ , $\mathfrak{p}(n)_{-1} \cong \wedge ^2((\mathbb C^n)^*)$ and $\mathfrak{p}(n)_{1} \cong Sym^2(\mathbb C^{n})$ as $\mathfrak{p}(n)_0$ -modules. It is well known that the derived superalgebra $\mathfrak{sp}(n)=\mathfrak{p}(n) \cap \mathfrak{sl}(n,n)=[\mathfrak{p}(n), \mathfrak{p}(n)]$ is simple for $n \geq 3$ . We have $\mathfrak{p}(n)=\mathbb CI' \oplus \mathfrak{sp}(n)$ , where $I'=\left (\begin {smallmatrix} I_n & 0 \\ 0 & -I_n \end {smallmatrix}\right )$ .

6.2

We note that the following element

$$\begin{align*} c=\sum_{i=1}^{2n}(-1)^{|i|}v_i\otimes v_i^*, \end{align*}$$

where $v_i^*=v_{i+n}$ for $1\leq i\leq n$ , and $v_i^*=v_{i-n}$ when $n<i\leq 2n$ , is $\mathfrak{p}(V)$ -invariant [Reference Moon9]. Then we have that

(6.1) $$ \begin{align} c^{\otimes k}=\sum_I (-1)^{|i_1|+|i_3|+\dots+|i_{2k-1}|}(v_{i_1}\otimes v_{i_2})\otimes (v_{i_3}\otimes v_{i_4})\otimes\dots\otimes (v_{i_{2k-1}}\otimes v_{i_{2k}}), \end{align} $$

where the sum runs over all $2k$ -multi-subsets $I=(i_1,i_2,\dots ,i_{2k})$ of the index set $\{1,2,\dots ,2n\}$ with the property that $|i_1-i_2|=|i_3-i_4|=\dots =|i_{2k-1}-i_{2k}|=n$ . As the action of $\operatorname {\mathfrak{p}}(V)$ on $V^{\otimes 2k}$ commutes with the action of $S_{2k}$ , all the $2k$ -tensors ${\theta _\sigma :=\sigma \circ c^{\otimes k}}$ are also $\operatorname {\mathfrak{p}}(V)$ invariants for every $\sigma \in S_{2k}$ . We have that

$$ \begin{align*} \begin{aligned} \theta_{\sigma^{-1}}=\sum_I (-1)^{\gamma(I,\sigma)+(|i_1|+|i_3|+\dots+|i_{2k-1}|)}(v_{i_{\sigma(1)}}\otimes v_{i_{\sigma(2)}}) &\otimes (v_{i_{\sigma(3)}}\otimes v_{i_{\sigma(4)}})\\ &\otimes\dots\otimes (v_{i_{\sigma(2k-1)}}\otimes v_{i_{\sigma(2k)}}), \end{aligned} \end{align*} $$

where I is as described above, and $\gamma (I,\sigma )$ is the sign resulting from the action of $\sigma ^{-1}$ on $v_I=v_{i_1}\otimes \dots \otimes v_{i_{2k}}$ . It is known that the space $(V^{\otimes 2k})^{\operatorname {\mathfrak{p}}(V)}$ is spanned by the set $\{\theta _\sigma \colon \sigma \in S_{2k}\}$ (see [Reference Deligne, Lehrer and Zhang4],[Reference Sergeev19]).

We have a $\operatorname {\mathrm{\mathfrak{p}}}(V)$ -module isomorphism $\phi _1\colon V\otimes V\cong \mathfrak{gl}(V)$ which sends $v_i\otimes v_{i+n}$ to $(-1)^{|i|}E_{i,i}$ and $v_{i+n}\otimes v_i$ to $(-1)^{|i+n|}E_{i+n,i+n}$ for $1\leq i\leq n$ . There is a surjective homomorphism $\phi _2\colon \mathfrak{gl}(V)\to \mathfrak{p}(V)$ , that sends a basis vector $E_{i,j}$ of $\mathfrak{gl}(V) $ to a certain basis vector of $\operatorname {\mathrm{\mathfrak{p}}}(V)$ by the following rule:

(6.2) $$ \begin{align} \phi_2(E_{i,j})&= \begin{cases} E_{i,j}-E_{j+n,i+n} \quad&\text{for}\quad 1\le i,j\le n\\ E_{i,j}-E_{j-n,i-n} \quad&\text{for}\quad n< i,j\le 2n\\ E_{i,j}+E_{j-n,i+n} \quad&\text{for}\quad 1\le i\le n\quad\text{and}\quad n<j\le 2n\\ E_{i,j}-E_{j+n,i-n} \quad&\text{for}\quad n< i\le 2n\quad\text{and}\quad 1\le j\le n. \end{cases} \end{align} $$

We denote the composition $\phi _2\circ \phi _1\colon V\otimes V\to \operatorname {\mathrm{\mathfrak{p}}}(V)$ by $\phi $ . Then,

(6.3) $$ \begin{align} \phi(v_i\otimes v_j)&= \begin{cases} (-1)^{|i|}E_{i,j+n}+E_{j,i+n}\quad&\text{for}\quad 1\le i,j\le n\\ (-1)^{|i|}E_{i,j-n}-E_{j,i-n}\quad&\text{for}\quad n< i,j\le 2n\\ (-1)^{|i|}E_{i,j-n}-E_{j,i+n} \quad&\text{for}\quad 1\le i\le n\quad\text{and}\quad n<j\le 2n\\ (-1)^{|i|}E_{i,j+n}-E_{j,i-n} \quad&\text{for}\quad n<i\le 2n\quad\text{and}\quad 1\le j\le n. \end{cases} \end{align} $$

The map $\phi $ induces an epimorphism $\phi ^{\otimes k}\colon (V\otimes V)^{\otimes k}=V^{\otimes 2k}\to \operatorname {\mathrm{\mathfrak{p}}}(V)^{\otimes k}$ defined by

$$ \begin{gather*} \phi^{\otimes k}((v_1\otimes v_2)\otimes\dots\otimes (v_{2k-1}\otimes v_{2k}))=\phi(v_1\otimes v_2)\otimes\dots\otimes\phi(v_{2k-1}\otimes v_{2k)}, \end{gather*} $$

which further induces a surjective map $\eta $ on the $\operatorname {\mathrm{\mathfrak{p}}}(V)$ -invariants:

$$ \begin{gather*} \eta\colon (V^{\otimes 2k})^{\operatorname{\mathrm{\mathfrak{p}}}(V)}\to (\operatorname{\mathrm{\mathfrak{p}}}(V)^{\otimes k})^{\operatorname{\mathrm{\mathfrak{p}}}(V)}. \end{gather*} $$

We now calculate the image of the element $c\in V\otimes V$ under $\phi $ . Using formula (6.3), we find that

$$ \begin{align*} \phi(c)&= \phi\Big(\sum_{i=1}^n(-1)^{|i|}v_i\otimes v_{i+n}\Big)+\phi\Big(\sum_{j=1}^n (-1)^{|j+n|}v_{j+n}\otimes v_j\Big)\\ &=\sum_{i=1}^n(E_{i,i}-E_{n+i,n+i})+\sum_{j=1}^n(E_{n+j,n+j}-E_{j,j})=0. \end{align*} $$

Therefore, $\eta (c^{\otimes k})=\phi ^{\otimes k}(c^{\otimes k})=0$ . Since the action of $\operatorname {\mathrm{\mathfrak{p}}}(V)$ is compatible with the action of the symmetric group $S_{2k}$ on $V^{\otimes 2k}$ we conclude that $\eta (\theta _{\sigma ^{-1}})=0$ for every $\sigma \in S_{2k}$ . This means $(\operatorname {\mathrm{\mathfrak{p}}}(V)^{\otimes k})^{\operatorname {\mathrm{\mathfrak{p}}}(V)}=0$ , for every positive integer k, and consequently, $T(\operatorname {\mathrm{\mathfrak{p}}}(V))^{\operatorname {\mathrm{\mathfrak{p}}}(V)}=\mathbb C$ . Thus by Proposition 3.1, we obtain the following.