1. Introduction

Let \(g\) be a complex Lie algebra. The Lie–Poisson bracket on the symmetric algebra \(S(g)\) is the unique Poisson bracket extending the Lie bracket,

figure 1

We suppose that \(\xi\) is an arbitrary element of the dual space \(g^*\) and let \(\bar\partial_\xi\) denote the constant vector field in the direction \(\xi\). We write \( \kern1.1pt\overline{\vphantom{C}\kern6.3pt}\kern-7.4pt C \) for the Poisson center of the symmetric algebra \(S(g)\). We define \( \kern1.1pt\overline{\vphantom{C}\kern6.3pt}\kern-7.4pt C _\xi\) as the algebra generated by the set \(\bigcup_{n=0}^\infty\bar\partial_\xi^n \kern1.1pt\overline{\vphantom{C}\kern6.3pt}\kern-7.4pt C \). Mishchenko and Fomenko [1] showed the following theorem.

Theorem 1.

The algebra \( \kern1.1pt\overline{\vphantom{C}\kern6.3pt}\kern-7.4pt C _\xi\) is Poisson commutative.

Vinberg [2] inquired whether the argument shift algebra \( \kern1.1pt\overline{\vphantom{C}\kern6.3pt}\kern-7.4pt C _\xi\) could be extended to a commutative subalgebra \(C_\xi\) of the universal enveloping algebra \(U(g)\). Nazarov and Olshanski [3] constructed the quantum argument shift algebra \(C_\xi\) for any regular semisimple \(\xi\) in terms of (i) the Yangian in the case \(g=gl_d(\mathbb C)\) and (ii) the twisted Yangians in the orthogonal and symplectic cases. Tarasov [4] constructed the same quantum argument shift algebra for \(g=gl_d(\mathbb C)\) via the symmetrization map. The quantum argument shift algebra \(C_\xi\) is also constructed via the Feigin–Frenkel center for (i) any simple complex Lie algebra \(g\) and any regular \(\xi\) [5], [6], and (ii) any simple complex Lie algebra of type \(A\) or \(C\) and any \(\xi\) [7], [8].

So far, the argument shift operator \(\bar\partial_\xi\) had not been quantized. Gurevich, Pyatov, and Saponov [9] defined the quantum derivations \(\partial^i_j\) on the universal enveloping algebra \(Ugl_d(\mathbb C)\). We found an explicit formula for the quantum derivations of appropriate elements [10] and showed a quantum analogue of the Mishchenko and Fomenko theorem [11].

In the following, we present an explicit formula for the quantum argument shifts of an arbitrary central element up to the second order (see Proposition 1). We also identify a reduced set of generators of the algebra generated by the quantum argument shifts up to the second order (see Corollary 1 and Theorem 5). This reduced set of generators provides an alternative to those given by Futorny and Molev [7]. Complex combinatorial formulas play an essential role here (see Theorem 4 and Proposition 4).

2. Preliminaries

We write \(\delta\) for the identity matrix and let \(x^{\mathrm T}\) be the transpose of a matrix \(x\). We suppose that \(d\) is a nonnegative integer and let \(M(d,A)\) denote the algebra of \(d\times d\) matrices with entries in an algebra \(A\). We write \(x^i_j\) for the \((i,j)\) element of a \(d\times d\) matrix \(x\) and

$$x^i=\begin{pmatrix} x^i_1\!&\!\ldots\!&\!x^i_d \end{pmatrix},\qquad x_j=\begin{pmatrix} x^1_j \\ \vdots \\ x^d_j \end{pmatrix}$$

for the \(i\)th row vector and the \(j\)th column vector of the matrix \(x\).

We define the generating matrix of the Lie algebra \(gl_d=gl_d(\mathbb C)\) as the \(d\times d\) matrix \(e\) composed of the indeterminates \(e^i_j\) (generators of the Lie algebra \(gl_d\)). The universal enveloping algebra of the Lie algebra \(gl_d\) is the quotient algebra

$$Ugl_d=\mathbb{C}\langle e^i_j\rangle/ \bigl(e^{i_1}_{j_1}e^{i_2}_{j_2}-e^{i_2}_{j_2}e^{i_1}_{j_1}-e^{i_2}_{j_1}\delta^{i_1}_{j_2}+\delta^{i_2}_{j_1}e^{i_1}_{j_2}\colon i_1,j_1,i_2,j_2=1,\ldots,d\bigr),$$

where \(\mathbb{C}\langle e^i_j\rangle\) denotes the free unital algebra on the indeterminates \(e^i_j\) and the denominator in the right-hand side denotes the ideal generated by the elements

$$\bigl\{e^{i_1}_{j_1}e^{i_2}_{j_2}-e^{i_2}_{j_2}e^{i_1}_{j_1}-e^{i_2}_{j_1}\delta^{i_1}_{j_2}+\delta^{i_2}_{j_1}e^{i_1}_{j_2}\colon i_1,j_1,i_2,j_2=1,\ldots,d\bigr\}.$$

The following relation holds in the universal enveloping algebra \(Ugl_d\):

$$ [(e^n)^{i_1}_{j_1},e^{i_2}_{j_2}]=[e^{i_1}_{j_1},(e^n)^{i_2}_{j_2}]= (e^n)^{i_2}_{j_1}\delta^{i_1}_{j_2}-\delta^{i_2}_{j_1}(e^n)^{i_1}_{j_2},\qquad n=0,1,2,\ldots{}\,.$$
(1)

This can be proved by induction.

Quantum derivations on the universal enveloping algebra \(Ugl_d\) were defined in [9]. We give a slightly modified definition of these operators as follows.

Definition 1.

The quantum derivations on the universal enveloping algebra \(Ugl_d\) are the matrix elements of a unique homomorphism of unital complex algebras

$$Ugl_d\to M(d,Ugl_d),\qquad x\mapsto\partial x$$

such that \(\partial \operatorname{tr} (\xi e)= \operatorname{tr} (\xi e)+\xi\) for any numerical matrix \(\xi\).

We define the polynomials

$$f^{(n)}_\pm(x)=\sum_{m=0}^{n+1}\frac{1\pm(-1)^{n-m}}2\binom{n}mx^m.$$

The following theorem is proved in [10].

Theorem 2.

The quantum derivations of the matrix elements \((e^n)^i_j\) are given by

$$\begin{aligned} \, \partial(e^n)^i_j&=\sum_{m=0}^{n}\bigl(f^{(n-m-1)}_{+}(e)_j(e^m)^i+f^{(n-m-1)}_{-}(e)(e^m)^i_j\bigr)= \\ &=\sum_{m=0}^{n}\bigl((e^m)_jf^{(n-m-1)}_{+}(e)^i+(e^m)^i_jf^{(n-m-1)}_{-}(e)\bigr). \end{aligned}$$

We write \(C\) for the center of the universal enveloping algebra \(Ugl_d\). The center \(C\) is generated by the elements \( \operatorname{tr} e, \operatorname{tr} e^2,\ldots{}\,\).

We suppose that \(\xi\) is an arbitrary numerical matrix. The map \(\partial_\xi= \operatorname{tr} (\xi\partial)\) is called the quantum argument shift operator in the direction \(\xi\). We define \(C_\xi\) as the algebra generated by the set \(\bigcup_{n=0}^\infty\partial_\xi^nC\). The following theorem is proved in [11], [12].

Theorem 3.

The algebra \(C_\xi\) is a quantum argument shift algebra in the direction \(\xi\).

3. Formulas for second-order quantum argument shifts

We present formulas for the second-order quantum argument shifts of central elements. Theorem 2 suffices for this purpose. We adopt the convention that \( \operatorname{tr} e^{-1}=1\) for simplicity of notation. The following formulas give the quantum argument shifts of an arbitrary central element up to the second order.

Proposition 1.

$$\partial\bigl( \operatorname{tr} e^{n_1} \operatorname{tr} e^{n_2}\ldots\bigr)= \sum_{m_1=-1}^{n_1} \operatorname{tr} e^{m_1}\sum_{m_2=-1}^{n_2} \operatorname{tr} e^{m_2}\ldots\prod_k f^{(n_k-m_k-1)}_{-}(e)$$

and

$$\begin{aligned} \, &\partial\partial_\xi\bigl( \operatorname{tr} e^{n_1} \operatorname{tr} e^{n_2}\ldots\bigr)= \sum_{m_1=-1}^{n_1} \operatorname{tr} e^{m_1}\sum_{m_2=-1}^{n_2} \operatorname{tr} e^{m_2}\ldots \sum_{k_1=-1}^{n_1-m_1-1}f^{(k_1)}_{-}(e)\sum_{k_2=-1}^{n_2-m_2-1}f^{(k_2)}_{-}(e)\ldots{} \nonumber\\ &\ldots\partial \operatorname{tr} \biggl(\xi\prod_\ell f^{(n_\ell-m_\ell-k_\ell-2)}_{-}(e)\biggr) \end{aligned}$$
(2)

for a finite product \( \operatorname{tr} e^{n_1} \operatorname{tr} e^{n_2}\ldots{}\,\).

Proof

is by direct computation. We have

$$\begin{aligned} \, \partial \operatorname{tr} e^n&=\sum_{m=0}^n\bigl(f^{(n-m-1)}_{+}(e)e^m+f^{(n-m-1)}_{-}(e) \operatorname{tr} e^m\bigr)= \\ &=f^{(n)}_{-}(e)+\sum_{m=0}^nf^{(n-m-1)}_{-}(e) \operatorname{tr} e^m=\sum_{m=-1}^nf^{(n-m-1)}_{-}(e) \operatorname{tr} e^m \end{aligned}$$

by Theorem 2 and the identity \(\sum_{m=0}^nf^{(n-m-1)}_{+}(x)x^m=f^{(n)}_{-}(x)\). We obtain

$$\begin{aligned} \, \partial( \operatorname{tr} e^{n_1} \operatorname{tr} e^{n_2}\ldots)&=\partial( \operatorname{tr} e^{n_1})\partial( \operatorname{tr} e^{n_2})\ldots= \nonumber\\ &=\sum_{m_1=-1}^{n_1} \operatorname{tr} e^{m_1}\sum_{m_2=-1}^{n_2} \operatorname{tr} e^{m_2}\ldots\prod_k f^{(n_k-m_k-1)}_{-}(e). \end{aligned}$$
(3)

We proceed to calculate the second-order quantum argument shifts

$$ \partial_\xi( \operatorname{tr} e^{n_1} \operatorname{tr} e^{n_2}\ldots)=\sum_{m_1=-1}^{n_1} \operatorname{tr} e^{m_1}\sum_{m_2=-1}^{n_2} \operatorname{tr} e^{m_2}\ldots \operatorname{tr} \biggl(\xi\prod_kf^{(n_k-m_k-1)}_{-}(e)\biggr)$$
(4)

and

$$\begin{aligned} \, &\partial\partial_\xi( \operatorname{tr} e^{n_1} \operatorname{tr} e^{n_2}\ldots)= \sum_{k_1=-1}^{n_1}\sum_{k_2=-1}^{n_2}\ldots \partial\biggl(\prod_\ell \operatorname{tr} e^{k_\ell}\biggr)\partial\biggl( \operatorname{tr} \biggl(\xi\prod_\ell f^{(n_\ell-k_\ell-1)}_{-}(e)\biggr)\!\biggr)= \\ &=\sum_{k_1=-1}^{n_1}\sum_{k_2=-1}^{n_2}\ldots\sum_{m_1=-1}^{k_1} \operatorname{tr} e^{m_1} \sum_{m_2=-1}^{k_2} \operatorname{tr} e^{m_2}\ldots\prod_\ell f^{(k_\ell-m_\ell-1)}_{-}(e)\partial \operatorname{tr} \biggl(\xi\prod_\ell f^{(n_\ell-k_\ell-1)}_{-}(e)\biggr) \end{aligned}$$

by formula (3). Because

$$\sum_{k_1=-1}^{n_1}\sum_{k_2=-1}^{n_2}\ldots\sum_{m_1=-1}^{k_1}\sum_{m_2=-1}^{k_2}\ldots= \sum_{m_1=-1}^{n_1}\sum_{m_2=-1}^{n_2}\ldots\sum_{k_1=m_1}^{n_1}\sum_{k_2=m_2}^{n_2}\ldots,$$

we arrive at formula (2). \(\blacksquare\)

We write \(A[S]\) for the algebra generated by an algebra \(A\) and a set \(S\) contained in the quantum argument shift algebra \(C_\xi\). We define

$$C_\xi^{(0)}=C,\qquad C_\xi^{(n)}=C_\xi^{(n-1)}[\partial_\xi^nC].$$

Formula (4) implies the following assertion.

Corollary 1.

\(C_\xi^{(1)}=C[ \operatorname{tr} \bigl(\xi e^n\bigr)\colon n=1,2,\ldots\,]\).

We have

$$\begin{aligned} \, \partial_\xi^2( \operatorname{tr} e^{n_1} \operatorname{tr} e^{n_2}\ldots)={}&\sum_{m_1=-1}^{n_1} \operatorname{tr} e^{m_1}\sum_{m_2=-1}^{n_2} \operatorname{tr} e^{m_2}\ldots{} \nonumber\\ &{}\ldots\sum_{k_1=-1}^{n_1-m_1-1}\sum_{k_2=-1}^{n_2-m_2-1}\ldots \operatorname{tr} \biggl(\xi\prod_\ell f^{(k_\ell)}_{-}(e)\,\partial \operatorname{tr} \biggl(\xi\prod_\ell f^{(n_\ell-m_\ell-k_\ell-2)}_{-}(e)\biggr)\!\biggr) \end{aligned}$$
(5)

by formula (2). Formula (5) implies the corollary.

Corollary 2.

The algebra \(C_\xi^{(2)}\) is contained in the algebra generated by the algebra \(C_\xi^{(1)}\) and the elements

$$\operatorname{tr} (\xi e^m\partial \operatorname{tr} (\xi e^n))+ \operatorname{tr} (\xi e^n\partial \operatorname{tr} (\xi e^m)),\qquad m,n=0,1,2,\ldots{}\,.$$

Proof.

The elements of the form

$$\sum_{m_1=-1}^{n_1+1}\sum_{m_2=-1}^{n_2+1}\ldots \operatorname{tr} \biggl(\xi\prod_k f^{(m_k)}_{-}(e)\,\partial \operatorname{tr} \biggl(\xi\prod_kf^{(n_k-m_k)}_{-}(e)\biggr)\!\biggr)$$

belong to the additive monoid generated by the elements

$$\operatorname{tr} (\xi e^n\partial \operatorname{tr} (\xi e^n)),\quad \operatorname{tr} (\xi e^m\partial \operatorname{tr} (\xi e^n))+ \operatorname{tr} (\xi e^n\partial \operatorname{tr} (\xi e^m)),\qquad m,n=0,1,2,\ldots{}\,.$$

Any element of \(C_\xi^{(2)}\) is contained in the algebra generated by the algebra \(C_\xi^{(1)}\) and the elements

$$\begin{aligned} \, \operatorname{tr} (\xi e^m\partial \operatorname{tr} (\xi e^n))+ \operatorname{tr} (\xi e^n\partial \operatorname{tr} (\xi e^m)),\qquad m,n=0,1,2,\ldots{}\,.\quad\blacksquare \end{aligned}$$

We suppose that \(m\) and \(n\) are nonnegative integers. We have

$$\begin{aligned} \, \operatorname{tr} (\xi e^m\partial \operatorname{tr} (\xi e^n))&= \operatorname{tr} \biggl(\xi e^m\sum_{j=1}^{n+1}\bigl(f^{(n-j)}_{+}(e)\xi e^{j-1}+f^{(n-j)}_{-}(e) \operatorname{tr} (\xi e^{j-1})\bigr)\biggr)= \nonumber\\ &=\sum_{j=1}^{n+1}\bigl( \operatorname{tr} (\xi e^mf^{(n-j)}_{+}(e)\xi e^{j-1})+ \operatorname{tr} (\xi e^mf^{(n-j)}_{-}(e)) \operatorname{tr} (\xi e^{j-1})\bigr), \end{aligned}$$
(6)

by Theorem 2 and thus

$$ \operatorname{tr} (\xi e^m\partial \operatorname{tr} (\xi e^n))=\sum_{j=1}^n \operatorname{tr} (\xi e^mf^{(n-j)}_{+}(e)\xi e^{j-1})\mod C_\xi^{(1)}$$
(7)

by Corollary 1.

Definition 2.

We define the \((m+n)\times n\) integer matrix \(P^{(m)}_n\) as the coefficients of the polynomials

$$x^mf^{(n-j)}_{+}(x)=\sum_{i=1}^{m+n}(P^{(m)}_n)^i_jx^{i-1}$$

and let \(P_n=P^{(0)}_n\).

The matrix \(P_n\) is the submatrix of the matrix \(P_{n+1}\) in the top right corner, \(P_{n+1}=\bigl(\begin{smallmatrix} * & P_n \\ 1 & 0 \end{smallmatrix}\bigr)\) and \(P^{(m)}_n=\binom{0}{P_n}\) (the first \(m\) row vectors are null). For instance, because

$$\begin{pmatrix} f^{(3)}_{+}(x) \!&\! f^{(2)}_{+}(x) \!&\! f^{(1)}_{+}(x) \!&\! f^{(0)}_{+}(x)\end{pmatrix} = \begin{pmatrix} 3x+x^3 \!&\! 1+x^2 \!&\! x \!&\! 1 \end{pmatrix}=\begin{pmatrix} x^0 \!&\! x^1 \!&\! x^2 \!&\! x^3 \end{pmatrix} \begin{pmatrix} 0 & 1 & 0 & 1 \\ 3 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix},$$

we have \(P_4=\biggl(\begin{smallmatrix} 0 & 1 & 0 & 1 \\ 3 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{smallmatrix}\biggr)\).

Definition 3.

We define

$$\tau_\xi(x)= \operatorname{tr} \left(\begin{pmatrix} \xi \!&\! \xi e \!&\! \ldots \!&\! \xi e^{m-1} \end{pmatrix}x \begin{pmatrix}\xi \\ \xi e \\ \vdots \\ \xi e^{n-1} \end{pmatrix}\right)= \sum_{i=1}^m\sum_{j=1}^n x^i_j \operatorname{tr} (\xi e^{i-1}\xi e^{j-1})$$

for any \(m\times n\) numerical matrix \(x\).

By formula (7), we now have

$$ \operatorname{tr} (\xi e^m\partial \operatorname{tr} (\xi e^n))=\tau_\xi(P^{(m)}_n)\mod C_\xi^{(1)}.$$
(8)

4. Generators of the algebra \(C_\xi^{(2)}\)

We give the reduced set of generators of the algebra \(C_\xi^{(2)}\). The generators given in Corollary 2 can be expressed in terms of lower triangular matrices.

Definition 4.

Let \(n\) be a nonnegative integer and \(x\) an \(n\times n\) numerical matrix. We define the \(n\times n\) lower triangular numerical matrix \(\sigma(x)\) by the formula

$$\sigma(x)=\begin{pmatrix} x^1_1 & 0 & \cdots & 0\\ x^2_1+x^1_2 & x^2_2 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ x^n_1+x^1_n & x^n_2+x^2_n & \cdots & x^n_n \end{pmatrix}=\sum_{i,j=1}^nx^i_j\delta_{\max\{i,j\}}\delta^{\min\{i,j\}}.$$

Proposition 2.

\((\tau_\xi \mathbin{\stackrel{\scriptscriptstyle{\circ}}{{}_{\vphantom{.}}}} \sigma)(x)=\tau_\xi(x)\) for any numerical square matrix \(x\).

Proof.

We suppose that \(m\) and \(n\) are nonnegative integers and let \((\zeta_1,\ldots,\zeta_n)\) be a finite sequence of elements of the set \(M(d,\mathbb{C})\sqcup\{e\}\). We have

$$\begin{aligned} \, \operatorname{tr} [\xi e^m,\zeta_1\ldots\zeta_n]=\smash{\sum_{\zeta_k=e}}\bigl( \operatorname{tr} (\zeta_1\ldots{}&\zeta_{k-1}e^m) \operatorname{tr} (\xi\zeta_{k+1}\ldots\zeta_n)- \operatorname{tr} (\zeta_1\ldots\zeta_{k-1}) \operatorname{tr} (\xi e^m\zeta_{k+1}\ldots\zeta_n)\bigr), \end{aligned}$$

by the commutation relation (1), and thus

$$\begin{aligned} \, \operatorname{tr} [\xi e^m,\xi e^n]&=\sum_{k=1}^n\bigl( \operatorname{tr} (\xi e^{m+k-1}) \operatorname{tr} (\xi e^{n-k})- \operatorname{tr} (\xi e^{k-1}) \operatorname{tr} (\xi e^{m+n-k})\bigr)= \nonumber\\ &=\sum_{k=1}^n[ \operatorname{tr} (\xi e^{m+k-1}), \operatorname{tr} (\xi e^{n-k})]=0, \end{aligned}$$
(9)

because the algebra \(C_\xi^{(1)}=C\bigl[ \operatorname{tr} (\xi e^n)\colon n=1,2,\ldots\,\bigr]\) (see Corollary 1) is commutative by Theorem 3. We have

$$(\tau_\xi \mathbin{\stackrel{\scriptscriptstyle{\circ}}{{}_{\vphantom{.}}}} \sigma)(x)=\sum_{i,j=1}^nx^i_j \operatorname{tr} (\xi e^{\max\{i,j\}-1}\xi e^{\min\{i,j\}-1})= \sum_{i,j=1}^nx^i_j \operatorname{tr} (\xi e^{i-1}\xi e^{j-1})=\tau_\xi(x)$$

for any \(n\times n\) numerical matrix \(x\) by formula (9). \(\blacksquare\)

Proposition 3.

For any nonnegative integers \(m\) and \(n\), we have

$$\operatorname{tr} \bigl(\xi e^m\partial \operatorname{tr} (\xi e^n)\bigr)+ \operatorname{tr} \bigl(\xi e^n\partial \operatorname{tr} (\xi e^m)\bigr)= (\tau_\xi \mathbin{\stackrel{\scriptscriptstyle{\circ}}{{}_{\vphantom{.}}}} \sigma)\begin{pmatrix} 0 & P_n^{\mathrm T} \\ P_m & 0 \end{pmatrix}\mod C_\xi^{(1)}.$$

Proof.

We have

by formula (8) and Proposition 2. \(\blacksquare\)

The following theorem plays an essential role in reducing the number of the generators given in Corollary 2 and Proposition 3. The proof is given in the Appendix.

Theorem 4.

For any nonnegative integers \(m\) and \(n\), we have

$$\sigma\begin{pmatrix} 0 & P_m^{\mathrm T} \\ P_{m+2n} & 0 \end{pmatrix}= \sum_{k=0}^n\biggl(\binom{2n-k}k+\binom{2n-k-1}{k-1}\biggr)P^{(m+k)}_{m+k},$$
(10)
$$\sigma \begin{pmatrix} 0 & P_m^{\mathrm T} \\ P_{m+2n+1} & 0 \end{pmatrix}= \sum_{k=0}^n\binom{2n-k}k\bigl(P^{(m+k)}_{m+k+1}+P^{(m+k+1)}_{m+k}\bigr).$$
(11)

The following theorem is the main result in this paper.

Theorem 5.

The algebra \(C_\xi^{(2)}\) is given by

$$C_\xi^{(2)}= C_\xi^{(1)}\bigl[\tau_\xi(P^{(n)}_n),\tau_\xi(P^{(n)}_{n+1})+\tau_\xi(P^{(n+1)}_n)\colon n=1,2,\ldots\,\bigr].$$

Proof.

The algebra \(C_\xi^{(2)}\) is contained in the algebra

$$C_\xi^{(1)}\bigl[\tau_\xi(P^{(n)}_n),\tau_\xi(P^{(n)}_{n+1})+\tau_\xi(P^{(n+1)}_n)\colon n=1,2,\ldots\,\bigr]$$

by Proposition 3 and Theorem 4. We prove that the elements \(\tau_\xi(P^{(n)}_n)\) and \(\tau_\xi(P^{(n)}_{n+1})+\tau_\xi(P^{(n+1)}_n)\) belong to the algebra

$$ C_\xi^{(1)}\bigl[\partial_\xi^2 \operatorname{tr} e^n\colon n=3,4,\ldots\,\bigr]$$
(12)

by induction on the nonnegative integer \(n\). Suppose that the integer \(n\) is positive and the elements \(\tau_\xi(P^{(m)}_m)\), \( \tau_\xi(P^{(m)}_{m+1})+\tau_\xi(P^{(m+1)}_m)\) belong to algebra (12) for any nonnegative integer \(m<n\). The element \(\tau_\xi(P^{(n)}_n)\) belongs to algebra (12) because the element \(\partial_\xi^2 \operatorname{tr} e^{2n+1}-(4n+2)\tau_\xi(P^{(n)}_n)\) belongs to the submodule

$$\operatorname{span} _C\bigl\{\tau_\xi(P^{(m)}_m)\bigr\}_{m=0}^{n-1}+ \operatorname{span} _C\bigl\{\tau_\xi(P^{(m)}_{m+1})+\tau_\xi(P^{(m+1)}_m)\bigr\}_{m=0}^{n-1}$$

modulo \(C_\xi^{(1)}\) by Theorem 4. Similarly, the element \(\tau_\xi(P^{(n)}_{n+1})+\tau_\xi(P^{(n+1)}_n)\) belongs to algebra (12). \(\blacksquare\)

We compute the first several elements of the generators:

$$\begin{aligned} \, &\tau_\xi(P^{(1)}_1) = \operatorname{tr} (\xi^2e), \\ &\tau_\xi(P^{(1)}_2)+\tau_\xi(P^{(2)}_1) = \operatorname{tr} (2\xi^2e^2+\xi e\xi e), \\ &\tau_\xi(P^{(2)}_2)= \operatorname{tr} (\xi^2e^3+\xi e\xi e^2), \\ &\tau_\xi(P^{(2)}_3)+\tau_\xi(P^{(3)}_2) = \operatorname{tr} (2\xi^2e^4+2\xi e\xi e^3+\xi e^2\xi e^2+\xi^2e^2), \\ &\tau_\xi(P^{(3)}_3)= \operatorname{tr} (\xi^2e^5+\xi e\xi e^4+\xi e^2\xi e^3+\xi^2e^3), \\ &\tau_\xi(P^{(3)}_4)+\tau_\xi(P^{(4)}_3) = \operatorname{tr} (2\xi^2e^6+2\xi e\xi e^5+2\xi e^2\xi e^4+\xi e^3\xi e^3+4\xi^2e^4+\xi e\xi e^3), \\ &\tau_\xi(P^{(4)}_4)= \operatorname{tr} (\xi^2e^7+\xi e\xi e^6+\xi e^2\xi e^5+\xi e^3\xi e^4+3\xi^2e^5+\xi e\xi e^4). \end{aligned}$$

They form a commutative family together with the elements \(\bigl\{ \operatorname{tr} (\xi e^n)\colon n=1,2,\dots\bigr\}\) (see Theorem 3 and Corollary 1).