Center at the critical level for centralizers in type $A$

TL;DR

This paper explicitly constructs generators of the center of an affine vertex algebra at the critical level associated with the centralizer of a nilpotent element in type A, linking it to quantum shift algebras and universal enveloping algebra centers.

Contribution

It provides an explicit construction of free generators of the center for the affine vertex algebra at the critical level related to centralizers in type A.

Findings

Constructed explicit free generators of the center.

Connected the generators to quantum shift of argument subalgebras.

Reproduced known generators of the universal enveloping algebra center.

Abstract

We consider the affine vertex algebra at the critical level associated with the centralizer of a nilpotent element in the Lie algebra $\mathfrak{gl}_N$. Due to a recent result of Arakawa and Premet, the center of this vertex algebra is an algebra of polynomials. We construct a family of free generators of the center in an explicit form. As a corollary, we obtain generators of the corresponding quantum shift of argument subalgebras and recover free generators of the center of the universal enveloping algebra of the centralizer produced earlier by Brown and Brundan.

Equations

$$\text{young}(12,345,6789)$$

$$E_{ij}^{(r)}=\sum _{\underset{\mathrm{col}(b)-\mathrm{col}(a)=r}{\mathrm{row}(a)=i,\mathrm{row}(b)=j}}{e}_{ab},$$

$$\big{[}E^{(r)}_{ij},E^{(s)}_{kl}\big{]}=\delta_{kj}\,E^{(r+s)}_{i\hskip 1.0ptl}-\delta_{i\hskip 1.0ptl}\,E^{(r+s)}_{kj},$$

$$E_{ij}^{(r)}\mapsto e_{ij}{v}^r,r=0,\dots ,p-1,1⩽i,j⩽n.$$

$$\big{\langle}E_{ii}^{(0)},E_{jj}^{(0)}\big{\rangle}=\min(\lambda_{i},\lambda_{j})-\delta_{ij}\big{(}\lambda_{1}+\dots+\lambda_{i-1}+(n-i+1)\lambda_{i}\big{)},$$

$$\big{\langle}E_{ij}^{(0)},E_{ji}^{(0)}\big{\rangle}=-\big{(}\lambda_{1}+\dots+\lambda_{i-1}+(n-i+1)\lambda_{i}\big{)},$$

$$\hat{\mathfrak{a}}=\mathfrak{a}[t,t^{-1}]\oplus \mathbb{C}1,$$

$$\big{[}X[r],Y[s]\big{]}=[X,Y][r+s]+r\,\delta_{r,-s}\langle X,Y\rangle\,\mathbf{1},\qquad X,Y\in\mathfrak{a},$$

$$V(\mathfrak{a})=\mathrm{U}(\hat{\mathfrak{a}})/\mathrm{I},$$

$$\mathfrak{z}(\hat{\mathfrak{a}})=\{v\in V(\mathfrak{a})|\mathfrak{a}[t]v=0\}.$$

$$T:X[r]\mapsto -rX[r-1],X\in \mathfrak{a},r<0.$$

$$\mathfrak{z}(\hat{\mathfrak{a}})=\mathbb{C}[T^r{S}_l|l=1,\dots ,N,r⩾0].$$

$$E_{ij}(u)=\{\begin{array}{ll}{E}_{ij}^{(0)}[-1]+\cdots +E_{ij}^{({\lambda }_j-1)}[-1]u^{{\lambda }_j-1}& \text{if}i⩾j,\\ E_{ij}^{({\lambda }_j-{\lambda }_i)}[-1]u^{{\lambda }_j-{\lambda }_i}+\cdots +E_{ij}^{({\lambda }_j-1)}[-1]u^{{\lambda }_j-1}& \text{if}i<j.\end{array}$$

$$\mathrm{cdet}\left[\begin{array}{cccc}x+{\lambda }_{1}T+E_{11}(u)& E_{12}(u)& \dots & E_{1n}(u)\\ E_{21}(u)& x+{\lambda }_{2}T+E_{22}(u)& \dots & E_{2n}(u)\\ ⋮& ⋮& \ddots & ⋮\\ E_{n1}(u)& E_{n2}(u)& \dots & x+{\lambda }_{n}T+E_{nn}(u)\end{array}\right]$$

$$\displaystyle{}=\sum_{\sigma\in\mathfrak{S}_{n}}{\rm sgn}\,\sigma\cdot\big{(}\delta_{\sigma(1)\hskip 1.0pt1}\,(x+\lambda_{1}\hskip 1.0ptT)+E_{\sigma(1)\hskip 1.0pt1}(u)\big{)}\dots\big{(}\delta_{\sigma(n)\hskip 1.0ptn}\,(x+{}$$

$$x^n+{\varphi }_1(u)x^{n-1}+\cdots +{\varphi }_n(u),{\varphi }_k(u)=\sum _r{\varphi }_k^{(r)}{u}^r.$$

$${\lambda }_{n-k+2}+\cdots +{\lambda }_n<r+k⩽{\lambda }_{n-k+1}+\cdots +{\lambda }_n$$

$${\varphi }_1^{(r)}=E_{11}^{(r)}[-1]=e_{1r+1}[-1]+\cdots +e_{N-rN}[-1],r=0,\dots ,N-1.$$

$${\rm cdet}\Bigg{[}\begin{matrix}x+\lambda_{1}\hskip 1.0ptT+E_{11}(u)&E_{12}(u)\\[3.99994pt] E_{21}(u)&x+\lambda_{2}\hskip 1.0ptT+E_{22}(u)\end{matrix}\Bigg{]}=x^{2}+\phi_{1}(u)\hskip 1.0ptx+\phi_{2}(u)$$

$${\varphi }_1(u)$$

$${\varphi }_2(u)$$

$${\varphi }_1^{(r)}=E_{11}^{(r)}[-1]+E_{22}^{(r)}[-1]\text{with}r=0,1,\dots ,{\lambda }_2-1,$$

$${\varphi }_2^{(r)}=\sum _{a+b=r}\left|\begin{array}{cc}{E}_{11}^{(a)}[-1]& E_{12}^{(b)}[-1]\\ E_{21}^{(a)}[-1]& E_{22}^{(b)}[-1]\end{array}\right|+{\lambda }_1{E}_{22}^{(r)}[-2]$$

$${\varphi }_1(u)$$

$${\varphi }_2(u)$$

$${\varphi }_1^{(0)}$$

$${\varphi }_1^{(1)}$$

$${\varphi }_2^{(1)}=\sum _{i=1}^{n-1}\left|\begin{array}{cc}{E}_{ii}^{(0)}[-1]& E_{in}^{(1)}[-1]\\ E_{ni}^{(0)}[-1]& E_{nn}^{(1)}[-1]\end{array}\right|+(n-1)E_{nn}^{(1)}[-2],$$

$$\big{[}\Delta,X[r]\hskip 1.0pt\big{]}=r\,X[r+1],\qquad\big{[}\Delta,\mathbf{1}\big{]}=0,$$

$$\Delta :\mathfrak{z}(\hat{\mathfrak{a}})\to \mathfrak{z}(\hat{\mathfrak{a}}).$$

$$\Delta :{\varphi }_k^{(r)}\mapsto 0$$

$$\Delta :{\varphi }_k^{(r)}\mapsto -(k-1)({\lambda }_1+\cdots +{\lambda }_{n-k+1}){\varphi }_{k-1}^{(r)}.$$

Full text

1 Introduction

Let $\mathfrak{a}$ be a finite-dimensional Lie algebra over $\mathbb{C}\hskip 1.0pt$ equipped with an invariant symmetric bilinear form. Consider the corresponding affine Kac–Moody algebra $\widehat{\mathfrak{a}}$ defined as a central extension of the Lie algebra of Laurent polynomials $\mathfrak{a}\hskip 1.0pt[t,t^{-1}]$; see (2.2) below. The vacuum module over $\widehat{\mathfrak{a}}$ is a vertex algebra whose center is a commutative associative algebra. In the case of a simple Lie algebra $\mathfrak{a}$ the structure of the center $\mathfrak{z}(\widehat{\mathfrak{a}})$ at the critical level was described by a celebrated theorem of Feigin and Frenkel [5] (see also [6]) which states that $\mathfrak{z}(\widehat{\mathfrak{a}})$ is an algebra of polynomials in infinitely many variables. This fact can be regarded as an affine version or chiralization of the well-known description of the center of the universal enveloping algebra ${\rm U}(\mathfrak{a})$ as a polynomial algebra.

On the other hand, in the case where $\mathfrak{a}=\mathfrak{g}^{e}$ is the centralizer of a nilpotent element $e$ in a simple Lie algebra $\mathfrak{g}$, the invariant algebra ${\rm S}(\mathfrak{a})^{\mathfrak{a}}$ admits free families of generators under certain additional conditions, due to the pioneering work of Panyushev, Premet and Yakimova [13]. In particular, these conditions hold for all nilpotents in types $A$ and $C$ which thus confirms Premet’s conjecture in those cases. Explicit generators of both the invariant algebra and the center of ${\rm U}(\mathfrak{a})$ in type $A$ were produced by Brown and Brundan [2] with the use of the shifted Yangians. It was shown by Yakimova [16] that the generators of ${\rm S}(\mathfrak{a})^{\mathfrak{a}}$ produced in [2] coincide with those previously conjectured in [13].

A recent work of Arakawa and Premet [1] provides a chiralization of these results for the centralizer $\mathfrak{a}$ which shows that the Feigin–Frenkel theorem extends to the affine vertex algebra at the critical level associated with the Kac–Moody algebra $\widehat{\mathfrak{a}}$. As a consequence, they showed the existence of regular quantum shift of argument subalgebras and proved that they are free polynomial algebras. Moreover, explicit formulas for generators of $\mathfrak{z}(\widehat{\mathfrak{a}})$ were produced in [1] in the case where $\mathfrak{a}$ is the centralizer of a minimal nilpotent in $\mathfrak{gl}_{N}$.

Our goal in this paper is to give explicit formulas for generators of $\mathfrak{z}(\widehat{\mathfrak{a}})$ for an arbitrary nilpotent element $e\in\mathfrak{gl}_{N}$ (Theorem 2.1). As a corollary, we get generators of the quantum shift of argument subalgebras $\mathcal{A}_{\chi}\in{\rm U}(\mathfrak{a})$ associated with an element $\chi\in\mathfrak{a}^{*}$ thus providing an explicit solution of Vinberg’s quantization problem [15] for centralizers (Corollary 2.6). It is based on the results of [1] showing that if the element $\chi$ is regular, then the classical limit ${\rm gr}\,\mathcal{A}_{\chi}$ coincides with the Mishchenko–Fomenko subalgebra $\overline{\mathcal{A}}_{\chi}\subset{\rm S}(\mathfrak{a})$; see [9]. We conjectured in [12, Conjecture 5.8] that the subalgebra $\mathcal{A}_{\chi}$ can also be obtained with the use of the symmetrization map. Note that the subalgebra $\mathcal{A}_{\hskip 1.0pt0}$ is the center of ${\rm U}(\mathfrak{a})$. In this case our generators essentially coincide with those found in [2]; see Corollary 2.7 below.

By taking $e=0$ in Theorem 2.1 we recover the formulas for the generators of the algebra $\mathfrak{z}(\widehat{\mathfrak{gl}}_{n})$ found in [3] and [4] which we also interpret with the use of an operator $\Delta$ preserving $\mathfrak{z}(\widehat{\mathfrak{gl}}_{n})$ (Corollary 2.4). For more details on this particular case and extensions to the other classical types see [11]. A different way to produce generators of the Feigin–Frenkel center $\mathfrak{z}(\widehat{\mathfrak{g}})$ was developed in a recent work by Yakimova [17].

2 Segal–Sugawara vectors

Using the notation of [2], suppose that $e\in\mathfrak{g}=\mathfrak{gl}_{N}$ is a nilpotent matrix with Jordan blocks of sizes $\lambda_{1},\dots,\lambda_{n}$, where $\lambda_{1}\leqslant\dots\leqslant\lambda_{n}$ and $\lambda_{1}+\dots+\lambda_{n}=N$. Consider the corresponding pyramid which is a left-justified array of rows of unit boxes such that the top row contains $\lambda_{1}$ boxes, the next row contains $\lambda_{2}$ boxes, etc. The row-tableau is obtained by writing the numbers $1,\dots,N$ into the boxes of the pyramid consecutively by rows from left to right. For instance, the row-tableau

[TABLE]

corresponds to the Jordan blocks of sizes $2,3,4$ and $N=9$. Let ${\rm row}(a)$ and ${\rm col}(a)$ denote the row and column number of the box containing the entry $a$.

Let $e_{a\hskip 1.0ptb}$ with $a,b=1,\dots,N$ be the standard basis elements of $\mathfrak{g}$. For any $1\leqslant i,j\leqslant n$ and $\lambda_{j}-\min(\lambda_{i},\lambda_{j})\leqslant r<\lambda_{j}$ set

[TABLE]

summed over $a,b\in\{1,\dots,N\}$. The elements $E_{ij}^{(r)}$ form a basis of the Lie algebra $\mathfrak{a}=\mathfrak{g}^{e}$. The commutation relations for the basis vectors have the form

[TABLE]

assuming that $E^{(r)}_{ij}=0$ for $r\geqslant\lambda_{j}$.

Note that in the particular case of a rectangular pyramid $\lambda_{1}=\dots=\lambda_{n}=p$ the Lie algebra $\mathfrak{a}$ is isomorphic to the truncated polynomial current algebra $\mathfrak{gl}_{n}[v]/(v^{p}=0)$ (also known as the Takiff algebra). The isomorphism is given by

[TABLE]

Equip the Lie algebra $\mathfrak{a}$ with the invariant symmetric bilinear form $\langle\,\,,\,\rangle$ defined as the normalized Killing form at the critical level on the [math]-th component of $\mathfrak{a}$; see [1, Sec. 5]. Explicitly, it is given by the formulas

[TABLE]

and if $\lambda_{i}=\lambda_{j}$ for some $i\neq j$ then

[TABLE]

whereas all remaining values of the form on the basis vectors are zero. Note that the sum in the brackets equals the number of boxes in the first $\lambda_{i}$ columns of the pyramid.

The corresponding affine Kac–Moody algebra $\widehat{\mathfrak{a}}$ is the central extension

[TABLE]

where $\mathfrak{a}[t,t^{-1}]$ is the Lie algebra of Laurent polynomials in $t$ with coefficients in $\mathfrak{a}$. For any $r\in\mathbb{Z}\hskip 1.0pt$ and $X\in\mathfrak{g}$ we will write $X[r]=X\,t^{r}$. The commutation relations of the Lie algebra $\widehat{\mathfrak{a}}$ have the form

[TABLE]

and the element $\mathbf{1}$ is central in $\widehat{\mathfrak{a}}$. The vacuum module at the critical level over $\widehat{\mathfrak{a}}$ is the quotient

[TABLE]

where ${\rm I}$ is the left ideal of ${\rm U}(\widehat{\mathfrak{a}})$ generated by $\mathfrak{a}[t]$ and the element $\mathbf{1}-1$. By the Poincaré–Birkhoff–Witt theorem, the vacuum module is isomorphic to the universal enveloping algebra {\rm U}\big{(}t^{-1}\mathfrak{a}[t^{-1}]\big{)}, as a vector space. This vector space is equipped with a vertex algebra structure; see [7], [8]. Denote by $\mathfrak{z}(\widehat{\mathfrak{a}})$ the center of this vertex algebra which is defined as the subspace

[TABLE]

It follows from the axioms of vertex algebra that $\mathfrak{z}(\widehat{\mathfrak{a}})$ is a unital commutative associative algebra. It can be regarded as a commutative subalgebra of {\rm U}\big{(}t^{-1}\mathfrak{a}[t^{-1}]\big{)}. This subalgebra is invariant with respect to the translation operator $T$ which is the derivation of the algebra {\rm U}\big{(}t^{-1}\mathfrak{a}[t^{-1}]\big{)} whose action on the generators is given by

[TABLE]

Any element of $\mathfrak{z}(\widehat{\mathfrak{a}})$ is called a Segal–Sugawara vector. By [1, Thm 1.4], there exists a complete set of Segal–Sugawara vectors $S_{1},\dots,S_{N}$, which means that all translations $T^{r}S_{l}$ with $r\geqslant 0$ and $l=1,\dots,N$ are algebraically independent and any element of $\mathfrak{z}(\widehat{\mathfrak{a}})$ can be written as a polynomial in the shifted vectors; that is,

[TABLE]

In the case $e=0$ this reduces to the Feigin–Frenkel theorem in type $A$ [5, 6].

To construct a complete set of Segal–Sugawara vectors, for all $i,j\in\{1,\dots,n\}$ introduce polynomials in a variable $u$ with coefficients in this algebra by

[TABLE]

For another variable $x$ calculate the column-determinant

[TABLE]

and write it as a polynomial in $x$ with coefficients in $V(\mathfrak{a})[u]$,

[TABLE]

Theorem 2.1**.**

The coefficients $\phi^{(r)}_{k}$ with $k=1,\dots,n$ and

[TABLE]

belong to the center $\mathfrak{z}(\widehat{\mathfrak{a}})$ of the vertex algebra $V(\mathfrak{a})$. Moreover, they form a complete set of Segal–Sugawara vectors for the Lie algebra $\mathfrak{a}$.

Examples 2.2*.*

In the principal nilpotent case with $n=1$ and $e=e_{1\hskip 1.0pt2}+\dots+e_{N-1\,N}$ we have

[TABLE]

For $n=2$ we have

[TABLE]

with

[TABLE]

where $X^{\prime}(u)=T\hskip 1.0ptX(u)$. Therefore, a complete set of Segal–Sugawara vectors for $\mathfrak{a}$ is given by

[TABLE]

and

[TABLE]

with   $r=\lambda_{2}-1,\dots,\lambda_{1}+\lambda_{2}-2$, where the vertical lines indicate the column-determinant.

The minimal nilpotent case $e=e_{n\,n+1}\in\mathfrak{gl}_{n+1}$ corresponds to the pyramid with the $n$ rows $1,\dots,1,2$. For the coefficients of $x^{n-1}$ and $x^{n-2}$ in the general expansion (2.4) we have

[TABLE]

Hence,

[TABLE]

whereas

[TABLE]

cf. [1, Sec. 5]. ∎

As with the minimal nilpotent case considered in [1], the proof of Theorem 2.1 follows the same approach as in the paper [3] which deals with the case $e=0$. We will give some necessary additional details in Section 3 which include the use of the derivation $\Delta$ of the algebra ${\rm U}(\widehat{\mathfrak{a}})$ determined by its commutation relations with the left multiplication operators

[TABLE]

for $X\in\mathfrak{a}$. We can regard $\Delta$ as an operator on the vacuum module $V(\mathfrak{a})$. It is immediate from the commutation relations that $\Delta$ preserves Segal–Sugawara vectors and thus gives rise to the operator

[TABLE]

Its use in the proof of Theorem 2.1 will be based on the following property of the coefficients $\phi^{(r)}_{k}\in V(\mathfrak{a})$, where we do not assume that they belong to $\mathfrak{z}(\widehat{\mathfrak{a}})$.

Lemma 2.3**.**

Under the action of $\Delta$ we have

[TABLE]

for all $r>\lambda_{n-k+2}+\dots+\lambda_{n}-k+1$, while for $r=\lambda_{n-k+2}+\dots+\lambda_{n}-k+1$ we have

[TABLE]

We will prove Lemma 2.3 in Section 3. In particular, it implies that a complete set of Segal–Sugawara vectors in the case $\lambda_{1}=\dots=\lambda_{n}=1$ (that is, for $\mathfrak{a}=\mathfrak{gl}_{n}$) can be produced from a single vector. Set

[TABLE]

which is the constant term of the polynomial in (2.3); see also [3] and [11, Ch. 7].

Corollary 2.4**.**

The elements $\Delta^{k}\phi$ for $k=0,1,\dots,n-1$ form a complete set of Segal–Sugawara vectors for $\mathfrak{gl}_{n}$. ∎

A slightly less explicit but equivalent expression for the coefficients $\phi^{(r)}_{k}$ is obtained in the following proposition, where we introduce $\tau$ as an element of the extended Lie algebra $\widehat{\mathfrak{a}}\oplus\mathbb{C}\hskip 1.0pt\tau$ which satisfies the commutation relations

[TABLE]

For all $i,j\in\{1,\dots,n\}$ define the elements $\mathcal{E}_{ij}(\tau)$ of the tensor product space $V(\mathfrak{a})\otimes\mathbb{C}\hskip 1.0pt[\tau]$ by

[TABLE]

Introduce elements $\phi^{\circ}_{1},\dots,\phi^{\circ}_{N}\in V(\mathfrak{a})$ by using the expansion of the column-determinant of the matrix $\mathcal{E}(\tau)=[\mathcal{E}_{ij}(\tau)]$,

[TABLE]

We will say that the element $E^{(b)}_{ij}[r]$ of the Lie algebra $t^{-1}\mathfrak{a}[t^{-1}]$ has weight $b$. This defines a grading on the universal enveloping algebra {\rm U}\big{(}t^{-1}\mathfrak{a}[t^{-1}]\big{)}.

Proposition 2.5**.**

For each $k=1,\dots,n$ and $r$ satisfying (2.5), the Segal–Sugawara vector $\phi^{(r)}_{k}$ coincides with the homogeneous component of maximal weight of the coefficient $\phi^{\circ}_{r+k}$.

Proof.

Expand the column-determinant

[TABLE]

and move all powers of $\tau$ to the right by commuting them with the elements $E^{(b)}_{ij}[r]$. It is straightforward to verify that a nonzero contribution to the maximum weight component of $\phi^{\circ}_{r+k}$ can only come from commutators involving the leading terms $\tau^{\lambda_{i}}$ of $\mathcal{E}_{i\hskip 1.0pti}(\tau)$ for $i=1,\dots,n$. Moreover, this component has weight equal to $r$ and coincides with $\phi^{(r)}_{k}$. ∎

By adapting Rybnikov’s construction [14] to the case of the affine Kac–Moody algebra $\widehat{\mathfrak{a}}$ as in [1], for any element $\chi\in\mathfrak{a}^{*}$ and a variable $z$ consider the homomorphism

[TABLE]

for any $X\in\mathfrak{a}$ and $r<0$. If $S\in\mathfrak{z}(\widehat{\mathfrak{a}})$ is a homogeneous element of degree $d$ with respect to the grading defined by $\deg X[r]=-r$ for $r<0$, define the elements $S_{(k)}\in{\rm U}(\mathfrak{a})$ (depending on $\chi$) by the expansion

[TABLE]

If the variable $z$ takes a particular nonzero value in $\mathbb{C}\hskip 1.0pt$, then the formula in (2.8) defines a homomorphism {\rm U}\big{(}t^{-1}\mathfrak{a}[t^{-1}]\big{)}\to{\rm U}(\mathfrak{a}). Since $\mathfrak{z}(\widehat{\mathfrak{a}})$ is a commutative subalgebra of {\rm U}\big{(}t^{-1}\mathfrak{a}[t^{-1}]\big{)}, its image under this homomorphism is a commutative subalgebra of ${\rm U}(\mathfrak{a})$ which we denote by $\mathcal{A}_{\chi}$. This subalgebra does not depend on the value of $z$.

It is clear from its definition that the degree of the coefficient $\phi^{(r)}_{k}$ in (2.4) equals $k$. Therefore, the respective degrees of the Segal–Sugawara vectors provided by Theorem 2.1 coincide with the degrees of the basic invariants of the symmetric algebra ${\rm S}(\mathfrak{a})$ given by

[TABLE]

as found in [13]; see also [2]. Define elements $\phi^{(a)}_{k\hskip 1.0pt(m)}\in{\rm U}(\mathfrak{a})$ by applying the homomorphism $\varrho_{\hskip 1.0pt\chi}$ to the Segal–Sugawara vectors:

[TABLE]

More explicitly, set $\chi_{ij}^{(r)}=\chi(E_{ij}^{(r)})$ and

[TABLE]

Furthermore, let

[TABLE]

The image of the column-determinant (2.3) under the homomorphism $\varrho_{\hskip 1.0pt\chi}$ will be written as a polynomial in $x$,

[TABLE]

with

[TABLE]

Recall that the Mishchenko–Fomenko subalgebra $\overline{\mathcal{A}}_{\chi}$ of the symmetric algebra ${\rm S}(\mathfrak{a})$ is a Poisson-commutative subalgebra generated by all $\chi$-shifts of all $\mathfrak{a}$-invariants $P\in{\rm S}(\mathfrak{a})^{\mathfrak{a}}$, as originally defined in [9]. By using the results of [1] we come to the following.

Corollary 2.6**.**

Suppose the element $\chi\in\mathfrak{a}^{*}$ is regular. Then the elements $\phi^{(r)}_{k\hskip 1.0pt(m)}$ for $k=1,\dots,n$ with $r$ satisfying conditions (2.5) and $m=0,\dots,k-1$ are algebraically independent generators of the algebra $\mathcal{A}_{\chi}$. Moreover, $\mathcal{A}_{\chi}$ is a quantization of the subalgebra $\overline{\mathcal{A}}_{\chi}$ so that ${\rm gr}\,\mathcal{A}_{\chi}=\overline{\mathcal{A}}_{\chi}$. ∎

As another corollary of Theorem 2.1, we recover the algebraically independent generators of the center of ${\rm U}(\mathfrak{a})$ constructed in [2] with the use of the shifted Yangians; see also [10] for the particular case of rectangular pyramids. The algebra $\mathcal{A}_{\hskip 1.0pt0}$ coincides with the center of ${\rm U}(\mathfrak{a})$ so that the formulas for the generators are found by taking $\chi=0$ in (2.8); cf. [11, Secs 6.5 & 7.2]. Write the column-determinant

[TABLE]

as a polynomial in $x$,

[TABLE]

Corollary 2.7**.**

The coefficients $\Phi^{(r)}_{k}$ with $k=1,\dots,n$ and $r$ satisfying conditions (2.5) are algebraically independent generators of the center of the algebra ${\rm U}(\mathfrak{a})$. ∎

The elements $\Phi^{(r)}_{k}$ are slightly different from the corresponding central elements $z_{r}$ given by [2, Eq. (1.3)]. The mapping $E_{ij}^{(p)}\mapsto E_{ij}^{(p)}+\delta_{p\hskip 1.0pt0}\hskip 1.0pt\delta_{ij}\hskip 1.0ptc\hskip 1.0pt\lambda_{i}$ defines an automorphism of the algebra ${\rm U}(\mathfrak{a})$ for any given constant $c$. The image of the Casimir element $\Phi^{(r)}_{k}$ under this automorphism with $c=-n+1$ coincides with $z_{r+k}$.

3 Proof of Theorem 2.1

To prove the first part of the theorem, we will verify that the coefficients $\phi^{(r)}_{k}\in V(\mathfrak{a})$ are annihilated by a family of elements which generate $\mathfrak{a}[t]$ as a Lie algebra. For such a family we take $E^{(0)}_{i+1\,i}[0]$, $E^{(\lambda_{i+1}-\lambda_{i})}_{i\,i+1}[0]$ for $i=1,\dots,n-1$, and $E^{(p)}_{i\,i}[s]$ for $p=0,\dots,\lambda_{i}-1$ all $i=1,\dots,n$ and $s\geqslant 0$.

We will apply the generators to the column-determinant ${\rm cdet}\,\mathcal{E}$ of the matrix $\mathcal{E}=[\mathcal{E}_{ij}]$, where we set $\mathcal{E}_{ij}=\delta_{ij}\hskip 1.0pt(x+\lambda_{i}\hskip 1.0ptT)+E_{ij}(u)$. The result of such an application is a polynomial in $x$ of the form

[TABLE]

It will be sufficient to verify that for $k=1,\dots,n$ the degree of the polynomial $\psi_{k}(u)$ in $u$ is less than

[TABLE]

(in particular, $\psi_{1}(u)$ must be equal to zero). We will consider the groups of generators case by case and use the notation $\tau_{i}=x+\lambda_{i}\hskip 1.0ptT$ for brevity. Note the commutation relations

[TABLE]

Generators $E^{(0)}_{i+1\,i}[0]$.

We will rely on some simple properties of column-determinants described in [3, Lemmas 4.1 & 4.2]. They allow us to write the element $E^{(0)}_{i+1\,i}[0]\,{\rm cdet}\,\mathcal{E}$ as the difference of two column-determinants

[TABLE]

obtained from ${\rm cdet}\,\mathcal{E}$ by replacing row $i$ and column $i+1$, respectively, as indicated. Here we set $\widetilde{\mathcal{E}}_{i+1\,j}=\mathcal{E}_{i+1\,j}$ for $j\leqslant i$ and

[TABLE]

for $j\geqslant i+1$, while $\widehat{\mathcal{E}}_{k\hskip 1.0pti}=\mathcal{E}_{k\hskip 1.0pti}$ for $k\geqslant i+1$ and

[TABLE]

for $k\leqslant i$.

Start with the first determinant and observe that it stays unchanged if we subtract row $i+1$ from row $i$. The first $i$ entries in the new row $i$ will be equal to zero and it will have the form

[TABLE]

with

[TABLE]

for $j=i+1,\dots,n$. Write the first determinant as a polynomial in $x$ and consider the coefficient of $x^{n-k}$. This coefficient is a polynomial in $u$, and if $n-k\leqslant i-1$, then its degree does not exceed

[TABLE]

This is clear since $E_{i\hskip 1.0ptl}(u)$ is a polynomial in $u$ of degree $\lambda_{l}-1$, while $\check{\mathcal{E}}_{i+1\,j}$ is a polynomial of degree $\lambda_{j}-\lambda_{i}-1$. However, the sum (3.4) is less than (3.2) because $\lambda_{n-k+1}-\lambda_{i}-1<0$.

Similarly, if $n-k\geqslant i$, then the coefficient of $x^{n-k}$ is a polynomial in $u$ whose degree does not exceed the sum

[TABLE]

which is less than (3.2) because $\lambda_{i}>0$. Therefore, the contribution to the polynomial $\psi_{k}(u)$ in (3.1) arising from the first determinant does not violate the required degree condition.

To reach the same conclusion for the second determinant, use its simultaneous expansion along columns $i$ and $i+1$. The expansion will involve $2\times 2$ column-minors of the form

[TABLE]

for $a<b$. First consider the case with $a=i$ and suppose that $\lambda_{i}=\lambda_{i+1}$. Then the minor (3.5) equals

[TABLE]

We have $E_{b\hskip 1.0pti}(u)(\tau_{i}-\tau_{i+1})=E_{b\hskip 1.0pti}(u)(\lambda_{i}-\lambda_{i+1})\hskip 1.0ptT=0$. Note that since $[E^{(r)}_{i\hskip 1.0pti},E^{(s)}_{b\hskip 1.0pti}]=0$ for the values with the condition $r+s\geqslant\lambda_{i}$, the degree of the polynomial \big{[}\tau_{i}+E_{i\hskip 1.0pti}(u),E_{b\hskip 1.0pti}(u)\big{]} in $u$ does not exceed $\lambda_{i}-1$. Furthermore, by (2.1) and (3.3) the coefficient of $u^{\lambda_{i}-1}$ equals $\lambda_{i}E^{(\lambda_{i}-1)}_{b\hskip 1.0pti}[-2]-\lambda_{i}E^{(\lambda_{i}-1)}_{b\hskip 1.0pti}[-2]=0$ so that the degree of the polynomial does not exceed $\lambda_{i}-2$.

Now consider the contribution of the $2\times 2$ minor under consideration to the coefficient of $x^{n-k}$ in the second determinant. If $n-k\leqslant i-1$ then the contribution is a polynomial in $u$ whose degree does not exceed

[TABLE]

which is less than the sum in (3.2) because $\lambda_{n-k+1}-1<\lambda_{i+1}$. If $n-k\geqslant i$ then the contribution to the coefficient of $x^{n-k}$ is a polynomial in $u$ whose degree does not exceed

[TABLE]

which is again less than (3.2).

Continuing with the case $a=i$ in (3.5), suppose now that $\lambda_{i}<\lambda_{i+1}$. Exactly as above we find that the minor equals $E_{b\hskip 1.0pti}(u)(\tau_{i}-\tau_{i+1})$ plus a polynomial in $u$ of degree $\leqslant\lambda_{i+1}-2$. This implies that the resulting contribution to the coefficient of $x^{n-k}$ is a polynomial in $u$ whose degree is less than (3.2).

The minor (3.5) is equal to zero for $i+1\leqslant a<b$, while the remaining values of $a$ and $b$ are considered in a way quite similar to the above arguments. This allows us to conclude that the coefficients $\phi^{(r)}_{k}\in V(\mathfrak{a})$ satisfying (2.5) are annihilated by all generators of the form $E^{(0)}_{i+1\,i}[0]$.

Generators $E^{(\lambda_{i+1}-\lambda_{i})}_{i\,i+1}[0]$.

We will argue in the same way as above. It will be convenient to multiply the expression $E^{(\lambda_{i+1}-\lambda_{i})}_{i\,i+1}[0]\,{\rm cdet}\,\mathcal{E}$ by $u^{\lambda_{i+1}-\lambda_{i}}$ so that the product will be written as the difference of two column-determinants

[TABLE]

obtained from ${\rm cdet}\,\mathcal{E}$ by replacing row $i+1$ and column $i$, respectively, as indicated, where now we use the notation $\widetilde{\mathcal{E}}_{i\hskip 1.0ptj}=\mathcal{E}_{i\hskip 1.0ptj}$ for $j\geqslant i+1$ and

[TABLE]

for $j\leqslant i$, while $\widehat{\mathcal{E}}_{k\,i+1}=\mathcal{E}_{k\,i+1}$ for $k\leqslant i$ and

[TABLE]

for $k\geqslant i+1$. The product $u^{\lambda_{i+1}-\lambda_{i}}\,E^{(\lambda_{i+1}-\lambda_{i})}_{i\,i+1}[0]\,{\rm cdet}\,\mathcal{E}$ is a polynomial in $x$ and we use calculations similar to the previous case to verify that for $k=1,\dots,n$ its coefficient of $x^{n-k}$ is a polynomial in $u$ of degree less than the sum of $\lambda_{i+1}-\lambda_{i}$ and the expression (3.2).

Generators $E^{(0)}_{i\hskip 1.0pti}[1]$.

Note the commutation relations \big{[}E^{(0)}_{i\hskip 1.0pti}[1],\tau_{j}\big{]}=\lambda_{j}\hskip 1.0ptE^{(0)}_{i\hskip 1.0pti}[0] implied by (3.3). Hence, we also have the relations

[TABLE]

where we set \varkappa_{i\hskip 1.0ptm}=\big{\langle}E_{ii}^{(0)},E_{mm}^{(0)}\big{\rangle} and use the notation

[TABLE]

By [3, Lemma 4.1], we get

[TABLE]

which equals the sum

[TABLE]

plus the difference of two column-determinants

[TABLE]

where $\mathcal{E}^{\widehat{\jmath}}_{\widehat{\jmath}}$ denotes the column-determinant of the matrix obtained from $\mathcal{E}$ by deleting row and column $j$. Now we proceed as in [3] relying on Lemma 4.2 therein to evaluate the action of the elements of the form $E^{(0)}_{i\hskip 1.0pti}[0]$, $\mathcal{E}_{i\hskip 1.0ptj}[0]$ and $\mathcal{E}_{m\hskip 1.0pti}[0]$. For the generator $E^{(0)}_{i\hskip 1.0pti}[0]$ occurring as the $(j,j)$-entry with $j\neq i$ we have

[TABLE]

where row and column $j$ are deleted in the column-determinants in the sums, and $\mathcal{E}_{i\hskip 1.0ptm}$ occurs in row $i$ and column $m$. Clearly, the left hand side of (3.10) is zero for $j=i$.

Next consider the first determinant in (3.9). For the expression $\mathcal{E}_{i\hskip 1.0ptj}[0]$ occurring as the $(i,j)$ entry, with the remaining entries in column $j$ equal to zero, we get an expansion into a sum of column-determinants,

[TABLE]

where row $i$ and column $j$ are deleted in the column-determinants in the sum. Note that nonzero commutators occur only for $i\neq j$. If $m\neq i$, then the commutator \big{[}\mathcal{E}_{i\hskip 1.0ptj}[0],\mathcal{E}_{j\hskip 1.0ptm}\big{]} is a polynomial in $u$ whose component of maximal degree equals $\min(\lambda_{i},\lambda_{j})\hskip 1.0ptE^{(\lambda_{m}-1)}_{i\hskip 1.0ptm}[-1]\,u^{\lambda_{m}-1}$. By writing the column-determinants as polynomials in $x$, we can see that the resulting contribution to each polynomial $\psi_{k}(u)$ in (3.1) of any components with the powers of $u$ less than $\lambda_{m}-1$, is a polynomial in $u$ whose degree is less than (3.2). For this reason we may replace the commutator \big{[}\mathcal{E}_{i\hskip 1.0ptj}[0],\mathcal{E}_{j\hskip 1.0ptm}\big{]} in the above expansion with the polynomial $\min(\lambda_{i},\lambda_{j})\hskip 1.0pt\mathcal{E}_{i\hskip 1.0ptm}$.

Furthermore, observe that if $s\neq j$ and $s<i$, then the commutator \big{[}\mathcal{E}_{i\hskip 1.0ptj}[0],\mathcal{E}_{s\hskip 1.0pti}\big{]} occurring for $i>j$, is zero for $\lambda_{i}-\lambda_{s}\geqslant\lambda_{j}$. Otherwise, if $\lambda_{i}-\lambda_{s}<\lambda_{j}$, then the commutator is a polynomial in $u$ whose component of maximal degree equals $(\lambda_{i}-\lambda_{j}-\lambda_{s})\hskip 1.0ptE^{(\lambda_{j}-1)}_{s\hskip 1.0ptj}[-1]\,u^{\lambda_{j}-1}$. Similarly, the maximal degree component of the commutator \big{[}\mathcal{E}_{i\hskip 1.0ptj}[0],\mathcal{E}_{s\hskip 1.0pti}\big{]} with $s>i>j$ equals $-\lambda_{j}\hskip 1.0ptE^{(\lambda_{j}-1)}_{s\hskip 1.0ptj}[-1]\,u^{\lambda_{j}-1}$.

By looking at the powers of $x$ we find that the contributions of these components to each polynomial $\psi_{k}(u)$ in (3.1) are polynomials in $u$ whose degrees are less than the expression in (3.2), unless $\lambda_{j}=\lambda_{i}$. Moreover, the same property is shared by the components with powers of $u$ less than $\lambda_{j}-1$. This allows us to replace all commutators \big{[}\mathcal{E}_{i\hskip 1.0ptj}[0],\mathcal{E}_{s\hskip 1.0pti}\big{]} with $s\neq j$ occurring in the column-determinant, with the polynomial $-\min(\lambda_{j},\lambda_{s})\hskip 1.0pt\mathcal{E}_{s\hskip 1.0ptj}$.

The same argument implies that we may replace the remaining commutators with $i>j$ by the rule

[TABLE]

Bringing the calculations together, we can write the first column-determinant in (3.9) as the sum of two expressions

[TABLE]

and

[TABLE]

Here row $i$ and column $j$ are considered to be deleted in the column-determinants so that the rows and columns are labelled, respectively, by the symbols $1,\dots,\widehat{\imath},\dots,n$ and $1,\dots,\widehat{\jmath},\dots,n$. The only nonzero entry $\mathcal{E}_{i\hskip 1.0ptm}-\delta_{i\hskip 1.0ptm}\tau_{i}$ in column $m$ in the column-determinants in (3.12) occurs in row $j$, while the displayed middle column in (3.13) occupies column $i$.

Note that permutation of rows in column-determinants results in a changed sign as for usual commutative determinants. Hence, by moving row $j$ in the expression (3.12) to the position of the $i$-th row we can present this expression in the form

[TABLE]

where the entry $\mathcal{E}_{i\hskip 1.0ptm}-\delta_{i\hskip 1.0ptm}\tau_{i}$ now occupies the $(i,m)$-position, while the rows and columns of the column-determinant are labelled by the symbols $1,\dots,\widehat{\jmath},\dots,n$.

In the expression (3.13) write each column-determinant as the difference

[TABLE]

and move row $j$ in the second determinant to the position of the $i$-th row so that the corresponding part of (3.13) will be written as the sum

[TABLE]

where the entry $\lambda_{j}\hskip 1.0pt\tau_{j}$ now occupies the $(i,i)$-position.

The next step is to show that taking signs into account, we can move column $i$ in the first column-determinant in (3.15) to the left by permuting it with columns $i-1,i-2,\dots,j+1$ consecutively without changing its contribution to the coefficients of the powers of $x$. Arguing by induction, suppose that column $i$ was swapped with a few columns on its left. As an induction step, we need to show that for any $l\in\{j+1,\dots,i-1\}$ the sum of two column determinants of the form

[TABLE]

does not contribute to any polynomial $\psi_{k}(u)$ in (3.1) beyond powers of $u$ less than (3.2). Using their expansions along the two displayed adjacent columns we note that the sum of any $2\times 2$ minors

[TABLE]

with $a<b$ is a polynomial in $u$ whose degree does not exceed $\lambda_{l}-1$. Therefore, the contribution of the sum (3.16) to $\psi_{k}(u)$ is a polynomial in $u$ whose degree is less than the sum in (3.2), except for the case where $\lambda_{i}=1$. However, in this case we must have $\lambda_{1}=\dots=\lambda_{i}$ so that the difference (3.17) is easily checked to be zero (which agrees with the Manin matrix property of [3, Lemma 4.3]). Thus, expression (3.13) can be replaced with

[TABLE]

where the middle column in the first sum occupies the position of column $j$, with the row and column $i$ considered to be deleted, while the middle column occurring in the second sum occupies the position of column $i$ with the row and column $j$ considered to be deleted.

Observe that the column expansion along column $i$ and the use of relations of the form (3.11) lead to a similar evaluation of the second determinant in (3.9). As in the above arguments we will only keep the terms which can contribute to the coefficients of the powers $x^{n-k}$ beyond powers of $u$ less than (3.2). By an appropriate adjustment of the labels of rows and columns of emerging column-determinants, as compared with the evaluation of the first determinant in (3.9), we can write the second determinant (taking into account the minus sign in front) as the sum of two expressions

[TABLE]

and

[TABLE]

where the rows and columns of the column-determinants in (3.19) and in the second sum in (3.20) are labelled by $1,\dots,\widehat{\imath},\dots,n$ with $\lambda_{i}\hskip 1.0pt\tau_{i}$ occurring as the $(j,j)$-entry, while the rows and columns in the first sum in (3.20) are labelled by $1,\dots,\widehat{\jmath},\dots,n$.

Bringing together the evaluations of the column-determinants in (3.8) and (3.9) by taking into account (3.10), (3.14), (3.18), (3.19) and (3.20), we can conclude that, modulo terms not contributing to the coefficients of the powers $x^{n-k}$ beyond powers of $u$ less than (3.2), the element $E^{(0)}_{i\hskip 1.0pti}[1]\hskip 1.0pt{\rm cdet}\,\mathcal{E}$ equals the sum of the following expressions:

[TABLE]

where the rows and columns of the column-determinants are labelled by $1,\dots,\widehat{\imath},\dots,n$ and we only displayed columns labelled by $j$; plus

[TABLE]

where the rows and columns of the column-determinants are labelled by $1,\dots,\widehat{\jmath},\dots,n$ and we displayed columns $i$ and $m$, respectively; together with

[TABLE]

where the rows and columns of the column-determinants are labelled by $1,\dots,\widehat{\jmath},\dots,n$ in the first sum and by $1,\dots,\widehat{\imath},\dots,n$ in the second, and $\tau_{i}-\tau_{j}$ occurs as the $(i,i)$ and $(j,j)$-entry, respectively. It is understood that the remaining non-displayed entries of all column-determinants coincide with the respective entries of the matrix $\mathcal{E}$.

The final part of the arguments is a straightforward verification that the contribution of each of the above column-determinants to the coefficient of $x^{n-k}$ in the polynomial $E^{(0)}_{i\hskip 1.0pti}[1]\hskip 1.0pt{\rm cdet}\,\mathcal{E}$ is a polynomial in $u$ whose degree is less than (3.2).

Generators $E^{(0)}_{i\hskip 1.0pti}[s]$ for $s\geqslant 2$.

Note that due to the fact that the element $E^{(0)}_{1\hskip 1.0pt1}+\dots+E^{(0)}_{n\hskip 1.0ptn}$ belongs to the kernel of the bilinear form $\langle\,\,,\,\rangle$, this step is not necessary for the Takiff case, where $\lambda_{1}=\dots=\lambda_{n}$. Indeed, all elements $E^{(0)}_{1\hskip 1.0pt1}[s]+\dots+E^{(0)}_{n\hskip 1.0ptn}[s]$ are central in the Lie algebra $\widehat{\mathfrak{a}}$, whereas in $\mathfrak{a}$ we have $E^{(0)}_{i\hskip 1.0pti}-E^{(0)}_{i+1\,i+1}=[E^{(0)}_{i\,i+1},E^{(0)}_{i+1\hskip 1.0pti}]$. Therefore, the required vanishing properties of ${\rm cdet}\,\mathcal{E}$ for the generators $E^{(0)}_{i\hskip 1.0pti}[s]$ for $s\geqslant 2$ follow from the previously considered cases.

Returning to the general values of the $\lambda_{i}$, note the relations

[TABLE]

implied by (2.6). Therefore the required properties $E^{(0)}_{i\hskip 1.0pti}[s]\hskip 1.0pt\phi^{(r)}_{k}=0$ for $s\geqslant 2$ follow from Lemma 2.3 by a straightforward induction on $s$ and $k$, where the cases $s=1$ and $k=1$ are taken as the induction bases.

Proof of Lemma 2.3.

The commutator of two derivations $\Delta$ and $T$ of V(\mathfrak{a})\cong{\rm U}\big{(}t^{-1}\mathfrak{a}[t^{-1}]\big{)} is another derivation $[\Delta,T]=2\hskip 1.0ptd,$ where the action of $d$ is determined by the relations

[TABLE]

We also have the commutation relation $[d,T]=-T$.

Applying the operator $\Delta$ to the column-determinant ${\rm cdet}\,\mathcal{E}$ we get the expansion

[TABLE]

where we use notation (3.6). The next step is to evaluate these column-determinants in the vacuum module in the same way as for the determinants (3.9) above, with the use of the relations $[d,\,\mathcal{E}_{ij}]=-\mathcal{E}^{\circ}_{ij}$ with $\mathcal{E}^{\circ}_{ij}=\mathcal{E}_{ij}-\delta_{ij}\hskip 1.0ptx$. The contribution arising from the action of $d$ equals

[TABLE]

where rows and columns of the column-determinants are labelled by the numbers $1,\dots,\widehat{\jmath},\dots,n$, and we displayed column $m$ in the middle.

To evaluate the column-determinants in (3.25) containing the terms $\mathcal{E}_{i\hskip 1.0ptj}[0]$, apply formulas (3.11). It is clear from these evaluations of the expression in (3.25) that the coefficient of $x^{n-k}$ is a polynomial in $u$ of degree not exceeding the number given in (3.2). This implies relations (2.7) thus proving the first part of the lemma. Furthermore, in proving its the second part, we only need to keep the leading component in $u$ of degree equal to (3.2). Apply the above formulas obtained for the evaluation of determinants (3.9), including suitable permutations of rows and columns of column-determinants, which are valid modulo lower degree terms in $u$. As a result, we can represent the contribution arising from the terms $\mathcal{E}_{i\hskip 1.0ptj}[0]$ as the sum of the expressions

[TABLE]

where the rows and columns of the column-determinants are labelled by $1,\dots,\widehat{\jmath},\dots,n$; plus the sum

[TABLE]

where the rows and columns of the column-determinants are labelled by $1,\dots,\widehat{\imath},\dots,n$; together with

[TABLE]

where the rows and columns of the column-determinants are labelled by $1,\dots,\widehat{\jmath},\dots,n$ and $\lambda_{j}\hskip 1.0pt(\tau_{j}-\tau_{i})$ occurs as the $(i,i)$-entry.

As with the determinants of the form (3.21) and (3.23) considered above, we can see that no contribution to the leading component in $u$ of degree equal to (3.2) comes from the expression (3.28), while modulo lower degree terms in $u$, both expressions (3.26) and (3.27) can be simplified by replacing all coefficients $\lambda_{1},\dots,\lambda_{j-1}$ with $\lambda_{j}$.

Thus, summarizing the arguments, we may conclude that for $r=\lambda_{n-k+2}+\dots+\lambda_{n}-k+1$ the image $\Delta(\phi^{(r)}_{k})$ coincides with the coefficient of $u^{r}$ of the polynomial in $u$ which occurs as the coefficient of $x^{n-k}$ in the expression

[TABLE]

where the rows and columns of the column-determinants are labelled by $1,\dots,\widehat{\jmath},\dots,n$ in the first sum and by $1,\dots,\widehat{\imath},\dots,n$ in the second.

The coefficient of $x^{n-k}$ in the above expression is a linear combination of the principal minors $\mathcal{E}^{a_{1},\dots,a_{k-1}}$ of the matrix $\mathcal{E}$ corresponding to the rows and columns $a_{1}<\dots<a_{k-1}$. In particular, observe that the linear combination contains the principal minor $\mathcal{E}^{n-k+2,\dots,n}$ with the coefficient

[TABLE]

To apply the formula for $\phi^{(r)}_{k-1}$ provided by (2.4) and complete the proof of the lemma we need to verify that all other principal minors containing $u^{r}$ occur in the linear combination with the same coefficient.

Define the parameters $1\leqslant s\leqslant n-k+2$ and $n-k+2\leqslant p\leqslant n$ by the conditions

[TABLE]

The principal minors containing $u^{r}$ must have the form $\mathcal{E}^{a_{1},\dots,a_{q},p+1,\dots,n}$ for some

[TABLE]

The coefficient of such a minor in the linear combination equals

[TABLE]

where we set $a_{0}=0$ and $a_{q+1}=p+1$. This coefficient coincides with (3.29) thus completing the proof of the lemma. ∎

To establish the required vanishing properties of the elements $\phi^{(r)}_{k}$ with respect to the remaining family of generators $E^{(p)}_{i\hskip 1.0pti}[s]$ with $s\geqslant 0$, observe that in view of Lemma 2.3 and relations (3.24), it is sufficient to consider the values $s=0$ and $s=1$.

Generators $E^{(p)}_{i\hskip 1.0pti}[0]$ with $p=0,1,\dots,\lambda_{i}-1$.

For a fixed value or $p$ we have the commutation relations

[TABLE]

where we set $\widetilde{\mathcal{E}}_{i\hskip 1.0pti}=\widehat{\mathcal{E}}_{i\hskip 1.0pti}=0$,

[TABLE]

and

[TABLE]

Hence the element $E^{(p)}_{i\hskip 1.0pti}[0]\,{\rm cdet}\,\mathcal{E}$ equals the difference of two column-determinants

[TABLE]

which simplifies to

[TABLE]

It remains to verify that each of these column-determinants is a polynomial in $x$ with the property that the coefficient of $x^{n-k}$ is a polynomial in $u$ whose degree is less that $p$ plus the expression in (3.2); the calculations are quite similar to the previous cases.

Generators $E^{(p)}_{i\hskip 1.0pti}[1]$ with $p=1,\dots,\lambda_{i}-1$.

We proceed as in the case $p=0$ considered above and use similar calculations with some simplifications due to the fact that they do not involve values of the bilinear form. We have the relations

[TABLE]

where we use the notation

[TABLE]

where empty sums are understood as being equal to zero. Relations (3.7) hold in the same form with $E^{(0)}_{i\hskip 1.0pti}[1]$ replaced by $u^{p}\hskip 1.0ptE^{(p)}_{i\hskip 1.0pti}[1]$ so that the commutator \big{[}u^{p}\hskip 1.0ptE^{(p)}_{i\hskip 1.0pti}[1],{\rm cdet}\,\mathcal{E}\big{]} equals the sum

[TABLE]

plus the difference of two column-determinants

[TABLE]

It turns out that in contrast to the case $p=0$, each of these column-determinants is a polynomial in $x$ with the property that the coefficient of $x^{n-k}$ is a polynomial in $u$ whose degree is less that $p$ plus the expression in (3.2). Indeed, this follows easily from the respective counterparts of the decompositions (3.10) and (3.11), because the resulting column-determinants retain the property that the degree of $u$ of any entry in column $l$ does not exceed $\lambda_{l}-1$.

We have thus proved that the coefficients $\phi^{(r)}_{k}$ with $k=1,\dots,n$ satisfying conditions (2.5) belong to the center $\mathfrak{z}(\widehat{\mathfrak{a}})$ of the vertex algebra $V(\mathfrak{a})$. To prove the second part of the theorem, equip the vector space $V(\mathfrak{a})\cong{\rm U}(t^{-1}\mathfrak{a}[t^{-1}])$ with the canonical filtration of the universal enveloping algebra and consider the symbols $\bar{\phi}^{(r)}_{k}$ of the coefficients in the associated graded algebra ${\rm S}(t^{-1}\mathfrak{a}[t^{-1}])={\rm gr}\,{\rm U}(t^{-1}\mathfrak{a}[t^{-1}])$. The symbols belong to the subalgebra of $\mathfrak{a}[t]$-invariants in ${\rm S}(t^{-1}\mathfrak{a}[t^{-1}])$, and it is clear from the definition of the elements $\phi^{(r)}_{k}\in V(\mathfrak{a})$ that each $\bar{\phi}^{(r)}_{k}$ coincides with the image of a certain element of the algebra of $\mathfrak{a}$-invariants ${\rm S}(\mathfrak{a})^{\mathfrak{a}}$ under the embedding $\imath:X\mapsto X[-1]$ for $X\in\mathfrak{a}$. More precisely, introduce polynomials in $u$ with coefficients in ${\rm S}(\mathfrak{a})$ by

[TABLE]

and write the determinant

[TABLE]

as a polynomial in $x$,

[TABLE]

Then under the embedding $\imath$ we have $\imath:\overline{\Phi}^{(r)}_{k}\mapsto\bar{\phi}^{(r)}_{k}$. By [2, Thm 4.1], the coefficients $\overline{\Phi}^{(r)}_{k}$ with $k=1,\dots,n$ and $r$ satisfying conditions (2.5) are algebraically independent generators of the algebra ${\rm S}(\mathfrak{a})^{\mathfrak{a}}$. The same generators were produced in an earlier work [13, Conj. 4.1] as a conjecture, which was later confirmed in [16, Thm 7]. In the terminology of [13], these generators arise from the good generating system of ${\rm S}(\mathfrak{gl}_{N})^{\mathfrak{gl}_{N}}$ for the nilpotent element $e$, formed by the coefficients of the characteristic polynomial of the matrix $[E_{ij}]_{i,j=1}^{N}$. This generating system satisfies the assumptions of [1, Thm 3.2] which implies that ${\rm S}(t^{-1}\mathfrak{a}[t^{-1}])^{\mathfrak{a}[t]}$ is a polynomial algebra in the variables $T^{j}\bar{\phi}^{(r)}_{k}$ with $j\geqslant 0$, where $k=1,\dots,n$ and $r$ satisfies conditions (2.5). This completes the proof of Theorem 2.1, since under these conditions on the indices, the elements $T^{j}\phi^{(r)}_{k}$ are algebraically independent generators of the algebra $\mathfrak{z}(\widehat{\mathfrak{a}})$.

Acknowledgments

The support of the Australian Research Council, grant DP180101825 is gratefully acknowledged.