1 Introduction↩︎

The Yangian \({\rm Y}(\mathfrak{gl}_{m|n})\) for the general linear Lie superalgebra \(\mathfrak{gl}_{m|n}\) was introduced by Nazarov [1] via an \(R\)-matrix presentation. It can be viewed as a deformation of the universal enveloping algebra \({\rm U}(\mathfrak{gl}_{m|n}[t])\). Since its introduction, its structure was extensively studied, in particular, due to its close connections with various areas of mathematical physics, such as Calogero-Sutherland systems [2], [3], non-linear super-Schrödinger equation [4] and superstrings on \(\text{AdS}_5\times \text{S}^5\) [5]. The super Yangian possesses two distinct families of central elements, established in [1], which consist of coefficients of certain formal power series, quantum Berezinian \(b(u)\) and quantum contraction \(z(u)\). These series can be regarded as super analogues of the quantum determinant \(\mathop{\rm qdet} T(u)\) for the ordinary Yangian \({\rm Y}(\mathfrak{gl}_{N})\) and the series \(\mathop{\rm qdet} T(u-1)/ \mathop{\rm qdet} T(u)\), respectively; see, e.g., [6]. As with their even counterparts, the coefficients of \(b(u)\) and \(z(u)\) generate the entire centre of \({\rm Y}(\mathfrak{gl}_{m|n})\), which was conjectured by Nazarov [1] and proved by Gow [7].

In this paper, we consider the double Yangian \({\rm DY}(\mathfrak{gl}_{m|n})\) for the general linear Lie superalgebra \(\mathfrak{gl}_{m|n}\) given by an \(R\)-matrix presentation, which is based on the definition of Zhang [8]. Our main goal is to establish the Poincaré–Birkhoff–Witt Theorem for \({\rm DY}(\mathfrak{gl}_{m|n})\). In the even case, the Poincaré–Birkhoff–Witt Theorem for \({\rm DY}(\mathfrak{gl}_N)\) goes back to the papers by Jing, Molev, Yang and the second author [9], Nazarov [10] and also the paper by Wendlandt [11], where it was proved for the double Yangian of an arbitrary finite-dimensional or simply laced affine Kac–Moody Lie algebra. The proof of this theorem in the super case, which we give in Section 2, relies, in particular, on the ideas of Etingof and Kazhdan [12], [13] and Nazarov [10], [14]. Furthermore, it employs certain preliminary results on the dual Yangian \({\rm Y}^+(\mathfrak{gl}_{m|n})\) and the level 0 double Yangian \({\rm DY}_0(\mathfrak{gl}_{m|n})\), so we study these algebras before proceeding towards \({\rm DY}(\mathfrak{gl}_{m|n})\).

Our second goal in Section 2 is to investigate possible generalizations of the quantum contraction \(z(u)\) for the Yangian \({\rm Y}(\mathfrak{gl}_{m|n})\) to the double Yangian \({\rm DY}(\mathfrak{gl}_{m|n})\). This research direction is partially motivated by the fact that the series \(z(u)\) has not been widely studied in the literature, in contrast with the quantum Berezinian \(b(u)\); see, e.g., the papers [15]–[18]. We introduce the dual Yangian analogue \(z^+(u)\) of the quantum contraction and we show that, as with \(z(u)\), its coefficients are algebraically independent elements of the centre of the suitably completed algebra \({\rm DY}(\mathfrak{gl}_{m|n})\).

In Section 3, we consider the so-called reflection algebras. The algebras associated with the reflection equation were originally introduced by Sklyanin [19] to describe integrable systems with the boundary conditions; see also [20]–[23] for more information on such algebras and their applications. A distinct class of reflection algebras, which are left coideal subalgebras in the Yangian \({\rm Y}(\mathfrak{gl}_{N})\), was studied by Molev and Ragoucy [24]. Moreover, its connection with Etingof–Kazhdan’s quantum affine vertex algebras [13] was investigated by the second author [25]. In this section, generalizing the approach from [24], [25] to the super case, we introduce double reflection algebras \({\rm DB} (\mathfrak{gl}_{m|n})\), which are left coideal subalgebras of (a suitable completion of) \({\rm DY}_0(\mathfrak{gl}_{m|n})\). As an application of the Poincaré–Birkhoff–Witt Theorem for \({\rm DY}(\mathfrak{gl}_{m|n})\), which implies that there exists an isomorphism of \(\mathbb{Z}_2\)-graded algebras \[\textstyle\mathop{\mathrm{gr}}_2 {\rm DY}(\mathfrak{gl}_{m|n})\cong {\rm U}(\widehat{\mathfrak{gl}}_{m|n}),\] we obtain the isomorphism \[\textstyle\mathop{\mathrm{gr}}_2 {\rm DB} (\mathfrak{gl}_{m|n})\cong {\rm U}(\mathfrak{gl}_{m|n}[t,t^{-1}]^{\sigma} ).\] Here \(\mathop{\mathrm{gr}}_2 A\) stands for the corresponding graded algebra of the algebra \(A\) with respect to a certain degree operator and \(\mathfrak{gl}_{m|n}[t,t^{-1}]^{\sigma}\) is a subalgebra of the loop Lie superalgebra \(\mathfrak{gl}_{m|n}[t,t^{-1}]\) which depends on the choice of involutive automorphism \(\sigma\) of \(\mathfrak{gl}_{m|n}\). The main result of Section 3 is an explicit presentation of the subalgebra \({\rm DB} (\mathfrak{gl}_{m|n})\subset {\rm DY}_0(\mathfrak{gl}_{m|n})\). More specifically, we show that it can be defined as an algebra in given generators subject to the family of three reflection relations and two unitarity constraints.

2 Double Yangian for the Lie superalgebra \(\mathfrak{gl}_{m|n}\)↩︎

In this section, we recall some properties of the Yangian \({\rm Y}(\mathfrak{gl}_{m|n})\). Next, we study the dual Yangian \({\rm Y}^+(\mathfrak{gl}_{m|n})\), the level 0 double Yangian \({\rm DY}_0(\mathfrak{gl}_{m|n})\) and the (centrally extended) double Yangian \({\rm DY}(\mathfrak{gl}_{m|n})\) with \(m\neq n\) and establish our main result, the Poincaré–Birkhoff–Witt Theorem for \({\rm DY}(\mathfrak{gl}_{m|n})\). Our definition of the double Yangian closely follows Zhang [8], but we use a different normalization of the Yang \(R\)-matrix which governs its defining relations. Finally, we introduce the dual Yangian analogue of the quantum contraction and study its properties.

2.1 Yangian for \(\mathfrak{gl}_{m|n}\)↩︎

Consider the Lie superalgebra \(\widehat{\mathfrak{gl}}_{m|n}=\mathfrak{gl}_{m|n}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}K .\) Its supercommutation relations are given by \[\begin{align} [e_{ij}(r),e_{kl}(s)] =&\, \delta_{kj}e_{il}(r+s) - \delta_{il}e_{kj}(r+s) (-1)^{(\bar{i}+\bar{j})(\bar{k}+\bar{l})}\nonumber\\ &+K\left( \delta_{kj}\delta_{il}(-1)^{\bar{i}} -\frac{\delta_{ij}\delta_{kl}}{m-n}(-1)^{\bar{i}+\bar{k}}\right)r\delta_{r+s0}\label{glmn} \end{align}\tag{1}\] for \(m\neq n\), where the element \(K\) is even and central, \(e_{ij}\in \mathfrak{gl}_{m|n}\) are matrix units and \(e_{ij}(r)=e_{ij}\otimes t^r\). The parity of the element \(e_{ij}(r)\) is \(\bar{i}+\bar{j}\), where \(\bar{i}=0\) for \(i=1,\ldots ,m\) and \(\bar{i}=1\) for \(i=m+1,\ldots ,m+n\). Note that we can rescale the central element by \(K=(n-m)K'\) so that the above relations apply to the case \(m=n\) as well.

We follow Nazarov [1] to define the Yangian for the general linear Lie superalgebra. The Yangian \({\rm Y}(\mathfrak{gl}_{m|n})\) is the \(\mathbb{Z}_2\)-graded unital associative algebra with generators \(t_{ij}^{(r)}\), where \(i,j =1,\ldots , m+n\) and \(r=1,2,\ldots ,\) subject to the defining relations \[\label{Y} [t_{ij}^{(r)},t_{kl}^{(s)}] =(-1)^{\bar{i}\bar{j}+\bar{i}\bar{k}+\bar{j}\bar{k}} \sum_{a=1}^{\min\left\{r,s\right\}}\left( t_{kj}^{(a-1)}t_{il}^{(r+s-a)}-t_{kj}^{(r+s-a)}t_{il}^{(a-1)}\right),\tag{2}\] where the square brackets denote the supercommutator and \(t_{ij}^{(0)}=\delta_{ij}\). The parity of the element \(t_{ij}^{(r)}\) is \(\bar{i}+\bar{j}\). Relations 2 can be expressed in terms of formal power series \[t_{ij}(u)=\delta_{ij}+\sum_{r\geqslant 1} t_{ij}^{(r)} u^{-r}\] as \[\label{Y2} [t_{ij} (u),t_{kl} (v)] =\frac{(-1)^{\bar{i}\bar{j}+\bar{i}\bar{k}+\bar{j}\bar{k}}}{u-v} \left(t_{kj}(u)t_{il}(v)-t_{kj}(v)t_{il}(u)\right).\tag{3}\]

Consider the rational \(R\)-matrix \(R(u)\in\mathop{\mathrm{End}}\mathbb{C}^{m|n}\otimes\mathop{\mathrm{End}}\mathbb{C}^{m|n} [u^{-1}]\) given by \[\label{Rmatrix} R(u)=1-Pu^{-1},\quad\text{where}\quad P=\sum_{i,j=1}^{m+n} e_{ij}\otimes e_{ji}(-1)^{\bar{j}}\tag{4}\] and \(1\) is the identity. The \(R\)-matrix satisfies the Yang–Baxter equation \[\label{ybe} R_{12}(u_1-u_2)R_{13}(u_1-u_3)R_{23}(u_2 -u_3)= R_{23}(u_2 -u_3)R_{13}(u_1-u_3)R_{12}(u_1-u_2).\tag{5}\] By using the \(R\)-matrix \(R(u)\), one can write the defining relations for the super Yangian in the so-called \(RTT\)-form as follows. Organize the series \(t_{ij}(u)\) into the matrix \[\label{teu} T(u)=\sum_{i,j=1}^{m+n} (-1)^{\bar{i}\bar{j}+\bar{j}} e_{ij}\otimes t_{ij}(u) .\tag{6}\] Relations 2 are then expressed as the identity of formal power series in the variables \(u\) and \(v\) such that their coefficients belong to \(\mathop{\mathrm{End}}\mathbb{C}^{m|n}\otimes\mathop{\mathrm{End}}\mathbb{C}^{m|n}\otimes {\rm Y}(\mathfrak{gl}_{m|n})\), \[\label{YRTT} R(u-v)T_1(u)T_2(v) =T_2(v)T_1(u)R(u-v).\tag{7}\] Note that in 7 we used the subscripts to indicate the tensor factors on which the corresponding matrices are applied, so that we have \[T_1(u)=\sum_{i,j=1}^{m+n} (-1)^{\bar{i}\bar{j}+\bar{j}} e_{ij}\otimes 1\otimes t_{ij}(u)\qquad\text{and}\qquad T_2(v)=\sum_{i,j=1}^{m+n} (-1)^{\bar{i}\bar{j}+\bar{j}} 1\otimes e_{ij}\otimes t_{ij}(v).\]

We now recall the Poincaré–Birkhoff–Witt Theorem for \({\rm Y}(\mathfrak{gl}_{m|n})\) which was proved by Gow [7]. First, introduce two different ascending filtrations on the Yangian by \[\label{deg21} \deg_1 t_{ij}^{(r)} = r\quad\text{and}\quad\deg_2 t_{ij}^{(r)}=r-1.\tag{8}\] Denote by \(\mathop{\mathrm{gr}}_1 {\rm Y}(\mathfrak{gl}_{m|n})\) and \(\mathop{\mathrm{gr}}_2 {\rm Y}(\mathfrak{gl}_{m|n})\) the corresponding graded algebras. We use the notation \(\hat{t}_{ij}^{(r)}\) (resp. \(\bar{t}_{ij}^{(r)}\)) for the images of generators in the respective components of the graded algebra \(\mathop{\mathrm{gr}}_1 {\rm Y}(\mathfrak{gl}_{m|n})\) (resp. \(\mathop{\mathrm{gr}}_2 {\rm Y}(\mathfrak{gl}_{m|n})\)). By 2 , \(\mathop{\mathrm{gr}}_1 {\rm Y}(\mathfrak{gl}_{m|n})\) is supercommutative, i.e. we have \([\hat{t}_{ij}^{(r)},\hat{t}_{kl}^{(s)}]=0.\) On the other hand, taking the images of the generators in the respective components of \(\mathop{\mathrm{gr}}_2 {\rm Y}(\mathfrak{gl}_{m|n})\) in 2 , one checks that the assignments \[\label{assignments1} e_{ij}(r-1)\mapsto (-1)^{\bar{i}}\bar{t}_{ij}^{(r)},\tag{9}\] where \(i,j=1,\ldots ,m+n\) and \(r\geqslant 1\), define a homomorphism \[\label{mapY} {\rm U}(\mathfrak{gl}_{m|n}[t])\to \textstyle\mathop{\mathrm{gr}}_2 {\rm Y}(\mathfrak{gl}_{m|n}).\tag{10}\] Let us recall [7], which implies that the map 10 is an algebra isomorphism.

Consider the transposition \(\tau\colon e_{ij}\mapsto (-1)^{\bar{i}\bar{j}+\bar{i}}e_{ji}\) on \(\mathop{\mathrm{End}}\mathbb{C}^{m|n}\). In [1], Nazarov introduced a certain family of central elements of the Yangian. They were defined as coefficients of the quantum contraction, a power series \(z (u)\in {\rm Y}(\mathfrak{gl}_{m|n})[[u^{-1}]]\) which is uniquely determined by the identity \[\label{qcontra} P^{\tau_2}T_1(u+m-n) \left(T_2(u)^{-1}\right)^\tau = P^{\tau_2}\otimes z (u).\tag{11}\] Here \(\tau_2\) indicates that the transposition \(\tau\) is applied on the second tensor factor. Finally, we recall that the quantum analogue of the Liouville theorem [1], along with the fact that the coefficients of the quantum Berezinian generate the centre of the Yangian [7], implies that the coefficients of \(z (u)\) also generate the centre of \({\rm Y}(\mathfrak{gl}_{m|n})\).

2.2 Dual Yangian for \(\mathfrak{gl}_{m|n}\)↩︎

Define the dual Yangian \({\rm Y}^+(\mathfrak{gl}_{m|n})\) as the \(\mathbb{Z}_2\)-graded unital associative algebra with generators \(t_{ij}^{(-r)}\), where \(i,j=1,\ldots , m+n\) and \(r=1,2,\ldots ,\) subject to the defining relations \[\begin{align} [t_{ij}^{(-r)},t_{kl}^{(-s)}] =&\,(-1)^{\bar{i}\bar{j}+\bar{i}\bar{k}+\bar{j}\bar{k}} \Big( \delta_{kj}t_{il}^{(-r-s)}-\delta_{il}t_{kj}^{(-r-s)} \Big.\nonumber\\ \Big. &+\sum_{a=1}^{\min\left\{r,s\right\}} \left( t_{kj}^{(-r-s+a-1)} t_{il}^{(-a)}-t_{kj}^{(-a)} t_{il}^{(-r-s+a-1)}\right)\Big).\label{Yd} \end{align}\tag{12}\] As before, the square brackets denote the supercommutator and the parity of \(t_{ij}^{(-r)}\) is \(\bar{i}+\bar{j}\). Defining relations 12 can be expressed in terms of formal power series \[t_{ij}^+ (u)=\delta_{ij}-\sum_{r\geqslant 1} t_{ij}^{(-r)} u^{r-1}\] as \[[t_{ij}^+ (u),t_{kl}^+ (v)] =\frac{(-1)^{\bar{i}\bar{j}+\bar{i}\bar{k}+\bar{j}\bar{k}}}{u-v} \left(t_{kj}^+(u) t_{il}^+(v)-t_{kj}^+(v)t_{il}^+(u)\right).\] Moreover, as with the Yangian, one can use the \(R\)-matrix 4 to write defining relations 12 in the \(RTT\)-form, i.e. as the identity of formal power series in the variables \(u\) and \(v\) such that their coefficients belong to \(\mathop{\mathrm{End}}\mathbb{C}^{m|n}\otimes\mathop{\mathrm{End}}\mathbb{C}^{m|n}\otimes {\rm Y}^+(\mathfrak{gl}_{m|n})\), \[\label{YdRTT} R(u-v)T_1^+ (u)T_2^+ (v) =T_2^+ (v)T_1^+ (u)R(u-v),\tag{13}\] where \(T^+(u)\) is given by \[\label{teud} T^+ (u)=\sum_{i,j=1}^{m+n} (-1)^{\bar{i}\bar{j}+\bar{j}} e_{ij}\otimes t^+_{ij}(u) .\tag{14}\]

To prove the Poincaré–Birkhoff–Witt Theorem for the dual Yangian we follow the approach of Nazarov [10], where such a result was proved in the even case. Introduce the ascending filtration on \({\rm Y}^+(\mathfrak{gl}_{m|n})\) by \[\label{deg22} \deg_2 t_{ij}^{(-r)}=-r.\tag{15}\] Denote by \(\mathop{\mathrm{gr}}_2 {\rm Y}^+(\mathfrak{gl}_{m|n})\) the corresponding graded algebra. We use the notation \(\bar{t}_{ij}^{(-r)}\) for the images of the generators in the respective components of \(\mathop{\mathrm{gr}}_2 {\rm Y}^+(\mathfrak{gl}_{m|n})\). Defining relations 12 imply the identities in \(\mathop{\mathrm{gr}}_2 {\rm Y}^+(\mathfrak{gl}_{m|n})\), \[[\bar{t}_{ij}^{(-r)},\bar{t}_{kl}^{(-s)}] = (-1)^{ \bar{i} \bar{j} + \bar{i}\bar{k} + \bar{j}\bar{k}} \left( \delta_{kj}\bar{t}_{il}^{(-r-s)}-\delta_{il} \bar{t}_{kj}^{(-r-s)}\right) .\] Therefore, by commutation relations 1 for the Lie superalgebra \(\widehat{\mathfrak{gl}}_{m|n}\), the assignments \[\label{assignments2} e_{ij}(-r)\mapsto (-1)^{\bar{i}}\bar{t}_{ij}^{(-r)},\tag{16}\] where \(i,j=1,\ldots ,m+n\) and \(r\geqslant 1\), define a homomorphism \[\label{mapYdx} {\rm U}(t^{-1}\mathfrak{gl}_{m|n}[t^{-1}])\to \textstyle\mathop{\mathrm{gr}}_2 {\rm Y}^+(\mathfrak{gl}_{m|n}).\tag{17}\]

The aforementioned proof in [10] employs a certain bilinear pairing motivated by [26]. In the super case, such a pairing \(\left<\cdot,\cdot\right>\colon {\rm Y}(\mathfrak{gl}_{m|n})\times {\rm Y}^+(\mathfrak{gl}_{m|n})\to \mathbb{C}\) is defined so that the corresponding linear map \({\rm Y}(\mathfrak{gl}_{m|n})\otimes {\rm Y}^+(\mathfrak{gl}_{m|n})\to \mathbb{C}\) satisfies \[T_1(u_1)\ldots T_k(u_k)T_{k+1}^+(v_1)\ldots T_{k+l}^+(v_l) \mapsto \prod_{i=1,\ldots, k }^{\longrightarrow} \prod_{j=1,\ldots, l }^{\longleftarrow} R_{ij+k}(u_i -v_j)\] in \((\mathop{\mathrm{End}}\mathbb{C}^{m|n})^{\otimes k} \otimes(\mathop{\mathrm{End}}\mathbb{C}^{m|n})^{\otimes l} \otimes {\rm Y}(\mathfrak{gl}_{m|n})\otimes {\rm Y}^+(\mathfrak{gl}_{m|n})\) for all integers \(k,l\geqslant 0\), where the arrows indicate the order of factors. In particular, we have \(\left<1,1\right>=1\). The fact that the pairing is well-defined is easily proved by using the Yang–Baxter equation 5 and the defining relations in the \(RTT\)-form, 7 and 13 . The following property of the pairing can be verified by arguing as in the proof of [10].

We can now define a bilinear pairing \(\mathop{\mathrm{gr}}_1 {\rm Y}(\mathfrak{gl}_{m|n})\times\mathop{\mathrm{gr}}_2 {\rm Y}^+(\mathfrak{gl}_{m|n})\to \mathbb{C}\) by \[\label{pairing} \big<\hat{t}_{i_1 j_1}^{(s_1)}\ldots \hat{t}_{i_k j_k}^{(s_k)}, \bar{t}_{i_{k+1} j_{k+1}}^{(-r_1)}\ldots \bar{t}_{i_{k+l} j_{k+l}}^{(-r_l)}\big> = \delta_{s_1+\ldots + s_k, r_1+\ldots +r_l} \big<t_{i_1 j_1}^{(s_1)}\ldots t_{i_k j_k}^{(s_k)}, t_{i_{k+1} j_{k+1}}^{(-r_1)}\ldots t_{i_{k+l} j_{k+l}}^{(-r_l)}\big>.\tag{18}\] For any \(r \geqslant 1\) denote by \(\mathop{\mathrm{gr}}_{1,r} {\rm Y}(\mathfrak{gl}_{m|n})\) (resp. \(\mathop{\mathrm{gr}}_{2,-r} {\rm Y}^+(\mathfrak{gl}_{m|n})\)) the subspace of degree \(r\) (resp. \(-r\)) in the graded algebra \(\mathop{\mathrm{gr}}_{1 } {\rm Y}(\mathfrak{gl}_{m|n})\) (resp. \(\mathop{\mathrm{gr}}_{2 } {\rm Y}^+(\mathfrak{gl}_{m|n})\)).

Proof. The proof of the theorem can be carried out in parallel with the case of dual Yangian for the general linear Lie algebra \(\mathfrak{gl}_N\), as given in [10]. In particular, it uses the nondegeneracy of restriction of bilinear pairing 18 to \(\mathop{\mathrm{gr}}_{1,s} {\rm Y}(\mathfrak{gl}_{m|n})\times \mathop{\mathrm{gr}}_{2,-s} {\rm Y}^+(\mathfrak{gl}_{m|n})\) for all \(s\geqslant 0\), which can be verified by similar arguments as [10]. Roughly speaking, this is due to the fact that the \(R\)-matrices which govern the defining relations for the (dual) Yangian for \(\mathfrak{gl}_N\) and for \(\mathfrak{gl}_{m|n}\) are of the same form. \(\qed\)

2.3 Double Yangian for \(\mathfrak{gl}_{m|n}\) at the level \(0\)↩︎

First, we shall derive a certain lemma on evaluation representations of the loop Lie superalgebra \(\mathcal{L}(\mathfrak{gl}_{m|n})=\mathfrak{gl}_{m|n}\otimes\mathbb{C}[t,t^{-1}]\), which we need in the proof of Theorem 6 below. Recall that for any representation \(\sigma\) of \(\mathfrak{gl}_{m|n}\) on \(\mathbb{C}^{m|n}\) and nonzero \(a\in \mathbb{C}\) one can define the evaluation representation \(\sigma_a \colon \mathcal{L}(\mathfrak{gl}_{m|n}) \to \mathop{\mathrm{End}}\mathbb{C}^{m|n}\) by \[\sigma_a \colon x\otimes t^r \mapsto a^r \sigma (x)\qquad\text{for all } x\in \mathfrak{gl}_{m|n}\text{ and }r\in\mathbb{Z}.\] Denote by \(\sigma_{a_1,\ldots ,a_k}\) the tensor product of evaluation representations \(\sigma_{a_1},\ldots ,\sigma_{a_k}\). We use the same notation for the extension of \(\sigma_{a_1,\ldots ,a_k}\) to the representation of enveloping algebra \({\rm U}(\mathcal{L}(\mathfrak{gl}_{m|n}))\). The next lemma is verified by arguing as in the proof of [14].

Throughout the rest of the paper, we assume that \(m\neq n\). The double Yangian \({\rm DY}_0(\mathfrak{gl}_{m|n})\) for \(\mathfrak{gl}_{m|n}\) at the level 0 is defined as the \(\mathbb{Z}_2\)-graded unital associative algebra generated by the elements \(t_{ij}^{(r)}\) and \(t_{ij}^{(-r)}\), where \(i,j=1,\ldots , m+n\) and \(r=1,2,\ldots,\) subject to the defining relations which are written in terms of the generator matrices 6 and 14 . They are given by 7 , 13 and \[\label{DYRTT} R(u-v)T_1 (u)T_2^+ (v) =T_2^+ (v)T_1 (u)R(u-v).\tag{19}\] The parity of the elements \(t_{ij}^{(r)}\) and \(t_{ij}^{(-r)}\) is again \(\bar{i}+\bar{j}\).

The degree operator \(\deg_2\), as given by 8 and 15 , defines an ascending filtration \[\label{filtration} \ldots \subseteq{\rm DY}_0(\mathfrak{gl}_{m|n})^{(r)}\subseteq {\rm DY}_0(\mathfrak{gl}_{m|n})^{(r+1)}\subseteq\ldots\subseteq{\rm DY}_0(\mathfrak{gl}_{m|n}),\tag{20}\] where \({\rm DY}_0(\mathfrak{gl}_{m|n})^{(r)}\) is the linear span of the elements of \({\rm DY}_0(\mathfrak{gl}_{m|n})\) whose degrees do not exceed \(r\). We shall write \(\bar{t}_{ij}^{(\pm r)}\) for the images of the generators in the respective components of the graded algebra \(\mathop{\mathrm{gr}}_2 {\rm DY}_0(\mathfrak{gl}_{m|n})\). A direct calculation shows that the assignments 9 and 16 define a homomorphism \[\label{mapYd} {\rm U}(\mathcal{L}(\mathfrak{gl}_{m|n}))\to \textstyle\mathop{\mathrm{gr}}_2 {\rm DY}_0(\mathfrak{gl}_{m|n}).\tag{21}\]

Let \({\rm Y}\) (resp. \({\rm Y}^+\)) be the unital subalgebra of the double Yangian generated by all elements \(t_{ij}^{(r)}\) (resp. \(t_{ij}^{(-r)}\)) with \(i,j=1,\ldots ,m+n\) and \(r=1,2,\ldots .\) Consider the descending filtration on \({\rm Y}^+\) defined by setting the degree of \(t_{ij}^{(-r)}\) to be \(r\) and denote by \(\widetilde{{\rm Y}}^+\) the corresponding completion of \({\rm Y}^+\). Introduce the extended double Yangian \(\widetilde{{\rm DY}}_0(\mathfrak{gl}_{m|n})\) at the level 0 as the space of all finite linear combinations of all products \(xy\) for \(x\in \widetilde{\rm Y}^+\) and \(y\in {\rm Y}\) with the multiplication extended by continuity from the double Yangian. The Hopf superalgebra structure on \(\widetilde{{\rm DY}}_0(\mathfrak{gl}_{m|n})\) is defined by the formulae \[\begin{align} &\Delta(t_{ij}(u))=\sum_{k=1}^{m+n} t_{ik}(u)\otimes t_{kj}(u),\quad \Delta(t^+_{ij}(u))=\sum_{k=1}^{m+n} t^+_{ik}(u)\otimes t^+_{kj}(u),\tag{22} \\ &S(T(u))=T(u)^{-1},\quad S(T^+(u))=T^+ (u)^{-1}, \qquad \varepsilon (T(u))= \varepsilon (T^+(u))=1. \tag{23} \end{align}\]

Proof. One easily checks that the map ?? preserves the ideal of defining relations for the double Yangian using the equivalent expressions for the above assignments, \[T(u)\mapsto R(-u+a)^{\tau_1}\quad\text{and}\quad T^+(u)\mapsto R(a-u )^{\tau_1},\] where \(\tau_1\) is the transposition \(e_{ij}\mapsto (-1)^{\bar{i}\bar{j}+\bar{i}}e_{ji}\) applied on the first tensor factor. \(\qed\)

Let us define a total order on the double Yangian generators as follows. For any \(i,j,k,l=1,\ldots ,m+n\) and \(r,s=1,2,\ldots\) set

1. \(t_{ij}^{(-r)}\prec t_{kl}^{(s)}\);

2. \(t_{ij}^{(\pm r)}\prec t_{kl}^{(\pm s)}\) if \((i,j)\) precedes \((k,l)\) in lexicographical order;

3. \(t_{ij}^{(r)}\prec t_{ij}^{(s)}\) and \(t_{ij}^{(-s)}\prec t_{ij}^{(-r)}\) if \(r<s\).

Let us equip the algebra \({\rm DY}_0(\mathfrak{gl}_{m|n})\) with the topology induced by filtration 20 .

Proof. Clearly, the double Yangian is spanned by all monomials in the generators. Hence, it is sufficient to check that for any integer \(p\) and a monomial \(\mu\) we can write \(\mu\) modulo \({\rm DY}_0(\mathfrak{gl}_{m|n})^{(p)}\) as a linear combination of ordered monomials, with at most power \(1\) for odd generators. First, as with the even case, one proves by induction using the defining relations that \(\mu\) can be expressed as a linear combination of monomials in generators satisfying [reef1] and [reef2]. These monomials are products of submonomials of the form \(t_{ij}^{(\pm r_1)}\ldots t_{ij}^{(\pm r_k)}\). If \(\bar{i}+\bar{j}\) is even, by relations 2 and 12 , we have \([t_{ij}^{(\pm r)},t_{ij}^{(\pm s)} ]=0\), so we can assume that [reef3] holds for such submonomials. If \(\bar{i}+\bar{j}\) is odd and \(r\neq s\), relations 2 and 12 take the form \[t_{ij}^{(\pm r)} t_{ij}^{(\pm s)} = - t_{ij}^{(\pm s)} t_{ij}^{(\pm r)} +\text{lower degree monomials}.\] Hence, if \(\pm r_{\sigma (1)}\leqslant\ldots \leqslant \pm r_{\sigma (k)}\) for some permutation \(\sigma\) of the indices \(1,\ldots ,k\), we have \[t_{ij}^{(\pm r_1)}\ldots t_{ij}^{(\pm r_k)} =\varepsilont_{ij}^{(\pm r_{\sigma(1)})}\ldots t_{ij}^{(\pm r_{\sigma(k)})} +\text{lower degree monomials}\quad\text{for some }\varepsilon\in\left\{-1,1\right\}.\] Finally, we can exclude all squares of odd generators. Indeed, by setting \(r=s\) and \((k,l)=(i,j)\) with \(\bar{i}+\bar{j}\) odd in relations 2 and 12 , on the left hand-side we obtain \(2(t_{ij}^{(\pm r)} )^2\), while the right hand-side consists of elements of lower degrees. Let us assume that the original monomial \(\mu\) is of degree \(d>p\). By the preceding discussion, we can express it modulo \({\rm DY}_0(\mathfrak{gl}_{m|n})^{(d-1)}\) as a linear combination of ordered monomials of degree \(d\) in the generators, with at most power \(1\) for odd generators. Clearly, we can now continue inductively to express \(\mu\) modulo \({\rm DY}_0(\mathfrak{gl}_{m|n})^{(p)}\) as a linear combination of such monomials of degrees \(d,d-1,\ldots , p+1\), as required.

As for the linear independence, it is verified by using the ideas of Etingof and Kazhdan [12] and Nazarov [14] which rely on the existence of evaluation representations. Suppose that some nontrivial linear combination of ordered monomials vanishes. Then its image under \(\pi_{a_1}\otimes\ldots \otimes\pi_{a_k}\) is trivial for any choice of \(k>0\) and nonzero \(a_1,\ldots ,a_k\in\mathbb{C}\). This leads to a contradiction by regarding the top degree components of the monomials with respect to the filtration 20 and arguing as in [14]. In particular, the argument employs the corresponding evaluation representations \(\sigma_{a_1,\ldots ,a_k}\) of \({\rm U}(\mathcal{L}(\mathfrak{gl}_{m|n}))\) and Lemma 4. \(\qed\)

2.4 Double Yangian for \(\mathfrak{gl}_{m|n}\)↩︎

Consider the \(R\)-matrix 4 . As \(P^2=1\), it satisfies \[\label{preuni1} R(u) R(-u)=1-u^{-2}.\tag{24}\] Let \(g(u)\) be the unique formal power series in \(1+u^{-1}\mathbb{C}[[u^{-1}]]\) such that \[\label{functg} g(u+m-n)=(1-u^{-2})g(u).\tag{25}\] The series \(g(u)\) also satisfies \[\label{preuni2} g(u) g(-u)(1-u^{-2})=1;\tag{26}\] see [9] for more details on \(g(u)\). By combining 24 and 26 we find that the normalized \(R\)-matrix \({\overline{R}}(u)=g(u)R(u)\) possesses the unitarity property, \[\label{uni} {\overline{R}}(u){\overline{R}}(-u)=1.\tag{27}\] Also, it is clear that it satisfies Yang–Baxter equation 5 . We explain the motivation for the use of this particular normalizing function \(g(u)\) in Subsection 2.5; cf. 35 .

The next lemma generalizes [13] to the super setting. As with its even counterpart, it is proved by a direct calculation relying on \(RTT\)-relations 7 and 13 .

The double Yangian \({\rm DY}(\mathfrak{gl}_{m|n})\) for \(\mathfrak{gl}_{m|n}\) is defined as the \(\mathbb{Z}_2\)-graded unital associative algebra generated by the elements \(C\), \(t_{ij}^{(r)}\) and \(t_{ij}^{(-r)}\), where \(i,j=1,\ldots , m+n\) and \(r=1,2,\ldots,\) subject to the defining relations which are written in terms of the generator matrices 6 and 14 . They are given by 7 , 13 and \[\label{DYRTTC} {\overline{R}}(u-v+C/2)T_1 (u)T_2^+ (v) =T_2^+ (v)T_1 (u) {\overline{R}}(u-v-C/2),\tag{28}\] where \(C\) is even central element and, as before, the parity of the elements \(t_{ij}^{(\pm r)}\) is \(\bar{i}+\bar{j}\).

There exists a natural epimorphism \(\phi\colon {\rm DY}(\mathfrak{gl}_{m|n})\to {\rm DY}_0(\mathfrak{gl}_{m|n})\) such that \(t_{ij}^{(\pm r)}\mapsto t_{ij}^{(\pm r)}\) and \(C\mapsto 0\). Due to Theorem 6, the subalgebra of the double Yangian \({\rm DY}(\mathfrak{gl}_{m|n})\) generated by all elements \(t_{ij}^{(r)}\) (resp. \(t_{ij}^{(-r)}\)) with \(i,j=1,\ldots ,m+n\) and \(r=1,2,\ldots\) coincides with the Yangian \({\rm Y}(\mathfrak{gl}_{m|n})\) (resp. dual Yangian \({\rm Y}^+(\mathfrak{gl}_{m|n})\)). Consider the descending filtration on the dual Yangian defined by setting the degree of \(t_{ij}^{(-r)}\) to be \(r\). Denote by \(\widetilde{{\rm Y}}^+(\mathfrak{gl}_{m|n})\) the corresponding completion of \({\rm Y}^+(\mathfrak{gl}_{m|n})\). We refer to \(\widetilde{{\rm Y}}^+(\mathfrak{gl}_{m|n})\) as the extended dual Yangian. Define the extended double Yangian \(\widetilde{{\rm DY}}(\mathfrak{gl}_{m|n})\) as the space of all finite \(\mathbb{C}[C]\)-linear combinations of all products \(xy\) for \(x\in \widetilde{\rm Y}^+(\mathfrak{gl}_{m|n})\) and \(y\in {\rm Y}(\mathfrak{gl}_{m|n})\) with the multiplication extended by continuity from the double Yangian. The Hopf superalgebra structure on \(\widetilde{{\rm DY}}(\mathfrak{gl}_{m|n})\) is given by \[\begin{align} &\Delta(t_{ij}(u))=\sum_{k=1}^{m+n} t_{ik}(u+C_2/4)\otimes t_{kj}(u-C_1/4),\\ &\Delta(t^+_{ij}(u))=\sum_{k=1}^{m+n} t^+_{ik}(u-C_2/4)\otimes t^+_{kj}(u+C_1/4), \\ &\Delta(C)=C_1 +C_2,\quad S(C)=-C,\quad \varepsilon ( C)= 0 \end{align}\] and 23 , where \(C_1=1\otimes C\) and \(C_2=1\otimes C\).

Extend the degree operator \(\deg_2\), given by 8 and 15 , by defining the degree of the central element \(C\) to be zero. Thus, we obtain the ascending filtration \[\label{filtration2} \ldots \subseteq {\rm DY}(\mathfrak{gl}_{m|n})^{(r)}\subseteq {\rm DY}(\mathfrak{gl}_{m|n})^{(r+1)}\subseteq\ldots\subseteq {\rm DY}(\mathfrak{gl}_{m|n}),\tag{29}\] where \({\rm DY}(\mathfrak{gl}_{m|n})^{(r)}\) is the linear span of the elements of \({\rm DY}(\mathfrak{gl}_{m|n})\) whose degrees do not exceed \(r\). Denote the corresponding graded algebra by \(\mathop{\mathrm{gr}}_2 {\rm DY}(\mathfrak{gl}_{m|n})\) and denote the images of generators in its respective components by \(\bar{t}_{ij}^{(\pm r)}\) and \(\bar{C}\). A direct calculation relying on the supercommutation relations 1 and the form of the series \(g(u)\), which goes in parallel with the proof of [9], implies that the assignments \[e_{ij}(r-1)\mapsto (-1)^{\bar{i}}\bar{t}_{ij}^{(r)},\qquad e_{ij}(-r)\mapsto (-1)^{\bar{i}}\bar{t}_{ij}^{(-r)} \qquad\text{and}\qquad K\mapsto \bar{C}\] with \(i,j=1,\ldots ,m+n\) and \(r\geqslant 1\) define a homomorphism \[\label{mapYYd} {\rm U}(\widehat{\mathfrak{gl}}_{m|n}) \to \textstyle\mathop{\mathrm{gr}}_2 {\rm DY}(\mathfrak{gl}_{m|n}).\tag{30}\]

Extend the total order [reef1]–[reef3] to \({\rm DY}(\mathfrak{gl}_{m|n})\) so that it includes the central element \(C\) in an arbitrary way. In the following theorem, we consider the topology over \({\rm DY}(\mathfrak{gl}_{m|n})\) induced by the filtration 29 .

Proof. As with the level \(0\) case from Theorem 6, one verifies by induction using the defining relations that the ordered monomials in generators, with at most power \(1\) for odd generators, topologically span \({\rm DY}(\mathfrak{gl}_{m|n})\). Next, Lemma 8 implies that, for any \(c\in\mathbb{C}\), the dual Yangian is naturally equipped with the structure of module for the double Yangian of level \(c\), i.e. such that \(C\) acts as a scalar multiplication by \(c\). Thus, the central element \(C\) is nonzero. Moreover, by arguing as in the proof of [9], one can show that its powers \(1, C,\ldots ,C^k\) are linearly independent for any positive \(k\). Finally, the linear independence of the ordered monomials in the generators with at most power \(1\) for odd generators is established by examining the image of their linear combination under the map \((1\otimes\phi)\circ\Delta\) and using Theorem 6, in parallel with the corresponding part of the proof of [9]. \(\qed\)

2.5 Quantum contraction↩︎

In the following lemma, we introduce the dual Yangian counterpart of the series \(z (u)\); recall 11 .

Proof. First of all, we observe that the coefficients of matrix entries of the shifted series \(T^+(u+m-n)\) and the inverse \(T^+(u)^{-1}\) are well-defined elements of the extended dual Yangian \(\widetilde{{\rm Y}}^+(\mathfrak{gl}_{m|n})\). The lemma is proved by analogous arguments as its Yangian counterpart established in [1], but we provide some details for completeness. First, consider the defining relation 13 . Multiplying it from the right and from the left by \(T_2^+(v)^{-1}\) and then applying the transposition \(\tau\) on the second tensor component we obtain \[R(u-v)^{\tau_2} \left(T_2^+ (v)^{-1}\right)^\tau T_1^+ (u) = T_1^+ (u) \left(T_2^+ (v)^{-1}\right)^\tau R(u-v)^{\tau_2}.\] Clearly, this is equivalent with \[\label{eeq1} \left(R(u-v)^{\tau_2}\right)^{-1} T_1^+ (u) \left(T_2^+ (v)^{-1}\right)^\tau = \left(T_2^+ (v)^{-1}\right)^\tau T_1^+ (u)\left(R(u-v)^{\tau_2}\right)^{-1}.\tag{31}\] As \((P^{\tau_2})^2=(m-n)P^{\tau_2}\), we have \[\left(R(u)^{\tau_2}\right)^{-1}=\sum_{l\geqslant 0}\frac{\left(P^{\tau_2}\right)^l}{u^l} =1+\sum_{l\geqslant 1}\frac{\left(m-n\right)^{l-1}}{u^l}P^{\tau_2} =1+(u-m+n)^{-1}P^{\tau_2}.\] Hence, multiplying 31 by \(u-v-m+n\) we get \[\left(u-v-m+n +P^{\tau_2}\right) T_1^+ (u) \left(T_2^+ (v)^{-1}\right)^\tau = \left(T_2^+ (v)^{-1}\right)^\tau T_1^+ (u)\left(u-v-m+n +P^{\tau_2}\right).\] Replacing the variables \((u,v)\) by \((u+m-n,u)\) the above identity becomes \[\label{eeq2} P^{\tau_2}T_1^+ (u+m-n) \left(T_2^+ (u )^{-1}\right)^\tau = \left(T_2^+ (u )^{-1}\right)^\tau T_1^+ (u+m-n)P^{\tau_2}.\tag{32}\] Finally, we observe that the image of \(P^{\tau_2}=\sum_{i,j=1}^{m+n} e_{ij}\otimes e_{ij}(-1)^{\bar{i}\bar{j}}\) is one-dimensional, so that the equality 32 implies the assertion of the lemma. \(\qed\)

As with the original quantum contraction, the series \(z^+(u)\) gives rise to a family of central elements in the extended dual Yangian \(\widetilde{{\rm Y}}^+(\mathfrak{gl}_{m|n})\).

Proof. The lemma can be verified by suitably modifying the proof of [1]. In the proof of Theorem 12 below we already present in detail such arguments in the case of the normalized \(R\)-matrix, so here we only sketch the major steps of the proof. First, by 13 and 31 we have \[\begin{align} &P_{12}^{\tau_2} \left(R_{02}(v-u )^{\tau_2}\right)^{-1} R_{01}(v-u-m+n) T^+_0(v)T_1^+(u+m-n) \left(T_2^+(u )^{-1}\right)^\tau\nonumber\\ &\quad= P_{12}^{\tau_2} T_1^+(u+m-n) \left(T_2^+(u )^{-1}\right)^\tau T^+_0(v) \left(R_{02}(v-u )^{\tau_2}\right)^{-1} R_{01}(v-u-m+n).\label{alfnjfd8} \end{align}\tag{33}\] Next, using the the property \[\label{rmatprop} P_{12}^{\tau_2} \left(R_{02}(u )^{\tau_2}\right)^{-1} R_{01}(u-m+n) =P_{12}^{\tau_2} \left(1- (u-m+n)^{-2}\right)\tag{34}\] of the \(R\)-matrix 4 and Lemma 10, we bring 33 to the form \[\left(1-(v-u-m+n)^{2}\right) P_{12}^{\tau_2}T^+_0(v)z^+ (u) = \left(1-(v-u-m+n)^{2}\right) P_{12}^{\tau_2}z^+ (u) T^+_0(v) .\] Finally, we cancel the terms \(1-(v-u-m+n)^{2}\) and conclude that \(T^+_0(v)\) and \(z^+ (u)\) commute, as required. \(\qed\)

Observe that the proof of Lemma 11 uses the property 34 of \(R(u)\). By combining 25 and 34 , one finds that the normalized \(R\)-matrix \({\overline{R}}(u)\) satisfies the identity \[\label{rmatpropc} P_{12}^{\tau_2} \left( {\overline{R}}_{02}(u )^{\tau_2}\right)^{-1} {\overline{R}}_{01}(u-m+n) =P_{12}^{\tau_2},\tag{35}\] a super analogue of the ordinary crossing symmetry property for the even Yang \(R\)-matrix. This property plays a key role in the proof of the next theorem.

Proof. Clearly, to prove the theorem, it is sufficient to verify the equalities \[\label{dydvaide} T (v)z^+(u) = z^+(u)T (v) \quad\text{and}\quad T^+(v)z(u) = z(u)T^+(v).\tag{36}\] Let us prove the first equality. Set \(x=v-u+C/2\). Consider the identity \[\begin{align} &P_{12}^{\tau_2} \left( {\overline{R}}_{02}(x)^{\tau_2}\right)^{-1} {\overline{R}}_{01}(x-m+n ) T_0(v)T_1^+(u+m-n) \left(T_2^+(u)^{-1}\right)^\tau\nonumber\\ =& P_{12}^{\tau_2} T_1^+(u+m-n) \left(T_2^+(u)^{-1}\right)^\tau T_0(v) \left( {\overline{R}}_{02}(x -C )^{\tau_2}\right)^{-1} {\overline{R}}_{01}(x-m+n-C ), \label{algnjfd8} \end{align}\tag{37}\] which is deduced from 28 . By using 35 and then Lemma 10, we rewrite the left-hand side of 37 as \[\begin{align} &P_{12}^{\tau_2} T_0(v)T_1^+(u+m-n) \left(T_2^+(u)^{-1}\right)^\tau = T_0(v)P_{12}^{\tau_2}T_1^+(u+m-n) \left(T^+_2(u)^{-1}\right)^\tau\nonumber\\ = & T_0(v)P_{12}^{\tau_2}z^+(u)=P_{12}^{\tau_2}T_0(v)z^+(u).\label{idd1} \end{align}\tag{38}\] As for the right-hand side of 37 , using Lemma 10 and then 35 we get \[\begin{align} & P_{12}^{\tau_2} z^+(u) T_0(v) \left( {\overline{R}}_{02}(x -C )^{\tau_2}\right)^{-1} {\overline{R}}_{01}(x-m+n-C ) \nonumber\\ =&z^+(u) T_0(v)P_{12}^{\tau_2} \left( {\overline{R}}_{02}(x -C )^{\tau_2}\right)^{-1} {\overline{R}}_{01}(x-m+n-C )\nonumber\\ =& z^+(u) T_0(v)P_{12}^{\tau_2} = P_{12}^{\tau_2} z^+(u) T_0(v).\label{idd2} \end{align}\tag{39}\] Due to 37 , the expressions in 38 and 39 coincide, which implies the first equality in 36 . Regarding the second equality, it is verified by an analogous argument, relying on Lemma 10 and 35 . However, instead of 37 , its proof starts with the identity \[\begin{align} &P_{12}^{\tau_2} \left( {\overline{R}}_{02}(y)^{\tau_2}\right)^{-1} {\overline{R}}_{01}(y-m+n ) T^+_0(v)T_1 (u+m-n) \left(T_2 (u)^{-1}\right)^\tau \\ =& P_{12}^{\tau_2} T_1 (u+m-n) \left(T_2 (u)^{-1}\right)^\tau T^+_0(v) \left( {\overline{R}}_{02}(y +C )^{\tau_2}\right)^{-1} {\overline{R}}_{01}(y-m+n+C ) \end{align}\] with \(y=-u+v-C/2\), which again follows from the defining relation 28 . \(\qed\)

We now give a simple application of Theorem 12. Introduce the extended vacuum module \(\widetilde{\rm V}_{c}(\mathfrak{gl}_{m|n})\) at the level \(c\in\mathbb{C}\) as the quotient of the algebra \(\widetilde{{\rm DY}}(\mathfrak{gl}_{m|n})\) by its left ideal generated by \(C-c\cdot 1\) and the elements \(t_{ij}^{(r)}\), where \(i,j=1,\ldots ,m+n\) and \(r=1,2,\ldots .\) Denote by \(\mathop{\mathrm{\boldsymbol{1}}}\) the image of the unit \(1\in \widetilde{{\rm DY}}(\mathfrak{gl}_{m|n})\) in the extended vacuum module. Let \[\mathfrak{z} (\widetilde{\rm V}_{c}(\mathfrak{gl}_{m|n})) =\left\{ v\in \widetilde{\rm V}_{c}(\mathfrak{gl}_{m|n})\,:\, t_{ij}^{(r)}v=0\text{ for all }i,j=1,\ldots ,m+n,\, r=1,2,\ldots \right\}\] be the subspace of invariants of \(\widetilde{\rm V}_{c}(\mathfrak{gl}_{m|n})\). Theorem 12 implies

At the end, we point out some similarities and differences between the series \(z (u)\) and \(z^+(u)\). First of all, we remark that Lemmas 10, 11 and the following discussion hold for \(m=n\) as well. As we recall in Section 2.1, the coefficients of \(z (u)\) generate the centre of the Yangian. However, its dual Yangian counterpart does not exhibit such a property. To see this, write \(t^+(u)=1-T^+(u)\). Using the formal Taylor Theorem we find \[T_1^+(u+m-n)\left(T_2^+(u)^{-1}\right)^{\tau} =\left( 1-\sum_{r\geqslant 0} \frac{(m-n)^r}{r!}\frac{d^r}{du^{r}}t_1^+(u) \right) \left( \sum_{r\geqslant 0} t_2^+(u)^r \right)^\tau .\] Note that for all \(r\geqslant 1\) the degree of the coefficient of \(u^{r-1}\) in \(\frac{d^r}{du^{r}}t_1^+(u)\) and in \(t_2^+(u)^r\) is less than or equal to \(-r-1\). Furthermore, the degree of the coefficient of \(u^{r-1}\) in \(t_1^+(u)t_2^+(u)^\tau\) is \(-r-2\). Thus, the above expression is of the form \[T_1^+(u+m-n)\left(T_2^+(u)^{-1}\right)^{\tau} = 1-t_1^+(u)+t_2^+(u)^\tau+\text{lower degree terms}.\] As \(P^{\tau_2}t_1^+(u)=P^{\tau_2}t_2^+(u)^\tau\), this implies that the degree of the coefficient \(z^{-r}\) in \[z^+(u)=1-\sum_{r\geqslant 1}z^{(-r)} u^{r-1}\] does not exceed \(-r-1\). Hence, in particular, the coefficients of the series \(z^+(u)\) do not contain any central elements of degree \(-1\). On the other hand, they are algebraically independent. Indeed, extend the degree function \(\deg_2\) for the dual Yangian to the extended algebra \(\widetilde{{\rm Y}}^+(\mathfrak{gl}_{m|n})\) by allowing it to take the infinite value. Then the elements of finite degree form a subalgebra which we denote by \(\widetilde{{\rm Y}}^+(\mathfrak{gl}_{m|n})_{\text{fin}}\). All coefficients \(z^{(-r)}\) belong to \(\widetilde{{\rm Y}}^+(\mathfrak{gl}_{m|n})_{\text{fin}}\) and their images \(\bar{z}^{(-r)}\) in \(\mathop{\mathrm{gr}}_2 \widetilde{{\rm Y}}^+(\mathfrak{gl}_{m|n})_{\text{fin}}\cong {\rm U}(t^{-1}\mathfrak{gl}_{m|n}[t^{-1}])\) are given by \[\bar{z}^{(-r)}=r\sum_{i=1}^{m+n} e_{ii}\otimes t^{-r-1}\quad \text{for all }r\geqslant 1.\]

3 Reflection algebras for the Lie superalgebra \(\mathfrak{gl}_{m|n}\)↩︎

In this section, we study reflection algebras, a certain class of left coideal subalgebras of the extended double Yangian at the level \(0\). In particular, we give a presentation of such algebras by generators and relations.

3.1 Reflection subalgebras of the double Yangian↩︎

For any \(\ell=1,\ldots ,m+n\) let \(G=(g_{ij})\) be the diagonal matrix of order \(m+n\), \[\label{gmatricca} G=\mathop{\mathrm{diag}}(\varepsilon_1,\ldots, \varepsilon_{m+n}), \qquad\text{where}\qquad \varepsilon_i=\begin{cases} 1&\text{for }i=1,\ldots ,\ell,\\ -1&\text{for }i=\ell+1,\ldots,m+n. \end{cases}\tag{40}\] Introduce the matrices \[\label{bbplus} B(u)=T(u )GT(-u)^{-1}\qquad\text{and}\qquad B^+(u)=T^+(u)GT^+(-u)^{-1}\tag{41}\] and denote by \(b_{ij}(u)\) and \(b^+_{ij}(u)\) their matrix entries, \[\label{serieess} B(u)=\sum_{i,j=1}^{m+n} (-1)^{\bar{i}\bar{j}+\bar{j}} e_{ij}\otimes b_{ij}(u) \qquad\text{and}\qquad B^+(u)=\sum_{i,j=1}^{m+n} (-1)^{\bar{i}\bar{j}+\bar{j}} e_{ij}\otimes b^+_{ij}(u).\tag{42}\] Finally, let \(b_{ij}^{(r)}\) and \(b_{ij}^{(-r)}\) be the coefficients of the series \(b_{ij}(u)\in {\rm Y}(\mathfrak{gl}_{m|n})[[u^{-1}]]\) and \(b^+_{ij}(u)\in \widetilde{\rm Y}^+(\mathfrak{gl}_{m|n})[[u]]\), so that we have \[\label{seriees} b_{ij}(u)=g_{ij}+\sum_{r\geqslant 1} b_{ij}^{(r)} u^{-r} \qquad\text{and}\qquad b^+_{ij}(u)=g_{ij}-\sum_{r\geqslant 1} b_{ij}^{(-r)} u^{r-1}.\tag{43}\] From now on, we regard \(b_{ij}^{(r)}\) and \(b_{ij}^{(-r)}\) as elements of the extended double Yangian at the level \(0\). Using the \(RTT\)-relations 7 , 13 and 19 , along with the identity \[R(u)G_1 R(v)G_2=G_2 R(v) G_1 R(u),\] one can show that the matrices 41 satisfy the reflection relations \[\begin{align} R(u-v)B_1(u)R(u+v )B_2(v) &=B_2(v)R(u+v )B_1(u)R(u-v),\tag{44}\\ R(u-v)B^+_1(u)R(u+v)B^+_2(v) &=B^+_2(v)R(u+v)B^+_1(u)R(u-v),\tag{45}\\ R(u-v )B_1(u)R(u+v )B^+_2(v) &=B^+_2(v)R(u+v )B_1(u)R(u-v ).\tag{46} \end{align}\] Furthermore, they possess the unitarity properties \[\label{unidentities} B(u)B(-u)=1 \qquad\text{and}\qquad B^+(u)B^+(-u)=1.\tag{47}\]

Extend the degree function \(\deg_2\) for the double Yangian, as defined by 8 , 15 and \(\deg_2 C=0\), to the algebra \(\widetilde{{\rm DY}}_0(\mathfrak{gl}_{m|n})\) by allowing it to take the infinite value. Then the elements of finite degree form a subalgebra which we denote by \(\widetilde{{\rm DY}}_0(\mathfrak{gl}_{m|n})_{\text{fin}}\). One easily checks that \(b_{ij}^{(r)}\) and \(b_{ij}^{(-r)}\) belong to \(\widetilde{{\rm DY}}_0(\mathfrak{gl}_{m|n})_{\text{fin}}\). Let \({\rm DB} (\mathfrak{gl}_{m|n})\) be its unital subalgebra generated by all elements \(b_{ij}^{(r)}\) and \(b_{ij}^{(-r)}\). Moreover, let \({\rm B}(\mathfrak{gl}_{m|n})\) (resp. \({\rm B}^+(\mathfrak{gl}_{m|n})\)) be its unital subalgebra generated by all \(b_{ij}^{(r)}\) (resp. \(b_{ij}^{(-r)}\)) with \(r=1,2,\ldots.\) We shall refer to all these subalgebras as reflection algebras.

Consider the families \(I_0,I_1\subset \left\{1,\ldots ,m+n\right\}^{\times 2}\) of pairs of indices \[\begin{align} &I_0=\left\{(i,j)\,:\,1\leqslant i,j\leqslant\ell\text{ or }\ell +1\leqslant i,j\leqslant m+n\right\},\\ &I_1=\left\{(i,j)\,:\,1\leqslant i \leqslant\ell <j\leqslant m+n\text{ or }1\leqslant j \leqslant\ell <i\leqslant m+n\right\}. \end{align}\] Define the families of elements \[\begin{align} &\Gamma =\left\{b_{ij}^{(2r-1)}\,:\, (i,j)\in I_0\text{ and }r\geqslant 1\right\} \cup \left\{b_{ij}^{( 2r )}\,:\, (i,j)\in I_1\text{ and }r\geqslant 1\right\} \subset {\rm B}(\mathfrak{gl}_{m|n}),\\ &\Gamma^+=\left\{b_{ij}^{(-2r)}\,:\, (i,j)\in I_0\text{ and }r\geqslant 1\right\}\cup \left\{b_{ij}^{(-2r+1)}\,:\, (i,j)\in I_1\text{ and }r\geqslant 1\right\} \subset {\rm B}^+(\mathfrak{gl}_{m|n}). \end{align}\] By a direct computation we obtain from 41 the equalities \[\label{slikeizo} b_{ij}^{(r)}=\left((-1)^{r-1}\varepsilon_i +\varepsilon_j\right)t_{ij}^{(r)}+\ldots \qquad\text{and}\qquad b_{ij}^{(-r)}=((-1)^r\varepsilon_i +\varepsilon_j)t_{ij}^{(-r)}+\ldots\tag{48}\] for all \(i,j=1,\ldots ,m+n\) and \(r=1,2,\ldots ,\) where the ellipses stand for the lower degree terms, i.e the terms of degree strictly less than \(r-1\) (resp. \(-r\)), of parity \(\bar{i}+\bar{j}\). Hence, in particular, all elements \(b_{ij}^{(\pm r)}\) are homogeneous. Moreover, for all \(b_{ij}^{(\pm r)}\in \Gamma\cup \Gamma^+\) we have \[\label{stupnjevvi} \deg_2b_{ij}^{(r)}=r-1\quad\text{and}\quad \deg_2b_{ij}^{(-r)}=-r,\tag{49}\] while for the remaining elements \(b_{ij}^{(\pm r)}\not\in \Gamma\cup \Gamma^+\) we have \[\label{stupnjevvi2} \deg_2b_{ij}^{(r)}<r-1\quad\text{and}\quad \deg_2b_{ij}^{(-r)}<-r.\tag{50}\] Introduce the following subsets of \(\mathop{\mathrm{gr}}_2 \widetilde{{\rm DY}}_0(\mathfrak{gl}_{m|n})_{\text{fin}} =\mathop{\mathrm{gr}}_2 {\rm DY}_0(\mathfrak{gl}_{m|n})\cong {\rm U}(\mathcal{L}(\mathfrak{gl}_{m|n}))\): \[\overline{\Gamma}=\left\{\bar{b}_{ij}^{(r)}\,:\, b_{ij}^{(r)}\in \Gamma\right\} \quad\text{and}\quad \overline{\Gamma}^+=\left\{\bar{b}_{ij}^{(-r)}\,:\, b_{ij}^{(-r)}\in \Gamma^+\right\}.\]

Proof. Let us prove that the family \(\overline{\Gamma}^+\) generates \(\mathop{\mathrm{gr}}_2 {\rm B}^+(\mathfrak{gl}_{m|n})\). By extracting the constant term of the second identity in 47 we find \[\label{lkmbghdf} \left(\varepsilon_i +\varepsilon_j\right) b_{ij}^{(-1)} =\sum_{a=1}^{\ell}b_{ia}^{(-1)}b_{aj}^{(-1)} + \sum_{a=\ell+1}^{m+n}b_{ia}^{(-1)}b_{aj}^{(-1)}.\tag{51}\] Therefore, if \(b_{ij}^{(-1)}\) does not belong to \(\Gamma^+\), we see by 49 and 50 that the degree of the right-hand side of 51 is \(-2\). Moreover, we have the equality \[\bar{b}_{ij}^{(-1)} =\begin{cases} \frac{1}{2}\sum_{a=\ell+1}^{m+n}\bar{b}_{ia}^{(-1)}\bar{b}_{aj}^{(-1)},&\text{if }1\leqslant i,j\leqslant \ell,\\ -\frac{1}{2}\sum_{a=1}^{\ell}\bar{b}_{ia}^{(-1)}\bar{b}_{aj}^{(-1)},&\text{if }\ell<i,j\leqslant m+n \end{cases}\] in the corresponding graded algebra \(\mathop{\mathrm{gr}}_2 {\rm B}^+(\mathfrak{gl}_{m|n})\). In other words, we proved that all elements \(\bar{b}_{ij}^{(-1)}\) belong to the subalgebra generated by \(\overline{\Gamma}^+\).

By extracting the coefficient of \(u\) in the second identity in 47 we find \[\label{lkmbghdf2} \left(\varepsilon_i -\varepsilon_j\right) b_{ij}^{(-2)} = \sum_{a=1}^{\ell}b_{ia}^{(-1)}b_{aj}^{(-2)} + \sum_{a=\ell+1}^{m+n}b_{ia}^{(-1)}b_{aj}^{(-2)} -\sum_{a=1}^{\ell}b_{ia}^{(-2)}b_{aj}^{(-1)} - \sum_{a=\ell+1}^{m+n}b_{ia}^{(-2)}b_{aj}^{(-1)}.\tag{52}\] Therefore, if \(b_{ij}^{(-2)}\) does not belong to \(\Gamma^+\), we see by 49 and 50 that the degree of the right-hand side of 52 is \(-3\). Moreover, we have the equality \[\bar{b}_{ij}^{(-2)} =\begin{cases} -\frac{1}{2}\sum_{a=1}^{\ell}\bar{b}_{ia}^{(-2)}\bar{b}_{aj}^{(-1)} +\frac{1}{2} \sum_{a=\ell+1}^{m+n}\bar{b}_{ia}^{(-1)}\bar{b}_{aj}^{(-2)},&\text{if }1\leqslant i \leqslant\ell <j\leqslant m+n,\\ \frac{1}{2}\sum_{a=\ell+1}^{m+n}\bar{b}_{ia}^{(-2)}\bar{b}_{aj}^{(-1)} -\frac{1}{2}\sum_{a=1}^{\ell}\bar{b}_{ia}^{(-1)}\bar{b}_{aj}^{(-2)},&\text{if }1\leqslant j \leqslant\ell <i\leqslant m+n \end{cases}\] in \(\mathop{\mathrm{gr}}_2 {\rm B}^+(\mathfrak{gl}_{m|n})\). Thus, all elements \(\bar{b}_{ij}^{(-2)}\) belong to the subalgebra generated by \(\overline{\Gamma}^+\).

Suppose that the elements \(\bar{b}_{ij}^{(-r)}\), where \(i,j=1,\ldots ,m+n\) and \(r=1,\ldots,s\), belong to the subalgebra generated by \(\overline{\Gamma}^+\). Then, by arguing as above, one can show that all elements \(\bar{b}_{ij}^{(-s-1)}\) belong to the subalgebra generated by \(\overline{\Gamma}^+\) as well. Hence, we conclude by induction that the \(\overline{\Gamma}^+\) generates the entire algebra \(\mathop{\mathrm{gr}}_2 {\rm B}^+(\mathfrak{gl}_{m|n})\).

The fact that the family \(\overline{\Gamma}\) generates \(\mathop{\mathrm{gr}}_2 {\rm B}(\mathfrak{gl}_{m|n})\) can be verified by suitably modifying the above arguments and employing the first identity in 47 . In fact, the constant term of \(B(u)\) is 1, which slightly simplifies the proof in this case, so we omit it. The analogous statement for \(\mathop{\mathrm{gr}}_1 {\rm B}(\mathfrak{gl}_{m|0})\) was established by Molev and Ragoucy; see [24]. \(\qed\)

Consider the involutive automorphism \(\sigma\) of \(\mathfrak{gl}_{m|n}\) given by \[\begin{align} \sigma\colon e_{ij} \mapsto \varepsilon_i\varepsilon_j e_{ij},\quad\text{where } i,j=1,\ldots ,m+n. \end{align}\] It induces the decomposition \(\mathfrak{gl}_{m|n}=\mathfrak{gl}_{m|n} (-1) \oplus\mathfrak{gl}_{m|n} (1)\), where \(\mathfrak{gl}_{m|n} (\pm 1)\) denotes the eigenspace of \(\sigma\) corresponding to the eigenvalue \(\pm 1\). Let \(\mathfrak{gl}_{m|n}[t,t^{-1}]^{\sigma} \subset \widehat{\mathfrak{gl}}_{m|n}\) be the Lie superalgebra of all Laurent polynomials of the form \[\sum_{i=-p}^q a_i \otimes t^i ,\quad\text{where }p,q\in\mathbb{Z}_{\geqslant 0},\,a_{i}\in \mathfrak{gl}_{m|n}((-1)^i) .\] By using the isomorphism 30 and the identities 48 we find \[\bar{b}_{ij}^{(r)}=(-1)^{\bar{i}}\left((-1)^{r-1}\varepsilon_i +\varepsilon_j\right)e_{ij}(r-1) \quad\text{and}\quad \bar{b}_{ij}^{(-r)}=(-1)^{\bar{i}}((-1)^r\varepsilon_i +\varepsilon_j)e_{ij} (-r)\] for all \(b_{ij}^{(\pm r)}\in \Gamma\cup \Gamma^+\). Therefore, by combining Theorems 9 and 14 we obtain

The next proposition is a generalization of [24], which states that, in the even case, i.e. for \(n=0\), the subalgebra of the Yangian \({\rm Y}(\mathfrak{gl}_{m })={\rm Y}(\mathfrak{gl}_{m|0})\) generated by all \(b_{ij}^{(r)}\), where \(i,j=1,\ldots ,m\) and \(r=1,2,\ldots ,\) is a left coideal in \({\rm Y}(\mathfrak{gl}_{m})\).

Proof. It is sufficient to check that the images of the generators \(b_{ij}^{(\pm r)}\) under the coproduct \(\Delta\) belong to \(\widetilde{{\rm DY}}_0(\mathfrak{gl}_{m|n})\otimes {\rm DB} (\mathfrak{gl}_{m|n})\). As with the aforementioned result of Molev and Ragoucy, this is proved by a direct calculation which relies on the explicit formulae 22 for the coproduct. First, as the coproduct maps \(T^+(u)T^+(u)^{-1}\) to \(1\), we derive from the second formula in 22 that for all \(i,j\) we have \[\Delta (t'^+_{ij}(u))=\sum_{k=1}^{m+n}t'^+_{kj}(u)\otimes t'^+_{ik}(u),\qquad\text{where}\qquad T^+(u)^{-1}=\sum_{i,j=1}^{m+n} (-1)^{\bar{i}\bar{j}+\bar{j}}e_{ij}\otimes t'^+_{ij}(u).\] Hence, using the second formula in 41 , we get \[\begin{align} \Delta(b_{ij}^+(u)) &=\Delta\left(\sum_{k=1}^{m+n}\varepsilon_k t_{ik}^+(u)t'^+_{kj}(-u)\right) = \sum_{k,r,s=1}^{m+n}\varepsilon_kt_{ir}^+(u)t'^+_{sj}(u)\otimes t_{rk }^+(u)t'^+_{ks}(-u) \\ &= \sum_{ r,s=1}^{m+n} t_{ir}^+(u)t'^+_{sj}(u)\otimes\sum_{ k=1}^{m+n}t_{rk }^+(u)\varepsilon_kt'^+_{ks}(-u) =\sum_{ r,s=1}^{m+n} t_{ir}^+(u)t'^+_{sj}(u)\otimes b^+_{rs}( u), \end{align}\] which belongs to \(\widetilde{{\rm DY}}_0(\mathfrak{gl}_{m|n})\otimes {\rm DB} (\mathfrak{gl}_{m|n})[[u]]\), as required. The statement for the series \(b_{ij}(u)\) is verified analogously; cf. [24]. \(\qed\)

The proof of Proposition 17 implies

3.2 Presentation of reflection algebras↩︎

Let \(G=(g_{ij})\) be the diagonal matrix as in 40 . Define \(\mathcal{ DB} (\mathfrak{gl}_{m|n})\) as the \(\mathbb{Z}_2\)-graded unital associative algebra generated by the elements \(\beta_{ij}^{(r)}\) and \(\beta_{ij}^{(-r)}\), where \(i,j=1,\ldots ,m+n\) and \(r=1,2,\ldots .\) The parity of \(\beta_{ij}^{(\pm r)}\) is \(\bar{i}+\bar{j}\) and the generators are subject to the defining relations which are given as follows. First, we introduce the series \[\beta_{ij}(u)=g_{ij}+\sum_{r\geqslant 1} \beta_{ij}^{(r)} u^{-r} \qquad\text{and}\qquad \beta^+_{ij}(u)=g_{ij}-\sum_{r\geqslant 1} \beta_{ij}^{(-r)} u^{r-1}\] with \(i,j=1,\ldots ,m+n\). Next, we organize the series into matrices \[\mathcal{B}(u)=\sum_{i,j=1}^{m+n} (-1)^{\bar{i}\bar{j}+\bar{j}} e_{ij}\otimes\beta_{ij}(u) \qquad\text{and}\qquad \mathcal{B}^+(u)=\sum_{i,j=1}^{m+n} (-1)^{\bar{i}\bar{j}+\bar{j}} e_{ij}\otimes\beta^+_{ij}(u).\] Finally, the defining relations consist of three reflection equations, \[\begin{align} R(u-v)\mathcal{B}_1(u)R(u+v )\mathcal{B}_2(v) &=\mathcal{B}_2(v)R(u+v )\mathcal{B}_1(u)R(u-v),\tag{53}\\ R(u-v)\mathcal{B}^+_1(u)R(u+v)\mathcal{B}^+_2(v) &=\mathcal{B}^+_2(v)R(u+v)\mathcal{B}^+_1(u)R(u-v),\tag{54}\\ R(u-v )\mathcal{B}_1(u)R(u+v )\mathcal{B}^+_2(v) &=\mathcal{B}^+_2(v)R(u+v )\mathcal{B}_1(u)R(u-v ) \tag{55} \end{align}\] and two unitarity conditions, \[\label{bunidentities} \mathcal{B}(u)\mathcal{B}(-u )=1 \qquad\text{and}\qquad \mathcal{B}^+(u)\mathcal{B}^+(-u)=1.\tag{56}\]

Denote by \(\mathcal{ B} (\mathfrak{gl}_{m|n})\) (resp. \(\mathcal{ B}^+ (\mathfrak{gl}_{m|n})\)) a unital subalgebra of \(\mathcal{ DB} (\mathfrak{gl}_{m|n})\) generated by all \(\beta_{ij}^{(r)}\) (resp. \(\beta_{ij}^{(-r)}\)) with \(i,j=1,\ldots ,m+n\) and \(r=1,2,\ldots.\) We shall need the following analogues of the families \(\Gamma\) and \(\Gamma^+\): \[\begin{align} &\mathcal{G} =\left\{\beta_{ij}^{(2r-1)}\,:\, (i,j)\in I_0\text{ and }r\geqslant 1\right\} \cup \left\{b_{ij}^{( 2r )}\,:\, (i,j)\in I_1\text{ and }r\geqslant 1\right\} \subset \mathcal{ B} (\mathfrak{gl}_{m|n}),\\ &\mathcal{G}^+=\left\{\beta_{ij}^{(-2r)}\,:\, (i,j)\in I_0\text{ and }r\geqslant 1\right\}\cup \left\{b_{ij}^{(-2r+1)}\,:\, (i,j)\in I_1\text{ and }r\geqslant 1\right\}\subset \mathcal{ B}^+ (\mathfrak{gl}_{m|n}). \end{align}\] Consider the ascending filtration over \(\mathcal{ DB} (\mathfrak{gl}_{m|n})\) defined by the degree operator \(\deg_2\), \[\deg_2\beta_{ij}^{(r)}=r-1 \quad\text{and}\quad\deg_2\beta_{ij}^{(-r)}=-r\] for all \(i,j=1,\ldots ,m+n\) and \(r=1,2,\ldots .\) As before, we write \(\bar{\beta}_{ij}^{(r)}\) for the image of the generator \(\beta_{ij}^{(r)}\) in the respective component of the corresponding graded algebra \(\mathop{\mathrm{gr}}_2 \mathcal{ DB} (\mathfrak{gl}_{m|n})\) and we use the notation \[\overline{\mathcal{G}}=\left\{\bar{\beta}_{ij}^{(r)}\,:\,\beta_{ij}^{(r)}\in \mathcal{G}\right\} \quad\text{and}\quad \overline{\mathcal{G}}^+=\left\{\bar{\beta}_{ij}^{(-r)}\,:\, \beta_{ij}^{(-r)}\in \mathcal{G}^+\right\}.\]

By comparing the relations 44 –47 and 53 –56 , we conclude that the assignments \(\beta_{ij}^{(\pm r)}\mapsto b_{ij}^{(\pm r)}\), where \(i,j=1,\ldots ,m+n\) and \(r\geqslant 1\), define the algebra epimorphism \[\label{bbmapY} \mathcal{ DB} (\mathfrak{gl}_{m|n})\to {\rm DB} (\mathfrak{gl}_{m|n}).\tag{57}\] Furthermore, the map given by 57 is filtration-preserving and it gives rise to the epimorphism of the corresponding graded algebras \[\label{grbbmapY} \textstyle\mathop{\mathrm{gr}}_2 \mathcal{ DB} (\mathfrak{gl}_{m|n})\to \mathop{\mathrm{gr}}_2 {\rm DB} (\mathfrak{gl}_{m|n}).\tag{58}\] Indeed, the surjectivity follows by the last assertion of Theorem 14. Our goal is to show that 57 is an isomorphism. In order to do so, we shall need the next two lemmas.

Proof. In parallel with the proof of Theorem 14, one can employ the first (resp. second) unitarity identity in 56 to show that the family \(\overline{\mathcal{G}}\) (resp. \(\overline{\mathcal{G}}^+\)) generates the algebra \(\mathop{\mathrm{gr}}_2 \mathcal{ B} (\mathfrak{gl}_{m|n})\) (resp. \(\mathop{\mathrm{gr}}_2 \mathcal{ B}^+ (\mathfrak{gl}_{m|n})\)). Hence, any element of \(\mathop{\mathrm{gr}}_2 \mathcal{ B} (\mathfrak{gl}_{m|n})\) (resp. \(\mathop{\mathrm{gr}}_2 \mathcal{ B}^+ (\mathfrak{gl}_{m|n})\)) can be written as a linear combination of monomials in the elements of \(\overline{\mathcal{G}}\) (resp. \(\overline{\mathcal{G}}^+\)). From now on we refer to them more briefly as monomials in \(\overline{\mathcal{G}}\) (resp. \(\overline{\mathcal{G}}^+\)). To prove that it is sufficient to consider ordered monomials only, one needs to employ the corresponding reflection equation 53 or 54 . We sketch the proof in the case of \(\mathop{\mathrm{gr}}_2 \mathcal{ B}^+ (\mathfrak{gl}_{m|n})\). The analogous arguments can be applied to \(\mathop{\mathrm{gr}}_2 \mathcal{ B} (\mathfrak{gl}_{m|n})\) as well.

Clearly, it is sufficient to check that any monomial in \(\overline{\mathcal{G}}^+\) can be written as a linear combination of ordered monomials in \(\overline{\mathcal{G}}^+.\) Suppose \(\mu\) is a monomial in \(\mathcal{G}^+\) of the form \[\mu =\mu_1\beta_{ij}^{(-r)}\beta_{kl}^{(-s)}\mu_2\] such that \(\bar{\beta}_{kl}^{(-s)}\) precedes \(\bar{\beta}_{ij}^{(-r)}\) with respect to the chosen ordering (so that \(\bar{\mu}\) is not ordered), while \(\mu_1\) and \(\mu_2\) are some monomials in \(\mathcal{G}^+\). By extracting the coefficients of \(u^{r-1}v^{s-1}\) of the matrix entry \(e_{ij}\otimes e_{kl}\) in 54 we get \[\label{imgaeof} \beta_{ij}^{(-r)}\beta_{kl}^{(-s)}= \pm\beta_{kl}^{(-s)}\beta_{ij}^{(-r)} +\gamma_1 +\gamma_2,\tag{59}\] where the sign \(\pm\) on the right-hand side depends on the parity of the given generators, \(\gamma_1\) stands for a linear combination of some elements \(\beta_{pq}^{(-t)}\) such that \(-t\leqslant -r-s\) and \(\gamma_2\) is a linear combination of some monomials of length two, \(\beta_{p_1q_1}^{(-t_1)}\beta_{p_2 q_2}^{(-t_2)}\) such that \(-t_1-t_2 < -r-s\). Hence, by taking the image of 59 in the \((-r-s)\)-component of the corresponding graded algebra, we express \(\bar{\mu}\) as a linear combination \[\label{ertz6} \bar{\mu}= \pm\bar{\mu}_1\bar{\beta}_{kl}^{(-s)}\bar{\beta}_{ij}^{(-r)}\bar{\mu}_2 + \bar{\mu}_1\bar{\gamma_1}\bar{\mu}_2.\tag{60}\] Observe that the length of the monomials which appear in the linear combination \(\bar{\mu}_1\bar{\gamma_1}\bar{\mu}_2\) is strictly less than the length of \(\bar{\mu}\). Therefore, we can continue to apply such a procedure, now starting with the monomials on the right-hand side of 60 , until, after finitely many steps, we obtain a linear combination of ordered monomials in \(\overline{\mathcal{G}}^+\), as required. \(\qed\)

In the next lemma, we make use of the remaining reflection equation 55 .

Proof. It is sufficient to prove that any element \(z\in\mathop{\mathrm{gr}}_2 \mathcal{ DB} (\mathfrak{gl}_{m|n})\) can be written as a linear combination of elements of the form \(xy\) with \(x\in\mathop{\mathrm{gr}}_2 \mathcal{ B}^+ (\mathfrak{gl}_{m|n})\) and \(y\in\mathop{\mathrm{gr}}_2 \mathcal{ B} (\mathfrak{gl}_{m|n})\). Indeed, the assertion of the lemma then follows by applying Lemma 19 to \(x\) and \(y\). Moreover, by Lemma 19, the set \(\overline{\mathcal{G}}\cup\overline{\mathcal{G}}^+\) generates \(\mathop{\mathrm{gr}}_2 \mathcal{ DB} (\mathfrak{gl}_{m|n})\) so we can assume without loss of generality that \(z\) is a monomial in \(\overline{\mathcal{G}}\cup\overline{\mathcal{G}}^+\). As in the proof of Lemma 19, we shall describe a procedure which one can employ to express \(z\) in such a way.

Suppose \(\mu\) is a monomial in \(\mathcal{G} \cup\mathcal{G}^+\) of the form \[\mu =\mu_1\beta_{ij}^{(r)}\beta_{kl}^{(-s)}\mu_2,\] where \(i,j,k,l=1,\ldots ,m+n\), \(r,s=1,2,\ldots\) and \(\mu_1,\mu_2\) are some monomials in \(\mathcal{G} \cup\mathcal{G}^+\). As \(\bar{\beta}_{kl}^{(-s)}\) precedes \(\bar{\beta}_{ij}^{(r)}\) with respect to the chosen ordering, the monomial \(\bar{\mu}\) is not ordered. The \(R\)-matrices \(R(u\pm v)\), which appear in the reflection equation 55 , are of the form \[R(u\pm v )=1-\frac{P}{u}\sum_{l\geqslant 1}(\mp 1)^l\frac{v^l}{u^{l}} .\] For \(r,s\geqslant 2\) the degree of the coefficient of \(u^{-r}v^{s-1}\) in \(\frac{v^l}{u^l}\beta_{ij}^{kl}(u,v)\) (resp. \(\frac{1}{u}\beta_{ij}^{kl}(u,v)\)), where \[\beta_{ij}^{kl}(u,v)=\left(\beta_{ij}(u)-g_{ij}\right)\left(g_{kl}-\beta_{kl}^+(v)\right),\] is less than or equal to \(r-s-1\) (resp. \(r-s-2\)). Using these observations and extracting the coefficients of \(u^{-r}v^{s-1}\) of the matrix entry \(e_{ij}\otimes e_{kl}\) in 55 , we get \[\label{imgaeof2} \beta_{ij}^{(r)}\beta_{kl}^{(-s)}+\gamma_1+\gamma_2 = \pm\beta_{kl}^{(-s)}\beta_{ij}^{(r)}+\delta_1+\delta_2,\tag{61}\] where the sign \(\pm\) on the right-hand side depends on the parity of the given generators, \(\gamma_1,\delta_1\) are linear combinations of the elements \(\beta_{pq}^{(\pm t)}\) of degree less than or equal to \(r-s-1\) and \(\gamma_2,\delta_2\) are linear combinations of some monomials in \(\mathcal{G} \cup \mathcal{G}^+\) of degree less than or equal to \(r-s-2\). Hence, by taking the image of 61 in the \((r-s-1)\)-component of the corresponding graded algebra, we express \(\bar{\mu}\) as a linear combination \[\label{nuredjenimnomii} \bar{\mu}=\pm \bar{\mu}_1\bar{\beta}_{kl}^{(-s)}\bar{\beta}_{ij}^{(r)}\bar{\mu}_2 +\bar{\mu}_1\bar{\delta}_1\bar{\mu}_2 -\bar{\mu}_1\bar{\gamma_1}\bar{\mu}_2.\tag{62}\] As with the proof of Lemma 19, we can continue to apply such a procedure, now starting with the monomials on the right-hand side of 62 , until, after finitely many steps, we obtain a linear combination of the elements of the form \(xy\) with \(x\in\mathop{\mathrm{gr}}_2 \mathcal{ B}^+ (\mathfrak{gl}_{m|n})\) and \(y\in\mathop{\mathrm{gr}}_2 \mathcal{ B} (\mathfrak{gl}_{m|n})\), as required. \(\qed\)

We are now ready to prove the main result of this section.

Proof. It is sufficient to check that the map 57 is injective. Therefore, let us assume that its kernel contains some nonzero element \(x\). Then the image of \(\bar{x}\in \mathop{\mathrm{gr}}_2 \mathcal{ DB} (\mathfrak{gl}_{m|n})\) under the map 58 is trivial, which leads to contradiction. Indeed, by Lemma 20, the element \(\bar{x}\) can be expressed as a linear combination of ordered monomials in \(\overline{\mathcal{G}}\cup\overline{\mathcal{G}}^+\) with at most power 1 for odd generators. Hence, its image in \(\mathop{\mathrm{gr}}_2 {\rm DB} (\mathfrak{gl}_{m|n})\) under the map 58 is a linear combination of ordered monomials in \(\overline{\Gamma}\cup \overline{\Gamma}^+\) with at most power 1 for odd generators. However, by Corollary 15 and the Poincaré–Birkhoff–Witt Theorem for the enveloping algebra \({\rm U}(\mathfrak{gl}_{m|n}[t,t^{-1}]^{\sigma} )\), such a linear combination is nonzero. \(\qed\)

Acknowledgement↩︎

This work has been supported in part by Croatian Science Foundation under the project UIP-2019-04-8488.

References↩︎