#  Quantized Weyl algebras, the double centralizer property, and a new First
Fundamental Theorem for $U_{q}(\mathfrak{gl}_{n})$

Gail Letzter  Gail Letzter, Mathematics Research Group, National Security
Agency<EMAIL_ADDRESS>,  Siddhartha Sahi  Siddhartha Sahi, Department
of Mathematics, Rutgers University<EMAIL_ADDRESS>and  Hadi Salmasian
Hadi Salmasian, Department of Mathematics and Statistics, University of Ottawa
<EMAIL_ADDRESS>
###### Abstract.

Let $\mathscr{P}:=\mathscr{P}_{m\times n}$ denote the quantized coordinate
ring of the space of $m\times n$ matrices, equipped with natural actions of
the quantized enveloping algebras $U_{q}(\mathfrak{gl}_{m})$ and
$U_{q}(\mathfrak{gl}_{n})$. Let $\mathscr{L}$ and $\mathscr{R}$ denote the
images of $U_{q}(\mathfrak{gl}_{m})$ and $U_{q}(\mathfrak{gl}_{n})$ in
$\mathrm{End}(\mathscr{P})$, respectively. We define a $q$-analogue of the
algebra of polynomial-coefficient differential operators inside
$\mathrm{End}(\mathscr{P})$, henceforth denoted by $\mathscr{PD}$, and we
prove that $\mathscr{L}\cap\mathscr{PD}$ and $\mathscr{R}\cap\mathscr{PD}$ are
mutual centralizers inside $\mathscr{PD}$. Using this, we establish a new
First Fundamental Theorem of invariant theory for $U_{q}(\mathfrak{gl}_{n})$.
We also compute explicit formulas in terms of $q$-determinants for generators
of the algebras $\mathscr{L}_{\mathfrak{h}}\cap\mathscr{PD}$ and
$\mathscr{R}_{\mathfrak{h}}\cap\mathscr{PD}$, where
$\mathscr{L}_{\mathfrak{h}}$ and $\mathscr{R}_{\mathfrak{h}}$ denote the
images of the Cartan subalgebras of $U_{q}(\mathfrak{gl}_{m})$ and
$U_{q}(\mathfrak{gl}_{n})$ in $\mathrm{End}(\mathscr{P})$, respectively. Our
algebra $\mathscr{PD}$ and the algebra $\mathrm{Pol}(\mathrm{Mat}_{m,n})_{q}$
that is defined in [SSV04] are related by extension of scalars, but we give a
new construction of $\mathscr{PD}$ using deformed twisted tensor products.

##  1\. Introduction

The First Fundamental Theorem (FFT) is one of the pinnacles of invariant
theory with a history as old as Hermann Weyl’s influential book, _The
Classical Groups_ [Wy39]. In its original form, the FFT for the group
$\mathrm{GL}_{n}$ describes the generators of the subalgebra of
$\mathrm{GL}_{n}$-invariants in the polynomial algebra $\mathcal{P}(V^{\oplus
k}\oplus(V^{*})^{\oplus l})$, where $V:=\mathbb{C}^{n}$ denotes the standard
$\mathrm{GL}_{n}$-module.

It was pointed out by R. Howe [Ho95, Sec. 2.3] that the FFT has an equivalent
formulation as a double centralizer property, which we now recall. Let
$\mathrm{Mat}_{m\times n}$ denote the vector space of complex $m\times n$
matrices. Then $\mathrm{Mat}_{m\times n}$ has a natural
$\mathrm{GL}_{m}\times\mathrm{GL}_{n}$-module structure by left and right
matrix multiplication. We equip the algebra
$\mathcal{P}:=\mathcal{P}(\mathrm{Mat}_{m\times n})$ of polynomials on
$\mathrm{Mat}_{m\times n}$ and the algebra
$\mathcal{PD}:=\mathcal{PD}(\mathrm{Mat}_{m\times n})$ of polynomial-
coefficient differential operators on $\mathrm{Mat}_{m\times n}$ with their
canonical $\mathrm{GL}_{m}\times\mathrm{GL}_{n}$-module structures. Recall
that $\mathcal{P}$ is a $\mathcal{PD}$-module. The (infinitesimal) actions of
the Lie algebras $\mathfrak{gl}_{m}$ and $\mathfrak{gl}_{n}$ on $\mathcal{P}$
are given by certain differential operators of order one, which are usually
called _polarization operators_. It follows that there exists a homomorphism
of algebras $\phi:U_{m,n}\to\mathcal{PD}$, where
$U_{m,n}:=U(\mathfrak{gl}_{m})\otimes U(\mathfrak{gl}_{n})$ is the tensor
product of the universal enveloping algebras of $\mathfrak{gl}_{m}$ and
$\mathfrak{gl}_{n}$, such that the diagram

(1)
$\textstyle{U_{m,n}\otimes\mathcal{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\otimes
f\mapsto\phi(x)\otimes f\hskip 14.22636pt}$$\scriptstyle{\hskip
14.22636ptx\otimes f\mapsto x\cdot
f}$$\textstyle{\mathcal{P}}$$\textstyle{\mathcal{PD}\otimes\mathcal{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D\otimes
f\mapsto D\cdot f}$

commutes. The _operator commutant version_ of the FFT, according to [Ho95, Thm
2.3.3], states that the subalgebra $\mathcal{PD}^{\mathrm{GL}_{m}}$ of
$\mathrm{GL}_{m}$-invariants in $\mathcal{PD}$ is generated by the image of
$U(\mathfrak{gl}_{n})$. Since
$\mathcal{PD}^{\mathrm{GL}_{m}}=\mathcal{PD}^{U(\mathfrak{gl}_{m})}$, the
latter assertion is equivalent to the following: the images of
$U(\mathfrak{gl}_{m})$ and $U(\mathfrak{gl}_{n})$ in $\mathcal{PD}$ are mutual
centralizers.

In [LZZ11, Sec. 6] the authors extend the original form of the FFT to the
quantized enveloping algebra $U_{q}(\mathfrak{gl}_{n})$ by considering a
$q$-analogue of $\mathcal{P}(V^{\oplus k}\oplus(V^{*})^{l})$ that carries a
$U_{q}(\mathfrak{gl}_{n})$-action, and then describing the generators of the
subalgebra of invariants. It is then natural to ask if the operator commutant
version of the FFT also has a $q$-analogue. It turns out that in the quantized
setting, the situation for the operator commutant FFT is more subtle than in
the classical case. One major issue is how to quantize the Weyl algebra
$\mathcal{PD}$ and, more importantly, the map $\phi:U_{m,n}\to\mathcal{PD}$.
Indeed we provide some justification that the latter map cannot be fully
quantized (see Proposition 3.14.4). Nevertheless, our first main result
(Theorem A) is a positive answer to the above question.

From now on let $\Bbbk:=\mathbb{C}(q)$ be the field of rational functions in a
parameter $q$. For the operator commutant FFT in the quantized setting we need
a quantized Weyl algebra $\mathscr{PD}:=\mathscr{PD}_{m\times n}$. The
$\Bbbk$-algebra $\mathscr{PD}$ that we consider is closely related to the
algebra $\mathrm{Pol}(\mathrm{Mat}_{m,n})_{q}$ of [SSV04, BKV06] (see
Corollary 3.16.1). We give a different construction of $\mathscr{PD}$ as the
_deformed twisted tensor product_ of $\mathscr{P}:=\mathscr{P}_{m\times n}$,
the quantized coordinate ring of $\mathrm{Mat}_{m\times n}$, and
$\mathscr{D}:=\mathscr{D}_{m\times n}$, the quantized algebra of constant-
coefficient differential operators on $\mathrm{Mat}_{m\times n}$ (see Section
3 for precise definitions). The construction of $\mathscr{P}$ and
$\mathscr{D}$ is analogous to the FRT construction [KS97, Sec. 9.1].
Concretely, the algebra $\mathscr{PD}$ is generated by $2mn$ generators
$t_{i,j}$ and $\partial_{i,j}$, where $1\leq i\leq m$ and $1\leq j\leq n$,
modulo the relations that are described in Section 3 (see Definition 3.7.3).
From now on we set

$U_{L}:=U_{q}(\mathfrak{gl}_{m})\quad,\quad
U_{R}:=U_{q}(\mathfrak{gl}_{n})\quad,\quad U_{LR}:=U_{L}\otimes U_{R}.$

Both $\mathscr{P}$ and $\mathscr{PD}$ are $U_{LR}$-module algebras (the
explicit formulas for the $U_{LR}$-action on generators are given in Remark
3.6.2). Furthermore, $\mathscr{P}$ is naturally a $\mathscr{PD}$-module. In
particular, we have homomorphisms of associative algebras

$\phi_{U}:U_{LR}\to\mathrm{End}_{\Bbbk}(\mathscr{P})\quad\text{and}\quad\phi_{PD}:\mathscr{PD}\to\mathrm{End}_{\Bbbk}(\mathscr{P}).$

Since $\mathscr{P}$ is a faithful $\mathscr{PD}$-module (see Proposition
3.11.4), we can identify $\mathscr{PD}$ with $\phi_{PD}(\mathscr{PD})$. Using
the latter identification, we set

$\mathscr{L}:=\phi_{U}(U_{L}\otimes 1),\quad\mathscr{R}:=\phi_{U}(1\otimes
U_{R}),\quad{\mathscr{L}_{\bullet}}:=\mathscr{L}\cap\mathscr{PD},\quad{\mathscr{R}_{\bullet}}:=\mathscr{R}\cap\mathscr{PD}.$

Note that in general we have ${\mathscr{L}_{\bullet}}\subsetneq\mathscr{L}$
and ${\mathscr{R}_{\bullet}}\subsetneq\mathscr{R}$. In fact if $m\leq n$ then
the restriction of $\phi_{U}$ to $U_{L}\otimes 1$ yields an isomorphism
$U_{L}\cong\mathscr{L}$ but one can show that
$\phi_{U}^{-1}({\mathscr{L}_{\bullet}})$ is properly contained in the locally
finite part of $U_{L}$ (see Proposition 3.14.3 and Example 10.2.2). Set

(2) $\mathsf{L}_{i,j}:=\sum_{r=1}^{n}t_{i,r}\partial_{j,r}\ \text{ for }1\leq
i,j\leq m\ \text{ and }\
\mathsf{R}_{i,j}:=\sum_{r=1}^{m}t_{r,i}\partial_{r,j}\ \text{ for }1\leq
i,j\leq n.$

The $\mathsf{L}_{i,j}$ and the $\mathsf{R}_{i,j}$ are natural analogues in
$\mathscr{PD}$ of the polarization operators of the (non-quantized) Weyl
algebra $\mathcal{PD}$. We have $\mathsf{L}_{i,j}\in{\mathscr{L}_{\bullet}}$
and $\mathsf{R}_{i,j}\in{\mathscr{R}_{\bullet}}$ (see Corollary 6.1.6). Of
course, similar inclusions hold in the non-quantized case. However, the
situation with the preimages of the $\mathsf{L}_{i,j}$ and the
$\mathsf{R}_{i,j}$ in $U_{L}$ and $U_{R}$ is more complicated. For example,
the root vector $E_{\varepsilon_{i}-\varepsilon_{j}}$ of $U_{L}$
(respectively, $U_{R}$), where $i<j$, does not lie in the preimage of
$\mathsf{L}_{i,j}$ (respectively, $\mathsf{R}_{i,j}$).

Henceforth we adopt the following notation: for subsets
$\mathcal{Y},\mathcal{Z}$ of an algebra $\mathcal{X}$, we set

(3) $\mathcal{Y}^{\mathcal{Z}}:=\\{y\in\mathcal{Y}\,:\,yz=zy\text{ for all
}z\in\mathcal{Z}\\}.$

Our first main theorem is the following.

######  Theorem A.

Let $\mathscr{L}$, $\mathscr{R}$, ${\mathscr{L}_{\bullet}}$, and
${\mathscr{R}_{\bullet}}$ be the subalgebras of
$\mathrm{End}_{\Bbbk}(\mathscr{P})$ defined above. We identify $\mathscr{PD}$
with $\phi_{PD}(\mathscr{PD})\subseteq\mathrm{End}_{\Bbbk}(\mathscr{P})$. Then
the following statements hold.

  * (i)

$\mathscr{PD}^{{\mathscr{R}_{\bullet}}}=\mathscr{PD}^{\mathscr{R}}={\mathscr{L}_{\bullet}}$.
Furthermore, ${\mathscr{L}_{\bullet}}$ is generated by the $\mathsf{L}_{i,j}$
for $1\leq i,j\leq m$.

  * (ii)

$\mathscr{PD}^{{\mathscr{L}_{\bullet}}}=\mathscr{PD}^{\mathscr{L}}={\mathscr{R}_{\bullet}}$.
Furthermore, ${\mathscr{R}_{\bullet}}$ is generated by the $\mathsf{R}_{i,j}$
for $1\leq i,j\leq n$.

Let us elucidate the relation between Theorem A and the literature on Howe
duality and the FFT in the quantized setting. Quantized analogues of
$(\mathfrak{gl}_{m},\mathfrak{gl}_{n})$-duality have been established in
[Zh02] and [NYM93], but these works do not consider the double centralizer
property inside a quantized Weyl algebra. To compare our results with those of
Lehrer–Zhang–Zhang [LZZ11], we briefly explain their formulation of the FFT
for $U_{q}(\mathfrak{gl}_{n})$. In [LZZ11, Sec. 6] the authors define a
$q$-analogue of the algebra $\mathcal{P}(V^{\oplus k}\oplus(V^{*})^{\oplus
l})$, which they call $\mathcal{A}_{k,l}$ (this algebra tacitly depends on $n$
as well). The algebra $\mathcal{A}_{k,l}$ is isomorphic to a twisted tensor
product of $\mathscr{P}_{k\times n}$ and $\mathscr{D}_{l\times n}$, but the
twisting is only with respect to the universal $R$-matrix of
$U_{q}(\mathfrak{gl}_{n})$. In particular, in the special case $k=l$ the
relations on the generators of $\mathcal{A}_{k,k}$ are not symmetric with
respect to their indices. Because of this asymmetry, $\mathcal{A}_{k,k}$ does
not appear to be the desired object for proving a double centralizer
statement. The twisting that we consider to define $\mathscr{PD}$ uses the
universal $R$-matrices of both $U_{L}$ and $U_{R}$. In addition, unlike
$\mathcal{A}_{k,l}$ whose relations are homogeneous, the relation (R6) of
$\mathscr{PD}$ is not homogeneous. From this viewpoint, $\mathscr{PD}$
resembles the classical Weyl algebra more than $\mathcal{A}_{k,k}$.

Our second main result (Theorem B below) is a new FFT for
$U_{q}(\mathfrak{gl}_{n})$, in the spirit of the aforementioned result of
[LZZ11], for a family of algebras $\mathscr{A}_{k,l,n}$ where $k,l,n$ are
positive integers. The latter algebras generalize $\mathscr{PD}$ and indeed we
have $\mathscr{A}_{m,m,n}=\mathscr{PD}_{m\times n}$. An explicit presentation
of $\mathscr{A}_{k,l,n}$ by generators and relations is given in Proposition
3.9.2. To state Theorem B, we need some notation. For integers $1\leq a\leq m$
and $1\leq b\leq n$ there exists an embedding of associative algebras

(4) $\boldsymbol{e}=\boldsymbol{e}_{a\times b}^{m\times
n}:\mathscr{PD}_{a\times b}\hookrightarrow\mathscr{PD}_{m\times n}$

that is defined as follows. We relabel the generators of
$\mathscr{PD}_{a\times b}$ by setting

(5) $\tilde{t}_{i,j}:=t_{a+1-i,b+1-j}\ \ \text{ and }\ \
\tilde{\partial}_{i,j}:=\partial_{a+1-i,b+1-j}\quad\text{ for }1\leq i\leq a,\
1\leq j\leq b.$

We relabel the generators of $\mathscr{PD}_{m\times n}$ similarly, with $a$
and $b$ replaced by $m$ and $n$ respectively. The map $\boldsymbol{e}$ of (4)
is uniquely determined by the assignments
$\boldsymbol{e}(\tilde{t}_{i,j}):=\tilde{t}_{i,j}$ and
$\boldsymbol{e}(\tilde{\partial}_{i,j}):=\tilde{\partial}_{i,j}$ (see
Proposition 3.9.4). Concretely, the map $\boldsymbol{e}$ identifies the
$a\times b$ matrices $[t_{i,j}]$ and $[\partial_{i,j}]$ formed by the
generators of $\mathscr{PD}_{a\times b}$ with the intersections of the lowest
$a$ rows and the rightmost $b$ columns in the analogous $m\times n$ matrices
formed by the generators of $\mathscr{PD}_{m\times n}$. Now fix integers
$k,l,n\geq 1$ and set $m:=\max\\{k,l\\}$. We define $\mathscr{A}_{k,l,n}$ to
be the subalgebra of $\mathscr{PD}_{m\times n}$ that is generated by the
$\tilde{t}_{i,j}$ and the $\tilde{\partial}_{i^{\prime},j}$, where $1\leq
i\leq k$, $1\leq i^{\prime}\leq l$, and $1\leq j\leq n$. The $U_{R}$-action on
$\mathscr{PD}_{m\times n}$ leaves $\mathscr{A}_{k,l,n}$ invariant and thus
$\mathscr{A}_{k,l,n}$ is a $U_{R}$-module algebra. The standard degree
filtration of $\mathscr{A}_{k,l,n}$ (corresponding to setting
$\deg\tilde{t}_{i,j}=\deg\tilde{\partial}_{i^{\prime},j}=1$) is
$U_{R}$-stable, hence the associated graded algebra
$\mathrm{gr}(\mathscr{A}_{k,l,n})$ is also a $U_{R}$-module algebra. Let
$\epsilon_{R}$ be the counit of $U_{R}$ and denote the subalgebra of
$U_{R}$-invariants in $\mathscr{A}_{k,l,n}$ by
$\left(\mathscr{A}_{k,l,n}\right)_{(\epsilon_{R})}$, that is,

$\left(\mathscr{A}_{k,l,n}\right)_{(\epsilon_{R})}:=\left\\{D\in\mathscr{A}_{k,l,n}\,:\,x\cdot
D=\epsilon_{R}(x)D\text{ for }x\in U_{R}\right\\}.$

We denote the subalgebra of $U_{R}$-invariants in
$\mathrm{gr}(\mathscr{A}_{k,l,n})$ by
$\big{(}\mathrm{gr}(\mathscr{A}_{k,l,n})\big{)}_{(\epsilon_{R})}$ as well. For
$1\leq i\leq k$ and $1\leq j\leq l$ we define elements
$\tilde{\mathsf{L}}_{i,j}\in\mathscr{A}_{k,l,n}$ by the formula

(6)
$\tilde{\mathsf{L}}_{i,j}:=\sum_{r=1}^{n}\tilde{t}_{i,r}\tilde{\partial}_{j,r}=\sum_{r=1}^{n}t_{m-i+1,r}\partial_{m-j+1,r}.$

By the same formula we can define analogous elements in
$\mathrm{gr}(\mathscr{A}_{k,l,n})$. By a slight abuse of the symbol
$\mathrm{gr}(\cdot)$, we denote these elements of
$\mathrm{gr}(\mathscr{A}_{k,l,n})$ by $\mathrm{gr}(\tilde{\mathsf{L}}_{i,j})$.

######  Theorem B.

The algebras $\left(\mathscr{A}_{k,l,n}\right)_{(\epsilon_{R})}$ and
$\big{(}\mathrm{gr}(\mathscr{A}_{k,l,n})\big{)}_{(\epsilon_{R})}$ are
generated by the $\tilde{\mathsf{L}}_{i,j}$ and the
$\mathrm{gr}(\tilde{\mathsf{L}}_{i,j})$ respectively, for $1\leq i\leq k$ and
$1\leq j\leq l$.

It would be interesting to relate Theorem B to the quantized FFT of [LZZ11,
Thm 6.10] for example by a deformation argument. However, we are unable to
establish such a connection.

Our third main theorem (Theorem C below) explicitly describes the images in
$\mathscr{PD}=\mathscr{PD}_{m\times n}$ of the Cartan subalgebras of $U_{L}$
and $U_{R}$. To state Theorem C we need to define certain elements of
$\mathscr{PD}$ that are constructed using $q$-determinants. Let
$\mathbf{i}:=(i_{1},\ldots,i_{r})$ and $\mathbf{j}:=(j_{1},\ldots,j_{r})$ be
$r$-tuples of integers satisfying $1\leq i_{1}<\cdots<i_{r}\leq m$ and $1\leq
j_{1}<\cdots<j_{r}\leq n$. Define quantum minors
$M^{\mathbf{i}}_{\mathbf{j}}\in\mathscr{P}$ and
$\overline{M}^{\mathbf{i}}_{\mathbf{j}}\in\mathscr{D}$ by

(7)
$M^{\mathbf{i}}_{\mathbf{j}}:=\sum_{\sigma}(-q)^{\ell(\sigma)}t_{i_{\sigma(1)},j_{1}}\cdots
t_{i_{\sigma(r)},j_{r}}\quad\text{and}\quad\overline{M}^{\mathbf{i}}_{\mathbf{j}}:=\sum_{\sigma}(-q^{-1})^{\ell(\sigma)}\partial_{i_{\sigma(1)},j_{1}}\cdots\partial_{i_{\sigma(r)},j_{r}},$

where the summations are over permutations in $r$ letters, and $\ell(\sigma)$
denotes the length of $\sigma$ (in the sense of Coxeter groups). For
$a,b,r\geq 1$ define $\mathbf{D}(r,a,b)\in\mathscr{PD}_{a\times b}$ by

(8)
$\mathbf{D}(r,a,b):=\sum_{\mathbf{i}}\sum_{\mathbf{j}}M^{\mathbf{i}}_{\mathbf{j}}{\overline{M}}^{\mathbf{i}}_{\mathbf{j}},$

where the summation indices $\mathbf{i}:=(i_{1},\ldots,i_{r})$ and
$\mathbf{j}:=(j_{1},\ldots,j_{r})$ satisfy $1\leq i_{1}<\cdots<i_{r}\leq a$
and $1\leq j_{1}<\cdots<j_{r}\leq b$. We also set $\mathbf{D}(0,a,b)=1$. For
$0\leq r\leq n$ and $1\leq k\leq n$ we define
$\mathbf{D}_{k,r}\in\mathscr{PD}_{m\times n}$ by

$\mathbf{D}_{k,r}:=\boldsymbol{e}_{m\times k}^{m\times
n}\left({\mathbf{D}}(r,m,k)\right).$

Similarly, for $0\leq r\leq m$ and $1\leq k\leq m$ we define
$\mathbf{D}_{k,r}^{\prime}\in\mathscr{PD}_{m\times n}$ by

$\mathbf{D}^{\prime}_{k,r}:=\boldsymbol{e}_{k\times n}^{m\times
n}\left(\mathbf{D}(r,k,n)\right).$

Note that $\mathbf{D}_{k,r}=\mathbf{D}^{\prime}_{k,r}=0$ when
$r>\min\\{k,m,n\\}$ and $\mathbf{D}_{k,0}=\mathbf{D}^{\prime}_{k,0}=1$. Set

(9) $\mathbf{R}_{a}:=\sum_{r=0}^{a}(q^{2}-1)^{r}\mathbf{D}_{a,r}\ \text{ for
}1\leq a\leq
n\quad\text{and}\quad\mathbf{L}_{b}:=\sum_{r=0}^{b}(q^{2}-1)^{r}\mathbf{D}^{\prime}_{b,r}\
\text{ for }1\leq b\leq m.$

Furthermore, let $U_{\mathfrak{h},L}$ and $U_{\mathfrak{h},R}$ denote the
Cartan subalgebras of $U_{L}$ and $U_{R}$, respectively (see Subsection 3.1).
We set

(10) $\mathscr{L}_{\mathfrak{h}}:=\phi_{U}(U_{\mathfrak{h},L}\otimes
1)\quad,\quad\mathscr{L}_{\mathfrak{h},\bullet}:=\mathscr{PD}\cap\mathscr{L}_{\mathfrak{h}}\quad,\quad\mathscr{R}_{\mathfrak{h}}:=\phi_{U}(1\otimes
U_{\mathfrak{h},R})\quad,\quad\mathscr{R}_{\mathfrak{h},\bullet}:=\mathscr{PD}\cap\mathscr{R}_{\mathfrak{h}}.$

######  Theorem C.

The following statements hold.

  * (i)

$\mathscr{R}_{\mathfrak{h},\bullet}$ is generated by
$\mathbf{R}_{1},\ldots,\mathbf{R}_{n}$.

  * (ii)

$\mathscr{L}_{\mathfrak{h},\bullet}$ is generated by
$\mathbf{L}_{1},\ldots,\mathbf{L}_{m}$.

In the proofs of our theorems we borrow at least two key ideas from [LZZ11].
First, we use the bialgebra structure of $\mathscr{P}_{n\times n}$ to define a
map ${{}\Gamma}_{k,l,n}$ from $\mathscr{P}_{k\times l}$ onto the subalgebra of
$U_{R}$-invariants in $\mathrm{gr}(\mathscr{A}_{k,l,n})$ (assuming
$n\geq\max\\{k,l\\}$). The map ${{}\Gamma}_{k,l,n}$, given in Definition
7.1.3, is similar to the map introduced in [LZZ11, Lem. 6.11]. Second, we
define a new product on $\mathscr{P}_{k\times l}$ such that the map
${{}\Gamma}_{k,l,n}$ becomes an isomorphism of algebras (see Definition
8.2.4). This product is analogous to the one defined in [LZZ11, Lem. 6.13].
However our product is given by a more complicated (and asymmetric) formula,
because it needs to be simutaneously compatible with _two_ universal
$R$-matrices. As a consequence, establishing the desired properties of this
product requires new ideas (see Section 8). Because of this, and the fact that
unlike [LZZ11] the generators $\mathsf{L}_{i,j}$ (respectively,
$\mathsf{R}_{i,j}$) or even their graded analogues are not weight vectors for
the Cartan subalgebras of the two copies of $U_{L}$ (respectively, $U_{R}$)
that act on $\mathscr{PD}$, the proofs of Theorems A and B become
substantially more complicated than the analogous results of [LZZ11]. See
Subsection 9.2 for more details.

The results of this paper were obtained as part of a project on Capelli
operators for quantum symmetric spaces. From this standpoint, it is natural to
ask if one can define quantized Weyl algebras in the latter setting and then
realize the action of (a large subalgebra of) the quantized enveloping algebra
via elements of this Weyl algebra. We address this question and its connection
to Capelli operators in upcoming work [LSS22a, LSS22b].

The structure of this paper is as follows. In Section 2 we review the required
background material on Hopf algebras and twisted tensor products. In Section 3
we construct and study the quantized Weyl algebra $\mathscr{PD}$ and its
variations, namely $\mathscr{PD}^{\mathrm{gr}}$, $\mathscr{A}_{k,l,n}$ and
$\mathscr{A}_{k,l,n}^{\mathrm{gr}}$. The main goal of Section 4 is to prove
that under $\phi_{U}$ the elements $K_{\lambda_{L,a}}\otimes 1$ and $1\otimes
K_{\lambda_{R,b}}$, defined in (79), of the Cartan subalgebra of $U_{LR}$ are
mapped into $\mathscr{PD}$. In Section 5 we compute explicit formulas for
$\phi_{U}(K_{\lambda_{L,a}}\otimes 1)$ and $\phi_{U}(1\otimes
K_{\lambda_{R,b}})$. Section 6 is devoted to some general properties of the
polarization operators $\mathsf{L}_{i,j}$, $\mathsf{R}_{i,j}$ and their
variants. In Section 7 we define the map ${{}\Gamma}_{k,l,n}$ and establish
some of its properties. Section 8 defines the new product on
$\mathscr{P}_{k\times l}$ and establishes its properties. Theorems A and B are
proved in Section 9 and Theorem C is proved in Section 10. Finally, Section 11
lists the commonly used notation in the paper.

### Acknowledgement

The research of S.S. was partially supported by NSF grants DMS-1939600,
DMS-2001537, and Simons Foundation grant 509766. The research of H.S. was
partially supported by an NSERC Discovery Grant (RGPIN-2018-04044).

##  2\. Hopf algebras and deformed twisted tensor products

Throughout this section $\mathbb{K}$ will denote an arbitrary field and $H$
will be a Hopf algebra over $\mathbb{K}$. We denote the coproduct, counit, and
antipode of $H$ by $\Delta$, $\epsilon$, and $S$. The opposite and co-opposite
of $H$ are denoted by $H^{\mathrm{op}}$ and $H^{\mathrm{cop}}$ (we use the
same notation for bialgebras as well). Throughout the paper, our notation for
specific Hopf algebras $H$ will remain consistent with these choices.

If $A$ is an associative algebra and $V$ is an $A$-module, a subspace
$W\subseteq V$ is called _$A$ -stable_ if $A\cdot W\subseteq W$. Finally, an
associative algebra $A$ is called an _$H$ -module algebra_ if it is equipped
by an $H$-module structure such that the product of $A$ yields an $H$-module
homomorphism $A\otimes A\to A$.

###  2.1. Local finiteness modulo an ideal

Given a two-sided ideal $I$ of $H$ (considered as an associative algebra), we
set $E(x,I):=\left\\{\mathrm{ad}_{y}(x)+I\,:\,y\in H\right\\}$ for $x\in H$,
where $\mathrm{ad}_{y}(x):=\sum y_{1}xS(y_{2})$ is the left adjoint action of
$H$ (we use the Sweedler notation $\Delta(y)=\sum y_{1}\otimes y_{2}$ for the
coproduct). We set

(11) $\mathscr{F}(H,I):=\left\\{x\in
H\,:\,\dim_{\mathbb{K}}E(x,I)<\infty\right\\}.$

For $I=0$ this is the locally finite part of $H$ (in the sense of [JL94]),
which we will denote by $\mathscr{F}(H)$. We have $E(xx^{\prime},I)\subseteq
E(x,I)E(x^{\prime},I)$ for $x,x^{\prime}\in H$. Consequently,
$\mathscr{F}(H,I)$ is a subalgebra of $H$.

###  2.2. The finite dual of $H$

Given a finite dimensional left $H$-module $V$, by the _right dual_ of $V$ we
mean the dual space $V^{*}$ equipped with the $H$-action defined by $\langle
x\cdot v^{*},v\rangle:=\langle v^{*},S^{-1}(x)\cdot v\rangle$ for $v^{*}\in
V^{*}$ and $v\in V$, where $\langle\cdot,\cdot\rangle:V^{*}\otimes
V\to\mathbb{K}$ is the canonical pairing. The _matrix coefficients_ of $V$ are
the linear functionals $\mathsf{m}_{v^{*},v}\in H^{*}$ defined by

$\mathsf{m}_{v^{*},v}(x):=\langle v^{*},x\cdot v\rangle\quad\text{ for }x\in
H,\ v\in V,\ v^{*}\in V^{*}.$

Indeed $\mathsf{m}_{v^{*},v}\in H^{\circ}$, where $H^{\circ}$ denotes the
finite dual of $H$ (for the definition of $H^{\circ}$ see [KS97, Sec. 1.2.8]).
Recall that $H^{\circ}$ has a canonical Hopf algebra structure. The product of
$H^{\circ}$ is given by

(12) $\lambda\mu(x):=\sum\lambda(x_{1})\mu(x_{2})\quad\text{for
}\lambda,\mu\in H^{\circ}\text{ and }x\in H,$

where $\Delta(x)=\sum x_{1}\otimes x_{2}$. Given two finite dimensional
$H$-modules $V$ and $W$, we have

$\mathsf{m}_{v^{*}\otimes w^{*},v\otimes
w}=\mathsf{m}_{v^{*},v}\mathsf{m}_{w^{*},w}\quad\text{for }v,w\in V\text{ and
}v^{*},w^{*}\in W.$

Let $\Delta^{\circ}$ denote the coproduct of $H^{\circ}$, so that
$\Delta^{\circ}(\lambda)=\sum\lambda_{1}\otimes\lambda_{2}$ for $\lambda\in
H^{\circ}$, where $\sum\lambda_{1}\otimes\lambda_{2}$ is uniquely determined
by

(13)
$\langle\lambda,xy\rangle=\sum\langle\lambda_{1},x\rangle\langle\lambda_{2},y\rangle\quad\text{for
}x,y\in H.$

If $\\{v_{i}\\}_{i=1}^{d}$ is a basis of $V$ and $\\{v_{i}^{*}\\}_{i=1}^{d}$
is the dual basis of $V^{*}$, then
$\Delta^{\circ}(\mathsf{m}_{v^{*},v})=\sum_{i=1}^{d}\mathsf{m}_{v^{*},v_{i}}\otimes\mathsf{m}_{v_{i}^{*},v}$.
The following remark will be used in Section 3.

######  Remark 2.2.1.

Let $H^{\bullet}\subseteq H^{\circ}$ be a sub-bialgebra. Then $H^{\bullet}$ is
an $H$-module algebra with respect to _right translation_ , where the action
is defined by $\langle x\cdot\lambda,y\rangle:=\langle\lambda,yx\rangle$ for
$\lambda\in H^{\bullet}$ and $x,y\in H$. If $H$ is equipped with a
$\mathbb{K}$-linear map $x\mapsto x^{\natural}$ that yields an isomorphism of
Hopf algebras $H\to H^{\mathrm{op}}$, then $H^{\bullet}$ has another
$H$-module algebra structure defined by $\langle
x\cdot\lambda,y\rangle:=\langle\lambda,x^{\natural}y\rangle$, which we call
_left translation_. Given any homomorphism of associative algebras $\tau:H\to
H$, we can define $\tau$-twisted left and right translation actions of $H$ on
$H^{\bullet}$, given respectively by the formulas

$\langle
x\cdot\lambda,y\rangle:=\langle\lambda,\tau(x)^{\natural}y\rangle\quad\text{and}\quad\langle
x\cdot\lambda,y\rangle:=\langle\lambda,y\tau(x)\rangle.$

When $H^{\bullet}$ is equipped with either one of the two $\tau$-twisted
actions, the following statements hold.

  * (i)

If $\tau:H\to H$ is a homomorphism of coalgebras, then $H^{\bullet}$ is an
$H$-module algebra.

  * (ii)

If $\tau:H\to H$ is an anti-homomorphism of coalgebras, then $H^{\bullet}$ is
an $H^{\mathrm{cop}}$-module algebra.

###  2.3. The isotypic component of the trivial $H$-module

For any $H$-module $V$ we set

$V_{(\epsilon)}:=\\{v\in V\,:\,h\cdot v=\epsilon(h)v\\},$

where as before $\epsilon$ denotes the counit of $H$. Let
$\psi:H\to\mathrm{End}_{\mathbb{K}}(V)$ be the algebra homomorphism
corresponding to this module structure (hence $h\cdot v=\psi(h)v$ for $h\in H$
and $v\in V$). We equip $\mathrm{End}_{\mathbb{K}}(V)$ with an $H$-module
structure, defined by $h\cdot T:=\sum\psi(h_{1})T\psi(S(h_{2}))$ for $h\in H$
and $T\in\mathrm{End}_{\mathbb{K}}(V)$, where $\Delta(h)=\sum h_{1}\otimes
h_{2}$.

######  Lemma 2.3.1.

$\mathrm{End}_{\mathbb{K}}(V)_{(\epsilon)}=\mathrm{End}_{\mathbb{K}}(V)^{\psi(H)}$,
where the right hand side is defined as in (3).

###### Proof.

The inclusion $\supseteq$ follows from

$h\cdot
T=\sum\psi(h_{1})T\psi(S(h_{2}))=T\sum\psi(h_{1})\psi(S(h_{2}))=T\psi\left(\sum
h_{1}S(h_{2})\right)=\epsilon(h)T,$

for $T\in\mathrm{End}_{\mathbb{K}}(V)^{\psi(H)}$. For the inclusion
$\subseteq$ note that if $T\in\mathrm{End}_{\mathbb{K}}(V)_{(\epsilon)}$ then

$\displaystyle\psi(h)T=\sum\psi(h_{1})\epsilon(h_{2})T$
$\displaystyle=\sum\psi(h_{1})T\psi(S(h_{2}))\psi(h_{3})$
$\displaystyle=\sum\epsilon(h_{1})T\psi(h_{2})=T\psi\left(\sum\epsilon(h_{1})h_{2}\right)=T\psi(h).\qed$

###  2.4. Braided triples, twisted tensor products and their deformations

Let $\mathcal{C}$ be a full subcategory of the category of $H$-modules that is
closed under direct sums and tensor products. To ensure that $\mathcal{C}$ is
a monoidal category we assume that the trivial $H$-module (the one-dimensional
vector space $\mathbb{K}$ equipped with the action $h\mapsto\epsilon(h)$ for
$h\in H$) belongs to $\mathrm{Obj}(\mathcal{C})$.

Assume that $\mathcal{C}$ is _braided_. The braiding $\check{R}$ of
$\mathcal{C}$ is a natural family of $H$-module isomorphisms

$\check{R}_{V,W}:V\otimes W\to W\otimes V\quad\text{for
$V,W\in\mathrm{Obj}(\mathcal{C})$}$

that satisfies the usual hexagon axioms (see for example [EGNO15, Def.
8.1.1]). Henceforth we call $(H,\mathcal{C},\check{R})$ a _braided triple_.

Let $A$ and $B$ be two $H$-module algebras, with products $m_{A}$ and $m_{B}$.
Assume that $A,B\in\mathrm{Obj}(\mathcal{C})$.

######  Lemma 2.4.1.

The map $\check{R}_{B,A}:B\otimes A\to A\otimes B$ satisfies the following
relations:

  * (i)

$\check{R}_{B,A}(1\otimes a)=a\otimes 1$ for $a\in A$ and
$\check{R}_{B,A}(b\otimes 1)=1\otimes b$ for $b\in B$.

  * (ii)

$\check{R}_{B,A}\circ(\mathrm{id}_{B}\otimes
m_{A})=(m_{A}\otimes\mathrm{id}_{B})(\mathrm{id}_{A}\otimes\check{R}_{B,A})(\check{R}_{B,A}\otimes\mathrm{id}_{A})$.

  * (iii)

$\check{R}_{B,A}\circ(m_{B}\otimes\mathrm{id}_{A})=(\mathrm{id}_{A}\otimes
m_{B})(\check{R}_{B,A}\otimes\mathrm{id}_{B})(\mathrm{id}_{B}\otimes\check{R}_{B,A})$.

###### Proof.

This is well known, but we supply a proof because we did not find a reference.

(i) Equip $\mathbb{K}$ with the canonical $H$-module structure induced by the
counit $\epsilon:H\to\mathbb{K}$. It is well known (see for example [EGNO15,
Exer. 8.1.6]) that one has a commuting triangle

---
$\textstyle{\mathbb{K}\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\check{R}_{\mathbb{K},A}}$$\textstyle{A\otimes\mathbb{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A}$

where the lower sides of the triangle are the canonical left and right unit
isomorphisms. It follows immediately that $\check{R}_{\mathbb{K},A}(1\otimes
a)=a\otimes 1$. Furthermore, by naturality of $\check{R}$ the diagram

$\textstyle{\mathbb{K}\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\check{R}_{\mathbb{K},A}}$$\scriptstyle{1_{B}\otimes\mathrm{id}_{A}}$$\textstyle{A\otimes\mathbb{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}_{A}\otimes
1_{B}}$$\textstyle{B\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\check{R}_{B,A}}$$\textstyle{A\otimes
B}$

is commutative, hence $\check{R}_{B,A}(1\otimes a)=a\otimes 1$. The other
relation is proved similarly.

(ii) Consider the diagram of maps below:

$\textstyle{A\otimes B\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}_{A}\otimes\check{R}_{B,A}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
B\otimes A\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
\check{R}_{B,A}\otimes\mathrm{id}_{A}}$$\scriptstyle{\check{R}_{B,A\otimes
A}}$$\scriptstyle{\ \ \mathrm{id}_{B}\otimes m_{A}}$$\textstyle{B\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\check{R}_{B,A}}$$\textstyle{A\otimes
A\otimes
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ \
m_{A}\otimes\mathrm{id}_{B}}$$\textstyle{A\otimes B}$

The square is commutative by naturality of $\check{R}$ (because $m_{A}$ and
$m_{B}$ are $H$-module homomorphisms) and the triangle is commutative by the
hexagon axiom. The assertion of (ii) follows from comparing two maps $B\otimes
A\otimes A\to A\otimes B$ in the diagram: one is obtained by composition of
the top and the right edges of the square, the other is obtained by the outer
edges of the triangle and the bottom edge of the square.

(iii) Similar to (ii). ∎

######  Definition 2.4.2.

Let $A,B\in\mathrm{Obj}(\mathcal{C})$ be $H$-module algebras. We denote the
products of $A$ and $B$ by $m_{A}$ and $m_{B}$, respectively. The _$\check{R}$
-twisted tensor product_ of $A$ and $B$, denoted by $A\otimes_{\check{R}}B$,
is the vector space $A\otimes B$ equipped with the binary operation

(14) $(m_{A}\otimes
m_{B})\circ(\mathrm{id}_{A}\otimes\check{R}_{B,A}\otimes\mathrm{id}_{B}).$

######  Proposition 2.4.3.

$A\otimes_{\check{R}}B$ is an associative algebra.

###### Proof.

This is well known and proved for example in [VV94, Prop. 2.2]. ∎

From Lemma 2.4.1(i) it follows that for $a,a^{\prime}\in A$ and
$b,b^{\prime}\in B$ the product (14) satisfies

(15) $(a\otimes 1)(a^{\prime}\otimes b)(1\otimes
b^{\prime})=(aa^{\prime}\otimes bb^{\prime}).$

######  Remark 2.4.4.

The vector space $A\otimes_{\check{R}}B=A\otimes B$ carries two module
structures: as the outer tensor product over $\mathbb{K}$ of $H$-modules $A$
and $B$, it is an $H\otimes H$-module. As the inner tensor product of $A$ and
$B$, it is an $H$-module. Of course the latter $H$-module structure is
obtained from the former one via restriction along the coproduct map $H\to
H\otimes H$.

######  Proposition 2.4.5.

$A\otimes_{\check{R}}B$ is an $H$-module algebra with the $H$-module structure
of Remark 2.4.4.

###### Proof.

Given $a,a^{\prime}\in A$ and $b,b^{\prime}\in B$, if we write
$\check{R}_{B,A}(b\otimes a^{\prime})=\sum a^{\prime\prime}\otimes
b^{\prime\prime}$ then for $x\in H$ we have

$\displaystyle x\cdot\big{(}(a\otimes b)(a^{\prime}\otimes b^{\prime})\big{)}$
$\displaystyle=x\cdot\sum aa^{\prime\prime}\otimes
b^{\prime\prime}b=\sum(x_{1}\cdot(aa^{\prime\prime}))\otimes(x_{2}\cdot(b^{\prime\prime}b)).$

Since $A$ and $B$ are $H$-module algebras, from the latter equalities and (15)
it follows that

$\displaystyle x\cdot\big{(}(a\otimes b)(a^{\prime}\otimes b^{\prime})\big{)}$
$\displaystyle=\sum(x_{1}\cdot a)(x_{2}\cdot
a^{\prime\prime})\otimes(x_{3}\cdot b^{\prime\prime})(x_{4}\cdot b)$
$\displaystyle=\sum((x_{1}\cdot a)\otimes 1)((x_{2}\cdot
a^{\prime\prime})\otimes\big{(}x_{3}\cdot
b^{\prime\prime})\big{)}(1\otimes(x_{4}\cdot b)).$

Since $\check{R}_{A,B}$ is an $H$-module isomorphism, from the latter
equalities and (15) we have

$\displaystyle x\cdot\big{(}(a\otimes b)(a^{\prime}\otimes b^{\prime})\big{)}$
$\displaystyle=\sum((x_{1}\cdot a)\otimes
1)\big{(}\check{R}_{B,A}\big{(}(x_{2}\cdot b)\otimes(x_{3}\cdot
a^{\prime})\big{)}\big{)}(1\otimes(x_{4}\cdot b^{\prime}))$
$\displaystyle=\sum((x_{1}\cdot a)\otimes(x_{2}\cdot b))((x_{3}\cdot
a^{\prime})\otimes(x_{4}\cdot b^{\prime}))=\sum(x_{1}\cdot(a\otimes
b))(x_{2}\cdot(a^{\prime}\otimes b^{\prime})).$

Thus $A\otimes_{\check{R}}B$ is an $H$-module algebra. ∎

Let $E_{A}\subseteq A$ and $E_{B}\subseteq B$ be subspaces that generate $A$
and $B$, respectively. Thus $A\cong T(E_{A})/I_{A}$ and $B\cong
T(E_{B})/I_{B}$, where $T(X)$ denotes the tensor algebra on $X$, and $I_{A}$
and $I_{B}$ denote the corresponding ideals of relations. We assume that
$E_{A}$ and $E_{B}$ are $H$-stable.

######  Remark 2.4.6.

The $H$-module structures on $E_{A}$ and $E_{B}$ equip $T(E_{A})$ and
$T(E_{B})$ with canonical $H$-module algebra structures. It is straightforward
to verify that the maps $T(E_{A})\to A$ and $T(E_{B})\to B$ are $H$-module
homomorphisms. In particular, $I_{A}$ and $I_{B}$ are $H$-stable subspaces of
$T(E_{A})$ and $T(E_{B})$, respectively.

Consider the linear map

$\gamma_{A,B}:E_{A}\otimes E_{B}\to T(E_{A}\oplus E_{B})\ ,\
\gamma_{A,B}(a\otimes b):=ab.$

Note that we can express $\gamma_{A,B}$ as the composition

(16) $E_{A}\otimes
E_{B}\xrightarrow{\mathsf{i}_{A}\otimes\mathsf{i}_{B}}T(E_{A}\oplus
E_{B})\otimes T(E_{A}\oplus E_{B})\xrightarrow{\ a\otimes b\mapsto ab\
}T(E_{A}\oplus E_{B}),$

where $\mathsf{i}_{A}:E_{A}\to T(E_{A}\oplus E_{B})$ and
$\mathsf{i}_{B}:E_{B}\to T(E_{A}\oplus E_{B})$ are canonical embeddings. For
the next lemma, recall that the $H$-module structure on $E_{A}\oplus E_{B}$
induces a canonical $H$-module algebra structure on $T(E_{A}\oplus E_{B})$.

######  Lemma 2.4.7.

$\gamma_{A,B}$ is an $H$-module homomorphism.

###### Proof.

Since $\mathsf{i}_{A}$, $\mathsf{i}_{B}$ and the product of $T(E_{A}\oplus
E_{B})$ are $H$-module homomorphisms, the assertion follows from the
description of $\gamma_{A,B}$ in (16). ∎

By the universal property of tensor algebras the map $E_{A}\oplus E_{B}\to
A\otimes_{\check{R}}B$ given by the assignment $a\oplus b\mapsto a\otimes
1+1\otimes b$ induces a homomorphism of algebras

$\pi:T(E_{A}\oplus E_{B})\to A\otimes_{\check{R}}B.$

######  Proposition 2.4.8.

Let $A$, $B$, $E_{A}$ and $E_{B}$ be as above. Then $\pi$ induces an
isomorphism of algebras

$T(E_{A}\oplus E_{B})/I_{A,B}\cong A\otimes_{\check{R}}B,$

where $I_{A,B}$ denotes the two-sided ideal of $T(E_{A}\oplus E_{B})$
generated by $I_{A}$, $I_{B}$ and the relations

(17) $ba-\gamma_{A,B}\circ\check{R}_{B,A}(b\otimes a)\quad\text{for }a\in
E_{A},\ b\in E_{B}.$

###### Proof.

We have $\pi(E_{A}\oplus E_{B})=E_{A,B}$ where $E_{A,B}:=(E_{A}\otimes
1)\oplus(1\otimes E_{B})$. Furthermore, $A\otimes_{\check{R}}B$ is generated
as an algebra by $E_{A,B}$. Thus $\pi$ is a surjection. Next we prove that
$\pi$ is an injection. From the definition of the product of
$A\otimes_{\check{R}}B$ it follows that $I_{A,B}\subseteq\ker\pi$. Thus, to
complete the proof it suffices to verify the reverse inclusion.

By the relations (17), every element of $T(E_{A}\oplus E_{B})/I_{A,B}$ can be
expressed as a linear combination of products of elements of $E_{A}$ and
$E_{B}$ in which elements of $E_{A}$ occur before elements of $E_{B}$. Thus,

(18) $T(E_{A}\oplus E_{B})=T(E_{A})T(E_{B})+I_{A,B}.$

Now choose $\\{a_{\alpha}\\}_{\alpha\in\mathcal{I}_{A}}\subseteq T(E_{A})$ and
$\\{b_{\beta}\\}_{\beta\in\mathcal{I}_{B}}\subseteq T(E_{B})$ such that
$\\{I_{A}+a_{\alpha}\\}_{\alpha\in\mathcal{I}_{A}}$ is a basis of
$T(E_{A})/I_{A}\cong A$ and $\\{I_{B}+b_{\beta}\\}_{\beta\in\mathcal{I}_{B}}$
is a basis of $T(E_{B})/I_{B}\cong B$. From (18) it follows that the elements
$I_{A,B}+a_{\alpha}b_{\beta}$ for $\alpha\in I_{A}$ and $\beta\in I_{B}$
constitute a spanning set of $T(E_{A}\oplus E_{B})/I_{A,B}$.

From (15) it follows that $\pi(a_{\alpha})=\overline{a}_{\alpha}\otimes 1$ for
some $\overline{a}_{\alpha}\in A$ and
$\pi(b_{\beta})=1\otimes\overline{b}_{\beta}$ for some
$\overline{b}_{\beta}\in B$. Also,
$\pi(I_{A,B}+a_{\alpha}b_{\beta})=\overline{a}_{\alpha}\otimes\overline{b}_{\beta}$.
The sets $\\{\overline{a}_{\alpha}\\}_{\alpha\in\mathcal{I}_{A}}$ and
$\\{\overline{b}_{\beta}\\}_{\beta\in\mathcal{I}_{B}}$ are bases of $A$ and
$B$, respectively. Thus $\pi$ maps a spanning set of $T(E_{A}\oplus
E_{B})/I_{A,B}$ bijectively onto the basis $\\{a_{\alpha}\otimes
b_{\beta}\,:\,\alpha\in\mathcal{I}_{A},\beta\in\mathcal{I}_{B}\\}$ of
$A\otimes_{\check{R}}B$. It follows that $\ker\pi\subseteq I_{A,B}$. ∎

Let $\psi:E_{B}\times E_{A}\to\mathbb{K}$ be an $H$-invariant bilinear form,
that is

(19) $\sum\psi(x_{1}\cdot b,x_{2}\cdot a)=\epsilon(x)\psi(b,a)\quad\text{for
}a\in E_{A},\ b\in E_{B},\ x\in H,$

where as before $\epsilon:H\to\mathbb{K}$ denotes the counit of $H$. Let
$I_{A,B,\psi}$ denote the two-sided ideal of $T(E_{A}\oplus E_{B})$ that is
generated by $I_{A}$, $I_{B}$, and relations of the form

(20) $ba-\gamma_{A,B}\circ\check{R}_{B,A}(b\otimes a)-\psi(b,a)\quad\text{for
}a\in E_{A},\ b\in E_{B}.$

######  Definition 2.4.9.

We call the algebra $A\otimes_{\check{R},\psi}B:=T(E_{A}\oplus
E_{B})/I_{A,B,\psi}$ the _$\psi$ -deformed $\check{R}$-twisted tensor product_
of $A$ and $B$ relative to $E_{A}$ and $E_{B}$.

The $H$-module structure of $E_{A}\oplus E_{B}$ equips $T(E_{A}\oplus E_{B})$
with a canonical $H$-module algebra structure. We have the following
statement.

######  Proposition 2.4.10.

The canonical $H$-module algebra structure on $T(E_{A}\oplus E_{B})$ descends
to an $H$-module algebra structure on $A\otimes_{\check{R},\psi}B$.

###### Proof.

It suffices to verify that $I_{A,B,\psi}$ is an $H$-stable subspace. By Remark
2.4.6 the subspaces $I_{A}$ and $I_{B}$ of $T(E_{A}\oplus E_{B})$ are
$H$-stable. Thus it remains to show that the relations (20) span an
$H$-submodule of $T(E_{A}\oplus E_{B})$.

Any $x\in H$ acts on $T^{0}(E_{A}\oplus E_{B})\cong\mathbb{K}$ by
$\epsilon(x)$. Next let $a\in E_{A}$, $b\in E_{B}$ and $x\in H$. By (19) and
Lemma 2.4.7 we have

$\displaystyle x\cdot\big{(}ba$
$\displaystyle-\gamma_{A,B}\circ\check{R}_{B,A}(b\otimes a)-\psi(b,a)\big{)}$
$\displaystyle=\sum\big{(}(x_{1}\cdot b)(x_{2}\cdot
a)-\gamma_{A,B}\circ\check{R}_{B,A}(x_{1}\cdot b\otimes x_{2}\cdot
a)-\psi(x_{1}\cdot b,x_{2}\cdot a)\big{)},$

which is a sum of relations of the form (20). ∎

By the universal property of tensor algebras the maps $\mathsf{i}_{A}$ and
$\mathsf{i}_{B}$ induce embeddings of associative algebras
$\mathsf{i}_{A}:T(E_{A})\to T(E_{A}\oplus E_{B})$ and
$\mathsf{i}_{B}:T(E_{B})\to T(E_{A}\oplus E_{B})$. The latter maps induce
$H$-equivariant homomorphisms of associative algebras

$A\cong T(E_{A})/I_{A}\xrightarrow{\ \ \bar{\mathsf{i}}_{A}\ \
}A\otimes_{\check{R},\psi}B\quad\text{ and }\quad B\cong
T(E_{B})/I_{B}\xrightarrow{\ \ \bar{\mathsf{i}}_{B}\ \
}A\otimes_{\check{R},\psi}B.$

By tensoring the latter maps and then composing with the products of the
algebras $T(E_{A}\oplus E_{B})$ and $A\otimes_{\check{R},\psi}B$ we obtain the
following commutative diagram:

(25)

In the above diagram the vertical maps are the canonical quotients.

######  Remark 2.4.11.

From Remark 2.4.6 it follows that $\mathsf{i}_{A}\otimes\mathsf{i}_{B}$ and
$\bar{\mathsf{i}}_{A}\otimes\bar{\mathsf{i}}_{B}$ are $H\otimes H$-module
homomorphisms. Also, the products of $T(E_{A}\oplus E_{B})$ and
$A\otimes_{\check{R},\psi}B$ are $H$-module homomorphisms (see Proposition
2.4.10). Thus the composition of the bottom horizontal maps in (25), which is
given by

(26) $A\otimes B\to A\otimes_{\check{R},\psi}B\ ,\ a\otimes b\mapsto
I_{A,B,\psi}+ab,$

is an $H$-module homomorphism (recall that an $H\otimes H$-module homomorphism
is also an $H$-module homomorphism by restriction along the coproduct map
$H\to H\otimes H$).

###  2.5. Locally finite braided triples and their products

Given a braided triple $(H,\mathcal{C},\check{R})$ we set

(27) $R_{V,W}:=\sigma_{W,V}\circ\check{R}_{V,W},$

where

$\sigma_{W,V}:W\otimes V\to V\otimes W\quad,\quad w\otimes v\mapsto v\otimes
w$

is the standard flip map.

######  Definition 2.5.1.

We say a braided triple $(H,\mathcal{C},\check{R})$ is _locally finite_ if it
satisfies the following conditions.

  * (i)

Every $V\in\mathrm{Obj}(\mathcal{C})$ is a sum of its finite dimensional
submodules that belong to $\mathrm{Obj}(\mathcal{C})$.

  * (ii)

For finite dimensional modules $V,W\in\mathrm{Obj}(\mathcal{C})$ there exist
$\omega_{V,W},\overline{\omega}_{V,W}\in H\otimes H$ such that

$R_{V,W}(v\otimes w)=\omega_{V,W}\cdot(v\otimes w)\quad\text{and}\quad
R_{V,W}^{-1}(v\otimes w)=\overline{\omega}_{V,W}\cdot(v\otimes w)\quad\text{
for }v\otimes w\in V\otimes W.$

Let us briefly explain the idea behind Definition 2.5.1. The braiding of the
category of modules of the Hopf algebra $U_{q}(\mathfrak{gl}_{n})$ is given by
a formal series that does not belong to $U_{q}(\mathfrak{gl}_{n})\otimes
U_{q}(\mathfrak{gl}_{n})$ (see Subsection 3.2). Thus, in order to give a
rigorous proof of Proposition 2.6.2 below, we need to be able to replace this
formal series locally by a finite 2-tensor.

######  Lemma 2.5.2.

Let $(H,\mathcal{C},\check{R})$ be a locally finite braided triple and let
$V_{1},\ldots,V_{r},W_{1},\ldots,W_{r}\in\mathrm{Obj}(\mathcal{C})$ be finite
dimensional modules. Let $\omega_{V,W},\overline{\omega}_{V,W}\in H\otimes H$
be chosen as in Definition 2.5.1(ii), where $V:=\bigoplus_{i=1}^{r}V_{i}$ and
$W:=\bigoplus_{i=1}^{r}W_{i}$. Then

$R_{V_{i},W_{j}}(v_{i}\otimes w_{j})=\omega_{V,W}\cdot(v_{i}\otimes
w_{j})\quad\text{and}\quad R^{-1}_{V_{i},W_{j}}(v_{i}\otimes
w_{j})=\overline{\omega}_{V,W}\cdot(v_{i}\otimes w_{j}),$

for all $v_{i}\in V_{i}$ and $w_{j}\in W_{j}$ where $1\leq i,j\leq r$.

###### Proof.

This follows from naturality of $\check{R}$ with respect to the canonical maps
$V_{i}\otimes W_{j}\hookrightarrow V\otimes W$. ∎

We need to work with the braided triples associated to tensor products of Hopf
algebras $U_{q}(\mathfrak{gl}_{n})$. To this end, we need the following
definition.

######  Definition 2.5.3.

Let $(H,\mathcal{C},\check{R})$ and
$(H^{\prime},\mathcal{C}^{\prime},\check{R}^{\prime})$ be locally finite
braided triples. Set $H^{\prime\prime}:=H\otimes H^{\prime}$ and equip
$H^{\prime\prime}$ with the canonical tensor product Hopf algebra structure.
Furthermore, let $\mathcal{C}^{\prime\prime}$ be the full subcategory of the
category of $H^{\prime\prime}$-modules whose objects are direct sums of
modules of the form $V\otimes V^{\prime}$ where
$V\in\mathrm{Obj}(\mathcal{C})$ and
$V^{\prime}\in\mathrm{Obj}(\mathcal{C}^{\prime})$. We define a braiding on
$\mathcal{C}^{\prime\prime}$ as follows. First, for
$V,W\in\mathrm{Obj}(\mathcal{C})$ and
$V^{\prime},W^{\prime}\in\mathrm{Obj}(\mathcal{C}^{\prime})$ we set

(28) $\check{R}^{\prime\prime}_{V\otimes V^{\prime},W\otimes
W^{\prime}}:=\left(\check{R}_{V,W}\right)_{13}\circ\left(\check{R}^{\prime}_{V^{\prime},W^{\prime}}\right)_{24}.$

Here $\left(\check{R}^{\prime}_{V^{\prime},W^{\prime}}\right)_{24}$ means that
$\check{R}^{\prime}_{V^{\prime},W^{\prime}}$ acts on the $2^{\mathrm{nd}}$ and
the $4^{\mathrm{th}}$ components of $(V\otimes V^{\prime})\otimes(W\otimes
W^{\prime})$, resulting in a map

$(V\otimes V^{\prime})\otimes(W\otimes W^{\prime})\to(V\otimes
W^{\prime})\otimes(W\otimes V^{\prime}),$

and $\big{(}\check{R}_{V,W}\big{)}_{13}$ is defined analogously. Next for
$\widetilde{V}:=\bigoplus_{i}V_{i}\otimes V^{\prime}_{i}$ and
$\widetilde{W}:=\bigoplus_{j}W_{j}\otimes W^{\prime}_{j}$ we define

$\check{R}^{\prime\prime}_{\widetilde{V},\widetilde{W}}:=\bigoplus_{i,j}\check{R}^{\prime\prime}_{V_{i}\otimes
V^{\prime}_{i},W_{j}\otimes W_{j}^{\prime}}.$

######  Proposition 2.5.4.

Let $H^{\prime\prime}$, $\mathcal{C}^{\prime\prime}$ and
$\check{R}^{\prime\prime}$ be as in Definition 2.5.3. Then
$(H^{\prime\prime},\mathcal{C}^{\prime\prime},\check{R}^{\prime\prime})$ is a
locally finite braided triple.

###### Proof.

It is trivial to check that
$(H^{\prime\prime},\mathcal{C}^{\prime\prime},\check{R}^{\prime\prime})$ meets
the condition of Definition 2.5.1(i). Next we show that
$(H^{\prime\prime},\mathcal{C}^{\prime\prime},\check{R}^{\prime\prime})$ is a
braided triple. It is straightforward to check that
$\mathcal{C}^{\prime\prime}$ is closed with respect to tensor products and
arbitrary direct sums. The hexagon axioms for $\check{R}^{\prime\prime}$
follow from those for $\check{R}$ and $\check{R}^{\prime}$. Next we verify
naturality of $\check{R}^{\prime\prime}$. Since $\check{R}^{\prime\prime}$ is
defined by expansion on direct sums of modules, it suffices to prove
commutativity of the diagram

$\textstyle{(V\otimes V^{\prime})\otimes(W\otimes
W^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\check{R}_{V\otimes
V^{\prime},W\otimes W^{\prime}}}$$\scriptstyle{f_{V}\otimes
f_{W}}$$\textstyle{(W\otimes W^{\prime})\otimes(V\otimes
V^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{W}\otimes
f_{V}}$$\textstyle{(\underline{V}\otimes\underline{V}^{\prime})\otimes(\underline{W}\otimes\underline{W}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\check{R}_{\underline{V}\otimes\underline{V}^{\prime},\underline{W}\otimes\underline{W}^{\prime}}}$$\textstyle{(\underline{W}\otimes\underline{W}^{\prime})\otimes(\underline{V}\otimes\underline{V}^{\prime})}$

for all choices of
$V,W,\underline{V},\underline{W}\in\mathrm{Obj}(\mathcal{C})$,
$V^{\prime},W^{\prime},\underline{V}^{\prime},\underline{W}^{\prime}\in\mathrm{Obj}(\mathcal{C}^{\prime})$,
$f_{V}\in\mathrm{Mor}_{\mathcal{C}^{\prime\prime}}(V\otimes
V^{\prime},\underline{V}\otimes\underline{V}^{\prime})$ and
$f_{W}\in\mathrm{Mor}_{\mathcal{C}^{\prime\prime}}(W\otimes
W^{\prime},\underline{W}\otimes\underline{W}^{\prime})$. The subtlety here is
that we cannot assume that $f_{V}$ and $f_{W}$ can be decomposed into tensor
products of maps on the tensor components. Since $\mathcal{C}^{\prime\prime}$
satisfies the condition of Definition 2.5.1(i), using naturality of
$\check{R}$ and $\check{R}^{\prime}$ we can assume that all of the modules in
the commutative diagram are finite dimensional. Using Lemma 2.5.2 for the
triples $(H,\mathcal{C},\check{R})$ and
$(H^{\prime},\mathcal{C}^{\prime},\check{R}^{\prime})$ it follows that
$R_{V\otimes V^{\prime},W\otimes W^{\prime}}$ and
$R_{\underline{V}\otimes\underline{V}^{\prime},\underline{W}\otimes\underline{W}^{\prime}}$
are given by the left action of the same 2-tensor in $H^{\prime\prime}\otimes
H^{\prime\prime}$. Commutativity of the diagram follows immediately.

Finally, we show that the condition of Definition 2.5.1(ii) holds. If
$R_{V,W}$ and $R^{\prime}_{V^{\prime},W^{\prime}}$ are equal to the actions of
$\omega_{V,W}\in H\otimes H$ and $\omega^{\prime}_{V^{\prime},W^{\prime}}\in
H^{\prime}\otimes H^{\prime}$ respectively, then $R^{\prime\prime}_{V\otimes
V^{\prime},W\otimes W^{\prime}}$ is equal to the action of

$(\omega_{V,W})_{13}(\omega^{\prime}_{V^{\prime},W^{\prime}})_{24}\in
H^{\prime\prime}\otimes H^{\prime\prime}.$

This verifies Definition 2.5.1(ii) for pairs of $H^{\prime\prime}$-modules in
$\mathcal{C}^{\prime\prime}$ of the form $V\otimes V^{\prime}$ and $W\otimes
W^{\prime}$. Using Lemma 2.5.2 for the triples $(H,\mathcal{C},\check{R})$ and
$(H^{\prime},\mathcal{C}^{\prime},\check{R}^{\prime})$, the claim follows for
general $H^{\prime\prime}$-modules in $\mathcal{C}^{\prime\prime}$. A similar
argument can be given for $\big{(}R^{\prime\prime}_{V\otimes
V^{\prime},W\otimes W^{\prime}}\big{)}^{-1}$. ∎

###  2.6. A result on $H^{\prime\prime}$-stable subalgebras of
$A\otimes_{\check{R}^{\prime\prime}}B$

We continue with the notation of Definition 2.5.3. Let $A$ and $B$ be two
$H^{\prime\prime}$-module algebras such that
$A,B\in\mathrm{Obj}(\mathcal{C}^{\prime\prime})$. Recall from Remark 2.4.4 and
Proposition 2.4.5 that $A\otimes_{\check{R}^{\prime\prime}}B$ is an
$H^{\prime\prime}$-module algebra and also an $H^{\prime\prime}\otimes
H^{\prime\prime}$-module (but not necessarily an $H^{\prime\prime}\otimes
H^{\prime\prime}$-module algebra). For $\omega\in H\otimes H$ let

$T_{\omega}:A\otimes_{\check{R}^{\prime\prime}}B\to
A\otimes_{\check{R}^{\prime\prime}}B$

denote the linear endomorphism obtained by the action of $\omega_{13}$. Here
by definition $(x\otimes y)_{13}:=x\otimes 1\otimes y\otimes 1$ for $x,y\in
H$.

######  Lemma 2.6.1.

Let $(H,\mathcal{C},\check{R})$ and
$(H^{\prime},\mathcal{C}^{\prime},\check{R}^{\prime})$ be locally finite
braided triples and let
$(H^{\prime\prime},\mathcal{C}^{\prime\prime},\check{R}^{\prime\prime})$ be
defined as in Definition 2.5.3. Let
$A,B\in\mathrm{Obj}(\mathcal{C}^{\prime\prime})$ be $H^{\prime\prime}$-module
algebras. Let $V^{\prime\prime}_{A}\subseteq A$ and
$V^{\prime\prime}_{B}\subseteq B$ be finite dimensional
$H^{\prime\prime}$-submodules such that
$V^{\prime\prime}_{A},V^{\prime\prime}_{B}\in\mathrm{Obj}(\mathcal{C}^{\prime\prime})$.
We decompose $V_{A}^{\prime\prime}$ and $V_{B}^{\prime\prime}$ as

$V_{A}^{\prime\prime}=\bigoplus_{i\in\Omega_{A}}V_{i,A}\otimes
V_{i,A}^{\prime}\quad\text{and}\quad
V_{B}^{\prime\prime}=\bigoplus_{j\in\Omega_{B}}V_{j,B}\otimes
V_{j,B}^{\prime},$

where $V_{i,A},V_{j,B}\in\mathrm{Obj}(\mathcal{C})$ and
$V^{\prime}_{i,A},V^{\prime}_{j,B}\in\mathrm{Obj}(\mathcal{C}^{\prime})$ for
$i\in\Omega_{A}$ and $j\in\Omega_{B}$ (here $\Omega_{A}$ and $\Omega_{B}$ are
index sets). Let $\omega_{V,W},\overline{\omega}_{V,W}\in H\otimes H$ be
chosen as in Definition 2.5.1(ii) where

$V:=\bigoplus_{i\in\Omega_{A}}V_{i,A}\quad\text{and}\quad
W:=\bigoplus_{j\in\Omega_{B}}V_{j,B}.$

For $x,y\in H$ let $\omega_{x,y}\in H\otimes H$ be defined by
$\omega_{x,y}:=\overline{\omega}_{V,W}(y\otimes x)\omega_{V,W}$. Then

$\left((\check{R}^{\prime\prime}_{B,A})^{-1}\circ T_{x\otimes
y}\circ\check{R}^{\prime\prime}_{B,A}\right)\Big{|}_{V^{\prime\prime}_{B}\otimes
V^{\prime\prime}_{A}}=T_{\omega_{x,y}}\quad\text{for }1\leq i\leq r.$

###### Proof.

By naturality of $\check{R}^{\prime\prime}$ it follows that

$\left(\check{R}^{\prime\prime}_{B,A}\right)\big{|}_{V^{\prime\prime}_{B}\otimes
V^{\prime\prime}_{A}}=\check{R}^{\prime\prime}_{V^{\prime\prime}_{B},V^{\prime\prime}_{A}}\quad\text{and}\quad\left(\check{R}^{\prime\prime}_{B,A}\right)^{-1}\big{|}_{V^{\prime\prime}_{A}\otimes
V^{\prime\prime}_{B}}=\left(\check{R}^{\prime\prime}_{V^{\prime\prime}_{B},V^{\prime\prime}_{A}}\right)^{-1}.$

Thus from (28) it follows that

$\left((\check{R}^{\prime\prime}_{B,A})^{-1}\circ T_{x\otimes
y}\circ\check{R}^{\prime\prime}_{B,A}\right)\big{|}_{V_{j,B}\otimes
V^{\prime}_{j,B}\otimes V_{i,A}\otimes
V^{\prime}_{i,A}}=\left(\check{R}_{V_{j,B},V_{i,A}}\right)^{-1}_{13}\circ
T_{x\otimes y}\circ\left(\check{R}_{V_{j,B},V_{i,A}}\right)_{13}.$

The relation (27) and Lemma 2.5.2 for the triple $(H,\mathcal{C},\check{R})$
imply the assertion of the lemma. ∎

######  Proposition 2.6.2.

Let $(H,\mathcal{C},\check{R})$ and
$(H^{\prime},\mathcal{C}^{\prime},\check{R}^{\prime})$ be locally finite
braided triples and let
$(H^{\prime\prime},\mathcal{C}^{\prime\prime},\check{R}^{\prime\prime})$ be
defined as in Definition 2.5.3. Let
$A,B\in\mathrm{Obj}(\mathcal{C}^{\prime\prime})$ be $H^{\prime\prime}$-module
algebras. Let $\mathcal{E}$ be a subspace of
$A\otimes_{\check{R}^{\prime\prime}}B$ and let $\mathcal{A}$ denote the
subalgebra of $A\otimes_{\check{R}^{\prime\prime}}B$ that is generated by
$\mathcal{E}$. Finally, let $\overline{H}$ be a sub-bialgebra of $H$.

  * (i)

Assume that for every pair of finite dimensional modules
$V,W\in\mathrm{Obj}(\mathcal{C})$, we can choose
$\omega_{V,W},\overline{\omega}_{V,W}\in\overline{H}\otimes H$ that satisfy
Definition 2.5.1(ii). If $\mathcal{E}$ is stable under the action of the
subalgebra $H\otimes 1\otimes\overline{H}\otimes 1$ of
$H^{\prime\prime}\otimes H^{\prime\prime}$, then so is $\mathcal{A}$.

  * (ii)

Assume that for every pair of finite dimensional modules
$V,W\in\mathrm{Obj}(\mathcal{C})$, we can choose
$\omega_{V,W},\overline{\omega}_{V,W}\in H\otimes\overline{H}$ that satisfy
Definition 2.5.1(ii). If $\mathcal{E}$ is stable under the action of the
subalgebra $\overline{H}\otimes 1\otimes H\otimes 1$ of
$H^{\prime\prime}\otimes H^{\prime\prime}$, then so is $\mathcal{A}$.

###### Proof.

For $r\geq 0$ set $\mathcal{A}_{r}:=\mathrm{Span}\left\\{w^{(1)}\cdots
w^{(r)}\,:\,w^{(i)}\in\mathcal{E}\text{ for }1\leq i\leq r\right\\}$, so that
$\mathcal{A}:=\sum_{r\geq 0}\mathcal{A}_{r}$. We prove by induction on $r$
that under the assumptions of (i) (respectively, of (ii)), the subspace
$\mathcal{A}_{r}$ is $(H\otimes 1\otimes\overline{H}\otimes 1)$-stable
(respectively, $(\overline{H}\otimes 1\otimes H\otimes 1)$-stable). For $r=0$
this is trivial and for $r=1$ this follows from the assumption on
$\mathcal{E}$. Next assume $r>1$.

Choose any $\alpha,\beta\in A\otimes_{\check{R}^{\prime\prime}}B$. We can
express $\alpha$ and $\beta$ as finite sums $\alpha=\sum a\otimes b$ and
$\beta=\sum a^{\prime}\otimes b^{\prime}$ where $a,a^{\prime}\in A$ and
$b,b^{\prime}\in B$. For each pair $(b,a^{\prime})$ that occurs in these
summations we also express $\check{R}^{\prime\prime}_{B,A}(b\otimes
a^{\prime})$ as a summation, that is,

(29) $\check{R}^{\prime\prime}_{B,A}(b\otimes a^{\prime})=\sum
a^{\prime\prime}\otimes b^{\prime\prime},$

where of course the $a^{\prime\prime}\in A$ and the $b^{\prime\prime}\in B$
depend on $b$ and $a^{\prime}$. For $h,h^{\prime}\in H$ we have

$\displaystyle(h\otimes 1\otimes h^{\prime}\otimes 1)\cdot\alpha\beta$
$\displaystyle=(h\otimes 1\otimes h^{\prime}\otimes 1)\cdot\left(\sum(a\otimes
b)(a^{\prime}\otimes b^{\prime})\right)$ $\displaystyle=(h\otimes 1\otimes
h^{\prime}\otimes 1)\cdot\left(\sum aa^{\prime\prime}\otimes
b^{\prime\prime}b^{\prime}\right)=\sum\big{(}(h\otimes
1)\cdot(aa^{\prime\prime})\big{)}\otimes\big{(}(h^{\prime}\otimes
1)\cdot(b^{\prime\prime}b^{\prime})\big{)}.$

Since both $A$ and $B$ are $H^{\prime\prime}$-module algebras, from the above
calculation and (15) we obtain

$\displaystyle(h\otimes 1\otimes h^{\prime}\otimes 1)\cdot\alpha\beta$
$\displaystyle=\sum((h_{1}\otimes 1)\cdot a)((h_{2}\otimes 1)\cdot
a^{\prime\prime})\otimes((h^{\prime}_{1}\otimes 1)\cdot
b^{\prime\prime})((h^{\prime}_{2}\otimes 1)\cdot b^{\prime})$ (30)
$\displaystyle=\sum\big{(}((h_{1}\otimes 1)\cdot a)\otimes
1\big{)}\big{(}(h_{2}\otimes 1\otimes h^{\prime}_{1}\otimes
1)\cdot(a^{\prime\prime}\otimes
b^{\prime\prime})\big{)}\big{(}1\otimes((h^{\prime}_{2}\otimes 1)\cdot
b^{\prime})\big{)},$

where $\Delta(h)=\sum h_{1}\otimes h_{2}$ and $\Delta(h^{\prime})=\sum
h_{1}^{\prime}\otimes h_{2}^{\prime}$. From Lemma 2.6.1 it follows that for
each pair $(h_{2},h_{1}^{\prime})$ that occurs on the right hand side of (2.6)
there exists a 2-tensor $\omega_{h_{2},h_{1}^{\prime}}=\sum
u_{h_{2},h_{1}^{\prime}}\otimes u^{\prime}_{h_{2},h_{1}^{\prime}}$ in
$H\otimes H$ such that

$\displaystyle(h_{2}\otimes 1\otimes h^{\prime}_{1}\otimes
1)\cdot\sum(a^{\prime\prime}\otimes b^{\prime\prime})$
$\displaystyle=\check{R}^{\prime\prime}_{B,A}\left(\big{(}\omega_{h_{2},h^{\prime}_{1}}\big{)}_{13}\cdot(b\otimes
a^{\prime})\right)$ (31)
$\displaystyle=\check{R}^{\prime\prime}_{B,A}\left(\big{(}(u_{h_{2},h^{\prime}_{1}}\otimes
1)\cdot b\big{)}\otimes\big{(}(u^{\prime}_{h_{2},h^{\prime}_{1}}\otimes
1)\cdot a^{\prime}\big{)}\right).$

Note that by Lemma 2.6.1 the 2-tensors $\omega_{h_{2},h_{1}^{\prime}}$ are of
the form

(32)
$\omega_{h_{2},h_{1}^{\prime}}=\overline{\omega}_{V,W}(h_{1}^{\prime}\otimes
h_{2})\omega_{V,W},$

where $\omega_{V,W},\overline{\omega}_{V,W}\in H\otimes H$ satisfy the
constraint of Definition 2.5.1(ii) for suitable $H$-modules $V,W$. By
comparing (2.6) and (2.6) and then using (15) we obtain

$\displaystyle(h\otimes 1\otimes h^{\prime}\otimes 1)\cdot\alpha\beta$
$\displaystyle=\sum\left(\big{(}(h_{1}\otimes 1)\cdot
a\big{)}\otimes\big{(}(u_{h_{2},h^{\prime}_{1}}\otimes 1)\cdot
b\big{)}\right)\left(\big{(}(u^{\prime}_{h_{2},h^{\prime}_{1}}\otimes 1)\cdot
a^{\prime}\big{)}\otimes\big{(}(h_{2}^{\prime}\otimes 1)\cdot
b^{\prime}\big{)}\right)$ $\displaystyle=\sum\big{(}(h_{1}\otimes 1\otimes
u_{h_{2},h^{\prime}_{1}}\otimes
1)\cdot\alpha\big{)}\big{(}(u^{\prime}_{h_{2},h^{\prime}_{1}}\otimes 1\otimes
h^{\prime}_{2}\otimes 1)\cdot\beta\big{)}.$

In what follows we complete the proofs of part (i) and part (ii).

(i) It suffices to prove that for $h\in H$ and $h^{\prime}\in\overline{H}$ and
$w^{(1)},\ldots,w^{(r)}\in\mathcal{E}$ we have

(33) $(h\otimes 1\otimes h^{\prime}\otimes 1)\cdot\left(w^{(1)}\cdots
w^{(r)}\right)\in\mathcal{A}_{r}.$

We set

$\alpha:=w^{(1)}\quad\text{and}\quad\beta:=w^{(2)}\cdots w^{(r)}.$

Since $\Delta(\overline{H})\subseteq\overline{H}\otimes\overline{H}$, we can
assume that $h^{\prime}_{1},h^{\prime}_{2}\in\overline{H}$, hence by
considering the first component on both sides of (32) we obtain
$u_{h_{2},h_{1}^{\prime}}\in\overline{H}$. Thus, by the induction hypothesis
we have

(34) $\big{(}(h_{1}\otimes 1\otimes u_{h_{2},h_{1}^{\prime}}\otimes
1)\cdot\alpha\big{)}\in\mathcal{E}\subseteq\mathcal{A}_{1}.\quad\text{and}\quad\big{(}(u^{\prime}_{h_{2},h^{\prime}_{1}}\otimes
1\otimes h^{\prime}_{2}\otimes 1)\cdot\beta\big{)}\in\mathcal{A}_{r-1}.$

The inclusion (33) follows from
$\mathcal{A}_{1}\mathcal{A}_{r-1}=\mathcal{A}_{r}$.

(ii) It suffices to prove (33) for $h\in\overline{H}$ and $h^{\prime}\in H$.
We define $\alpha$ and $\beta$ as in (i). As $h\in\overline{H}$, we can assume
that $h_{1},h_{2}\in\overline{H}$, hence
$u^{\prime}_{h_{2},h_{1}^{\prime}}\in\overline{H}$. Again the induction
hypothesis implies (34) and the inclusion (33) follows from
$\mathcal{A}_{1}\mathcal{A}_{r-1}=\mathcal{A}_{r}$. ∎

###  2.7. Braidings and matrix coefficients

Let $(H,\mathcal{C},\check{R})$ be a locally finite braided triple. As in (27)
we set $R_{V,W}:=\sigma_{W,V}\circ\check{R}_{V,W}$ for
$V,W\in\mathrm{Obj}(\mathcal{C})$. Let $H^{\circ}_{\mathcal{C}}\subseteq
H^{\circ}$ denote the $\mathbb{Z}$-span of matrix coefficients of the
$H$-modules that belong to $\mathrm{Obj}(\mathcal{C})$. Since $\mathcal{C}$ is
closed under direct sums and tensor products, indeed $H^{\circ}_{\mathcal{C}}$
is a sub-bialgebra of $H^{\circ}$.

Let $f\in H^{\circ}_{\mathcal{C}}$ be a sum of matrix coefficients of
$V_{1},\ldots,V_{N}\in\mathrm{Obj}(\mathcal{C})$, that is

$f=\sum_{i=1}^{N}\mathsf{m}_{v_{i}^{*},v_{i}},$

where $v_{i}\in V_{i}$ and $v_{i}^{*}\in V_{i}^{*}$. Similarly, let $g\in
H^{\circ}_{\mathcal{C}}$ be a sum of matrix coefficients of
$W_{1},\ldots,W_{N^{\prime}}\in\mathrm{Obj}(\mathcal{C})$, that is
$g=\sum_{i=1}^{N^{\prime}}\mathsf{m}_{w_{i}^{*},w_{i}}$. Choose
$\omega_{V,W},\overline{\omega}_{V,W}\in H\otimes H$ that satisfy the
condition of Definition 2.5.1(ii) for $V:=\bigoplus_{i=1}^{N}V_{i}$ and
$W:=\bigoplus_{i=1}^{N^{\prime}}W_{i}$. We define

(35) $R(f\otimes g):=\omega_{V,W}\cdot(f\otimes g)\quad\text{and}\quad
R^{-1}(f\otimes g):=\overline{\omega}_{V,W}\cdot(f\otimes g),$

where the actions of $\omega_{V,W}$ and $\overline{\omega}_{V,W}$ are by right
translation on tensor components. This means that for example if
$\omega_{V,W}=\sum r\otimes r^{\prime}\in H\otimes H$, then

$\big{(}\omega_{V,W}\cdot(f\otimes g)\big{)}(h\otimes h^{\prime}):=\sum
f(hr)g(h^{\prime}r^{\prime}).$

######  Remark 2.7.1.

(i) For $x,y\in H$ we have

(36)
$\displaystyle\left(\omega_{V,W}\cdot\left(\mathsf{m}_{v_{i}^{*},v_{i}}\otimes\mathsf{m}_{w_{j}^{*},w_{j}}\right)\right)(x\otimes
y)=\langle v_{i}^{*}\otimes w_{j}^{*},(x\otimes y)\cdot R_{V,W}(v_{i}\otimes
w_{j})\rangle,$

hence the left hand side of (36) is independent of the choice of
$\omega_{V,W}$. Summing over $i$ and $j$ it follows that $R(f\otimes g)$ is
independent of the choice of $\omega_{V,W}$ as well. The latter observation
and Lemma 2.5.2 imply that $R(f\otimes g)$ does not depend on how $f$ and $g$
are expressed as sums of matrix coefficients. An analogous statement holds for
$R^{-1}(f\otimes g)$.

(ii) From Lemma 2.5.2 it also follows that the formulas (35) extend to linear
maps

$R,R^{-1}:H^{\circ}_{\mathcal{C}}\otimes H^{\circ}_{\mathcal{C}}\to
H^{\circ}_{\mathcal{C}}\otimes H^{\circ}_{\mathcal{C}}.$

Indeed the latter maps $R$ and $R^{-1}$ are mutual inverses.

We define

(37) $\langle f\otimes g,R\rangle:=\left(R(f\otimes g)\right)(1\otimes 1).$

In the rest of this subsection $\Delta^{\circ}(f)=\sum f_{1}\otimes f_{2}$ and
$\Delta^{\circ}(g)=\sum g_{1}\otimes g_{2}$.

######  Lemma 2.7.2.

Let $f,g\in H^{\circ}_{\mathcal{C}}$. Then the following relations hold.

  * (i)

$R(f\otimes g)=\sum f_{1}\otimes g_{1}\langle f_{2}\otimes g_{2},R\rangle$.

  * (ii)

$f\otimes g=\sum\langle R^{-1}(f_{1}\otimes g_{1}),R\rangle f_{2}\otimes
g_{2}$.

###### Proof.

It suffices to verify the assertion when $f$ and $g$ are matrix coefficients
of finite dimensional $H$-modules $V,W\in\mathrm{Obj}(\mathcal{C})$. Suppose
that $f:=\mathsf{m}_{v^{*},v}$ and $g:=\mathsf{m}_{w^{*},w}$. If
$R_{V,W}(v\otimes w)=\sum\tilde{v}\otimes\tilde{w}$ then by (36) we have
$R(f\otimes
g)=\sum\mathsf{m}_{v^{*},\tilde{v}}\otimes\mathsf{m}_{w^{*},\tilde{w}}$.
Choose dual bases $\\{v_{i}\\}$ and $\\{v_{i}^{*}\\}$ for $V$ and $V^{*}$ and
dual bases $\\{w_{j}\\}$ and $\\{w_{j}^{*}\\}$ for $W$ and $W^{*}$. Using the
coproduct and counit identities of $H^{\circ}$ we have

$\displaystyle\sum f_{1}\otimes g_{1}\langle f_{2}\otimes g_{2},R\rangle$
$\displaystyle=\sum\mathsf{m}_{v^{*},v_{i}}\otimes\mathsf{m}_{w^{*},w_{j}}\left(\langle
v_{i}^{*},\tilde{v}\rangle\langle w_{j}^{*},\tilde{w}\rangle\right)$
$\displaystyle=\sum\langle
v_{i}^{*},\tilde{v}\rangle\mathsf{m}_{v^{*},v_{i}}\otimes\langle
w_{j}^{*},\tilde{w}\rangle\mathsf{m}_{w^{*},w_{j}}=\sum\mathsf{m}_{v^{*},\tilde{v}}\otimes\mathsf{m}_{w^{*},\tilde{w}}=R(f\otimes
g).$

This proves (i). For (ii) note that if $R^{-1}_{V,W}(v\otimes
w)=\sum\tilde{v}\otimes\tilde{w}$ then $R^{-1}(f\otimes
g)=\sum\mathsf{m}_{v^{*},\tilde{v}}\otimes\mathsf{m}_{w^{*},\tilde{w}}$. Next
we write $R^{-1}_{V,W}(v_{i}\otimes
w_{j})=\sum\tilde{v}^{i,j}\otimes\tilde{w}^{i,j}$ for each pair of indices
$i,j$. Then

$\displaystyle\sum\langle R^{-1}(f_{1}\otimes g_{1}),R\rangle f_{2}\otimes
g_{2}$
$\displaystyle=\sum\langle\mathsf{m}_{v^{*},\tilde{v}^{i,j}}\otimes\mathsf{m}_{w^{*},\tilde{w}^{i,j}},R\rangle\mathsf{m}_{v_{i}^{*},v}\otimes\mathsf{m}_{w_{j}^{*},w}$
$\displaystyle=\sum\langle
v^{*},v_{i}\rangle\mathsf{m}_{v_{i}^{*},v}\otimes\langle
w^{*},w_{j}\rangle\mathsf{m}_{w_{j}^{*},w}=\mathsf{m}_{v^{*},v}\otimes\mathsf{m}_{w^{*},w}=f\otimes
g.\qed$

Fix finite dimensional $H$-modules $V,W\in\mathrm{Obj}(\mathcal{C})$. Let
$\\{v_{i}\\}_{i=1}^{d}$ and $\\{w_{i}\\}_{i=1}^{d^{\prime}}$ be bases of $V$
and $W$. Also, let $\\{v_{i}^{*}\\}_{i=1}^{d}$ and
$\\{w_{i}^{*}\\}_{i=1}^{d^{\prime}}$ be the dual bases of $V^{*}$ and $W^{*}$.
We denote the matrix entries of $R_{V,W}$ in the basis $v_{i}\otimes w_{j}$ by
$R_{ij}^{kl}$, so that

$R_{V,W}(v_{i}\otimes w_{j})=\sum_{k,l}R_{ij}^{kl}v_{k}\otimes w_{l}.$

Set $\mathsf{t}^{V}_{a,b}:=\mathsf{m}_{v_{a}^{*},v_{b}}$ and
$\mathsf{t}^{W}_{a,b}:=\mathsf{m}_{w_{a}^{*},w_{b}}$. Then
$R_{ij}^{kl}=\langle\mathsf{t}^{V}_{k,i}\otimes\mathsf{t}^{W}_{l,j},R\rangle$,
so that as in [Ja96, Lem. 7.12] we have the well known relations

(38)
$\sum_{k,l}\langle\mathsf{t}^{V}_{k,i}\otimes\mathsf{t}^{W}_{l,j},R\rangle\mathsf{t}_{a,l}^{W}\mathsf{t}_{b,k}^{V}=\sum_{k,l}\langle\mathsf{t}^{V}_{b,k}\otimes\mathsf{t}^{W}_{a,l},R\rangle\mathsf{t}_{k,i}^{V}\mathsf{t}_{l,j}^{W}\quad\text{
for all }i,j,a,b.$

From (38) it follows that

(39) $\sum g_{1}f_{1}\langle f_{2}\otimes g_{2},R\rangle=\sum
f_{2}g_{2}\langle f_{1}\otimes g_{1},R\rangle\quad\text{for }f,g\in
H^{\circ}_{\mathcal{C}}.$

##  3\. The $q$-Weyl algebra $\mathscr{PD}$

In this section we construct $\mathscr{PD}$ as a deformed twisted tensor
product of the algebras $\mathscr{P}$ and $\mathscr{D}$ with respect to the
univeral $R$-matrix of $U_{LR}$. Recall that $\Bbbk:=\mathbb{C}(q)$.

###  3.1. The algebra $U_{q}(\mathfrak{gl}_{n})$

For $n\in\mathbb{N}$, the quantized enveloping algebra
$U_{q}(\mathfrak{gl}_{n})$ is the $\Bbbk$-algebra generated by $E_{i},F_{i}$
for $1\leq i\leq n-1$ and $K_{\varepsilon_{i}}^{\pm 1}$ for $1\leq i\leq n$,
that satisfy the relations
$K_{\varepsilon_{i}}K_{\varepsilon_{i}}^{-1}=K_{\varepsilon_{i}}^{-1}K_{\varepsilon_{i}}=1$,
$K_{\varepsilon_{i}}K_{\varepsilon_{j}}=K_{\varepsilon_{j}}K_{\varepsilon_{i}}$,

$K_{\varepsilon_{i}}E_{j}K_{\varepsilon_{i}}^{-1}=q^{{\llbracket
i,j\rrbracket}-\llbracket i,j+1\rrbracket}E_{j}\ ,\
K_{\varepsilon_{i}}F_{j}K_{\varepsilon_{i}}^{-1}=q^{-{\llbracket
i,j\rrbracket}+{\llbracket i,j+1\rrbracket}}F_{j}\ ,\
E_{i}F_{j}-F_{j}E_{i}=\llbracket
i,j\rrbracket\frac{K_{i}-K_{i}^{-1}}{q-q^{-1}},$

where $K_{i}:=K_{\varepsilon_{i}}K_{\varepsilon_{i+1}}^{-1}$ and

(40) $\llbracket a,b\rrbracket:=\begin{cases}1&\text{if }a=b\\\ 0&\text{if
}a\neq b,\end{cases}$

as well as the quantum Serre relations. For
$\lambda:=\sum_{i=1}^{n}m_{i}\varepsilon_{i}\in\mathbb{Z}\varepsilon_{1}+\cdots+\mathbb{Z}\varepsilon_{n}$
we set

(41) $K_{\lambda}:=\prod_{i=1}^{n}K_{\varepsilon_{i}}^{m_{i}}.$

The _Cartan subalgebra_ of $U_{q}(\mathfrak{gl}_{n})$ is the subalgebra
spanned by the $K_{\lambda}$ for
$\lambda\in\mathbb{Z}\varepsilon_{1}+\cdots+\mathbb{Z}\varepsilon_{n}$. We
denote the Cartan subalgebras of $U_{L}\cong U_{q}(\mathfrak{gl}_{m})$ and
$U_{R}\cong U_{q}(\mathfrak{gl}_{n})$ by $U_{\mathfrak{h},L}$ and
$U_{\mathfrak{h},R}$, respectively. Following [KS97] for the choice of the
coproduct $\Delta$ on $U_{q}(\mathfrak{gl}_{n})$, we set

$\Delta(E_{i}):=E_{i}\otimes K_{i}+1\otimes
E_{i}\quad,\quad\Delta(F_{i}):=F_{i}\otimes 1+K_{i}^{-1}\otimes
F_{i}\quad,\quad\Delta(K_{\varepsilon_{i}}):=K_{\varepsilon_{i}}\otimes
K_{\varepsilon_{i}}.$

The counit and antipode of $U_{q}(\mathfrak{gl}_{n})$ are given by

$\epsilon(E_{i})=\epsilon(F_{i})=0\ \ ,\ \ \epsilon(K_{\varepsilon_{i}}^{\pm
1})=1\ \ ,\ \ S(E_{i})=-E_{i}K_{i}^{-1}\ \ ,\ \ S(F_{i})=-K_{i}F_{i}\ \ ,\ \
S(K_{\varepsilon_{i}})=K_{\varepsilon_{i}}^{-1}.$

###  3.2. The universal $R$-matrix of $U_{q}(\mathfrak{gl}_{n})$

Recall that a Hopf algebra $H$ is called quasitriangular if it has a
_universal $R$-matrix_, i.e., if there exists an invertible 2-tensor
$\mathcal{R}\in H\otimes H$ satisfying

(42)
$\Delta^{\mathrm{cop}}=\mathcal{R}\Delta\mathcal{R}^{-1}\quad,\quad(\Delta\otimes\mathrm{id})(\mathcal{R})=\mathcal{R}_{13}\mathcal{R}_{23}\quad,\quad(\mathrm{id}\otimes\Delta)(\mathcal{R})=\mathcal{R}_{13}\mathcal{R}_{12}.$

Strictly speaking, $U_{q}(\mathfrak{gl}_{n})$ is not quasitriangular because
the formal series that is usually called the universal $R$-matrix of
$U_{q}(\mathfrak{gl}_{n})$ indeed belongs to a topological tensor product
$U_{h}(\mathfrak{gl}_{n})\widehat{\otimes}U_{h}(\mathfrak{gl}_{n})$ where
$U_{h}(\mathfrak{gl}_{n})$ denotes the $h$-adic Drinfeld–Jimbo quantum group.
However, it turns out that the setting of braided triples is a rigorous way to
work with this universal $R$-matrix.

Let $\mathcal{C}^{(n)}$ denote the full subcategory of the category of
$U_{q}(\mathfrak{gl}_{n})$-modules whose objects are direct sums of
irreducible finite dimensional $U_{q}(\mathfrak{gl}_{n})$-modules with highest
weight of the form $q^{\sum_{i=1}^{n}\lambda_{i}\varepsilon_{i}}$, where
$\lambda_{1}\geq\cdots\geq\lambda_{n}$ are integers. Such modules are
sometimes called modules of type $(1,\ldots,1)$. We define a braiding on
$\mathcal{C}^{(n)}$ as follows. First we fix a formal series description of
the universal $R$-matrix for $U_{q}(\mathfrak{gl}_{n})$. For more details see
[VY20, Thm 3.108] or [KS97, Sec. 8.3.2].

######  Definition 3.2.1.

Given $n\in\mathbb{N}$, the standard root vectors of
$U_{q}(\mathfrak{gl}_{n})$ are

$E_{\varepsilon_{i}-\varepsilon_{j}}:=(-1)^{j-i-1}[E_{i},[\ldots,E_{j-1}]_{q^{-1}}]_{q^{-1}}\quad\text{and}\quad
F_{\varepsilon_{i}-\varepsilon_{j}}:=(-1)^{j-i-1}[F_{j-1},[\ldots,F_{i}]_{q}]_{q},$

where $1\leq i<j\leq n$ and $[x,y]_{q^{\pm 1}}:=xy-q^{\pm 1}yx$. We set

(43) $\mathcal{R}^{(n)}:=\left(e^{h\sum_{i=1}^{n}H_{i}\otimes
H_{i}}\right)\prod_{i=1}^{n\choose
2}\mathrm{Exp}_{q}\left((q-q^{-1})E_{\beta_{i}}\otimes F_{\beta_{i}}\right),$

with the conventions $e^{hH_{i}}:=K_{\varepsilon_{i}}$, $e^{h}:=q$,
$\mathrm{Exp}_{q}(x):=\sum_{r\geq 0}q^{r\choose 2}\frac{x^{r}}{[r]_{q}!}$ and
$\beta_{i+\frac{j(j-1)}{2}}:=\varepsilon_{i}-\varepsilon_{j+1}$ for $1\leq
i\leq j\leq n-1$. Also, set

$\underline{\mathcal{R}}^{(n)}:=\left(\mathcal{R}^{(n)}\right)^{-1}_{21}.$

In what follows, we need $\mathcal{R}^{(n)}$ to define $\mathscr{P}$ and
$\mathscr{D}$, and we need $\underline{\mathcal{R}}^{(n)}$ to define
$\mathscr{PD}^{\mathrm{gr}}$ and $\mathscr{PD}$. The formal series
$\mathcal{R}^{(n)}$ and $\underline{\mathcal{R}}^{(n)}$ equip the category
$\mathcal{C}^{(n)}$ with two braidings which we describe below. Given
$V,W\in\mathrm{Obj}(\mathcal{C}^{(n)})$, the formal series (43) defines a
linear map

$\mathcal{R}^{(n)}_{V,W}:V\otimes W\to V\otimes W.$

To give sense to the action of $\mathcal{R}^{(n)}$ on $V\otimes W$ we make the
following two observations. First, for any $v\otimes w\in V\otimes W$ all but
finitely many terms of $\mathrm{Exp}_{q}\left((q-q^{-1})E_{\beta_{i}}\otimes
F_{\beta_{i}}\right)$ vanish on $v\otimes w$. Second, if $v\in V$ and $w\in W$
are weight vectors of weights $q^{\mu}$ and $q^{\nu}$ with
$\mu:=\sum_{i=1}^{n}\mu_{i}\varepsilon_{i}$ and
$\nu:=\sum_{i=1}^{n}\nu_{i}\varepsilon_{i}$ respectively, then the action of
$e^{h\sum_{i=1}^{n}H_{i}\otimes H_{i}}$ on $v\otimes w$ is by multiplication
by the scalar $q^{\langle\mu,\nu\rangle}$ where

(44) $\langle\mu,\nu\rangle:=\sum_{i=1}^{n}\mu_{i}\nu_{i}.$

By a similar reasoning, the action of $\underline{\mathcal{R}}^{(n)}$ yields
linear maps

$\underline{\mathcal{R}}^{(n)}_{V,W}:V\otimes W\to V\otimes W.$

It is well known that by setting

(45)
$\check{\mathcal{R}}^{(n)}_{V,W}:=\sigma_{V,W}\circ{\mathcal{R}}^{(n)}_{V,W}\quad\text{and}\quad\check{\underline{\mathcal{R}}}^{(n)}_{V,W}:=\sigma_{V,W}\circ\underline{\mathcal{R}}^{(n)}_{V,W}$

we obtain braidings on $\mathcal{C}^{(n)}$, which we will denote by
$\check{\mathcal{R}}^{(n)}$ and $\check{\underline{\mathcal{R}}}^{(n)}$

######  Proposition 3.2.2.

Set $H:=U_{q}(\mathfrak{gl}_{n})$, $\mathcal{C}:=\mathcal{C}^{(n)}$, and
$\check{R}:=\check{{\mathcal{R}}}^{(n)}$ or
$\check{\underline{\mathcal{R}}}^{(n)}$. Then $(H,\mathcal{C},\check{R})$ is a
locally finite braided triple.

###### Proof.

The only assertion that we need to prove is the property of Definition
2.5.1(ii). Fix finite dimensional
$V,W\in\mathrm{Obj}\left(\mathcal{C}^{(n)}\right)$. We construct an element of
$U_{q}(\mathfrak{gl}_{n})\otimes U_{q}(\mathfrak{gl}_{n})$ that acts on
$V\otimes W$ as $\mathcal{R}^{(n)}_{V,W}$. Since $E_{\beta_{i}}^{N_{i}}\otimes
F_{\beta_{i}}^{N_{i}}$ vanishes on $V\otimes W$ when $N_{i}$ is sufficiently
large, the exponential factor

$\mathrm{Exp}_{q}\left((q-q^{-1})E_{\beta_{i}}\otimes F_{\beta_{i}}\right)$

can be replaced by a finite sum. Next we provide a finite 2-tensor that
replaces $e^{h\sum_{i=1}^{n}H_{i}\otimes H_{i}}$. Let
$\mu^{(1)},\ldots,\mu^{(N)}$ be the distinct weights of $W$. Choose
$\nu\in\mathbb{Z}\varepsilon_{1}+\cdots+\mathbb{Z}\varepsilon_{n}$ such that
the values $\langle\nu,\mu^{(i)}\rangle$ are mutually distinct numbers. For
$1\leq i\leq N$ define $T_{i}\in U_{q}(\mathfrak{gl}_{n})$ by

$T_{i}:=\prod_{\tiny\begin{array}[]{c}1\leq j\leq N\\\ j\neq
i\end{array}}\left(\frac{K_{\nu}-q^{\langle\nu,\mu^{(j)}\rangle}}{q^{\langle\nu,\mu^{(i)}\rangle}-q^{\langle\nu,\mu^{(j)}\rangle}}\right).$

Then $T_{i}$ acts by 0 or 1 on the $\mu^{(j)}$-weight space of $W$, depending
on if $j\neq i$ or $j=i$ respectively. It follows that the action of
$\sum_{i=1}^{N}K_{\mu^{(i)}}\otimes T_{i}$ on $V\otimes W$ is identical to the
action of $e^{h\sum_{i=1}^{n}H_{i}\otimes H_{i}}$. Thus the action of
$\mathcal{R}^{(n)}$ on $V\otimes W$ is identical to the action of a (finite)
2-tensor in $U_{q}(\mathfrak{gl}_{n})\otimes U_{q}(\mathfrak{gl}_{n})$.
Analogous constructions can be given for
$\left(\mathcal{R}^{(n)}_{V,W}\right)^{-1}$ and
$\left(\underline{\mathcal{R}}^{(n)}_{V,W}\right)^{\pm 1}$. ∎

As in Subsection 2.2 let $U_{q}(\mathfrak{gl}_{n})^{\circ}$ be the finite dual
of $U_{q}(\mathfrak{gl}_{n})$.

######  Definition 3.2.3.

Let $U_{q}(\mathfrak{gl}_{n})^{\bullet}\subseteq
U_{q}(\mathfrak{gl}_{n})^{\circ}$ denote the sub-bialgebra that is spanned by
matrix coefficients of objects of $\mathcal{C}^{(n)}$.

For $f,g\in U_{q}(\mathfrak{gl}_{n})^{\bullet}$ we define
$\big{\langle}f\otimes g,\mathcal{R}^{(n)}\big{\rangle}$ and
$\big{\langle}f\otimes g,\underline{\mathcal{R}}^{(n)}\big{\rangle}$ as in
(37). For finite dimensional $V\in\mathrm{Obj}\left(\mathcal{C}^{(n)}\right)$
the right dual $V^{*}$ also belongs to
$\mathrm{Obj}\left(\mathcal{C}^{(n)}\right)$. From this and the fact that $S$
and $S^{-1}$ are conjugate by an element of the Cartan subalgebra it follows
that if $f\in U_{q}(\mathfrak{gl}_{n})^{\bullet}$ then $f\circ S^{\pm 1}\in
U_{q}(\mathfrak{gl}_{n})^{\bullet}$. It is well known (see for example [KS97,
Sec. 8.1.1]) that

(46)
$\left(\mathcal{R}^{(n)}\right)_{21}=\left(\underline{\mathcal{R}}^{(n)}\right)^{-1}=(1\otimes
S^{-1})\left(\underline{\mathcal{R}}^{(n)}\right)\quad\text{and}\quad(S\otimes
1)\left({\mathcal{R}}^{(n)}\right)=\left({\mathcal{R}}^{(n)}\right)^{-1}$

Consequently, for $f,g\in U_{q}(\mathfrak{gl}_{n})^{\bullet}$ we have

(47) $\big{\langle}f\otimes(g\circ
S^{-1}),\underline{\mathcal{R}}^{(n)}\big{\rangle}=\big{\langle}g\otimes
f,{\mathcal{R}}^{(n)}\big{\rangle}\quad\text{and}\quad\big{\langle}(f\circ
S)\otimes g,{\mathcal{R}}^{(n)}\big{\rangle}=\big{\langle}g\otimes
f,\underline{\mathcal{R}}^{(n)}\big{\rangle}.$

###  3.3. The involution $x\mapsto x^{\natural}$

It is well known (for example see [No96, Sec. 1.4]) that there exists a unique
$\Bbbk$-linear isomorphism of Hopf algebras

(48) $U_{q}(\mathfrak{gl}_{n})\to U_{q}(\mathfrak{gl}_{n})^{\mathrm{op}}\ ,\
x\mapsto x^{\natural},$

such that

(49) $E_{i}^{\natural}:=qK_{i}F_{i}\quad,\quad
F_{i}^{\natural}:=q^{-1}E_{i}K_{i}^{-1}\quad,\quad
K_{\varepsilon_{i}}^{\natural}:=K_{\varepsilon_{i}}.$

######  Lemma 3.3.1.

$S(x^{\natural})=S^{-1}(x)^{\natural}$ for $x\in U_{q}(\mathfrak{gl}_{n})$.

###### Proof.

Both sides are automorphisms of the algebra $U_{q}(\mathfrak{gl}_{n})$.
Therefore it suffices to verify that they agree on the $E_{i}$, the $F_{i}$,
and the $K_{\varepsilon_{i}}$. This is a straightforward calculation. ∎

By the canonical duality between $U_{q}(\mathfrak{gl}_{n})$ and
$U_{q}(\mathfrak{gl}_{n})^{\circ}$, the map (48) induces an isomorphism of
Hopf algebras
$U_{q}(\mathfrak{gl}_{n})^{\circ}\to\left(U_{q}(\mathfrak{gl}_{n})^{\circ}\right)^{\mathrm{cop}}$.
We denote the latter map by $u\mapsto u^{\natural}$ as well, so that

(50) $\langle u^{\natural},x\rangle=\langle
u,x^{\natural}\rangle\quad\text{for }u\in
U_{q}(\mathfrak{gl}_{n})^{\circ}\text{ and }x\in U_{q}(\mathfrak{gl}_{n}).$

###  3.4. The algebras $\mathscr{P}_{n\times n}$ and $\mathscr{D}_{n\times n}$

From now on we denote the standard positive system of the root system of
$\mathfrak{gl}_{n}$ by

$\Delta^{+}_{n}:=\\{\varepsilon_{i}-\varepsilon_{j}\,:\,1\leq i<j\leq n\\}$

Let $V^{(n)}$ denote the irreducible $U_{q}(\mathfrak{gl}_{n})$-module of
highest weight $q^{-\varepsilon_{n}}$ (all highest weights are considered with
respect to $\Delta_{n}^{+}$). Thus $V^{(n)}\cong\Bbbk^{n}$ as a vector space
and the homomorphism of algebras
$U_{q}(\mathfrak{gl}_{n})\to\mathrm{End}_{\Bbbk}\left(V^{(n)}\right)$ is
uniquely determined by the assignments

$K_{\varepsilon_{i}}\mapsto\mathsf{1}+(q^{-1}-1)\mathsf{E}_{i,i}\quad,\quad
E_{i}\mapsto\mathsf{E}_{i+1,i}\quad,\quad F_{i}\mapsto\mathsf{E}_{i,i+1},$

where the $\mathsf{E}_{i,j}$ are the elementary matrix units associated to the
standard basis $\\{e_{i}\\}_{i=1}^{n}$ of $V^{(n)}$ and
$\mathsf{1}:=\sum_{i=1}^{n}\mathsf{E}_{i,i}$. Using (43) the $R$-matrix of
$V^{(n)}\otimes V^{(n)}$ can be computed directly, and we obtain

$\mathcal{R}^{(n)}_{V^{(n)},V^{(n)}}=\sum_{1\leq i\leq
n}q\mathsf{E}_{i,i}\otimes\mathsf{E}_{i,i}+\sum_{1\leq i\neq j\leq
n}\mathsf{E}_{i,i}\otimes\mathsf{E}_{j,j}+(q-q^{-1})\sum_{1\leq j<i\leq
n}\mathsf{E}_{i,j}\otimes\mathsf{E}_{j,i}.$

For $1\leq i,j\leq n$ let $t_{i,j}$ denote the matrix coefficient
$\mathsf{m}_{e_{i}^{*},e_{j}}$ of $V^{(n)}$. By (39) the $t_{i,j}$ satisfy the
following relations:

  * (R1)

$t_{k,i}t_{k,j}=qt_{k,j}t_{k,i}$, $t_{i,k}t_{j,k}=qt_{j,k}t_{i,k}$ for $i<j$.

  * (R2)

$t_{i,l}t_{k,j}=t_{k,j}t_{i,l},\
t_{i,j}t_{k,l}-t_{k,l}t_{i,j}=(q-q^{-1})t_{i,l}t_{k,j}$ for $i<k$ and $j<l$.

Similarly, let $\breve{V}^{(n)}$ denote the irreducible
$U_{q}(\mathfrak{gl}_{n})$-module with highest weight $q^{\varepsilon_{1}}$.
Again $\breve{V}^{(n)}\cong\Bbbk^{n}$ as vector spaces, but the map
$U_{q}(\mathfrak{gl}_{n})\to\mathrm{End}_{\Bbbk}\big{(}\breve{V}^{(n)}\big{)}$
is uniquely determined by the assignments

$K_{\varepsilon_{i}}\mapsto\mathsf{1}+(q-1)\mathsf{E}_{i,i}\quad,\quad
E_{i}\mapsto\mathsf{E}_{i,i+1}\quad,\quad F_{i}\mapsto\mathsf{E}_{i+1,i}.$

Indeed $\breve{V}^{(n)}\cong\left(V^{(n)}\right)^{*}$. The $R$-matrix of
$\breve{V}^{(n)}\otimes\breve{V}^{(n)}$ is

${\mathcal{R}}_{\breve{V}^{(n)},\breve{V}^{(n)}}^{(n)}=\sum_{1\leq i\leq
n}q\mathsf{E}_{i,i}\otimes\mathsf{E}_{i,i}+\sum_{1\leq i\neq j\leq
n}\mathsf{E}_{i,i}\otimes\mathsf{E}_{j,j}+(q-q^{-1})\sum_{1\leq i<j\leq
n}\mathsf{E}_{i,j}\otimes\mathsf{E}_{j,i}.$

If $\partial_{i,j}$ for $1\leq i,j\leq n$ denotes the matrix coefficient
$\mathsf{m}_{e_{i}^{*},e_{j}}$ of $\breve{V}^{(n)}$, then again from (39) it
follows that the $\partial_{i,j}$ satisfy relations similar to those between
the $t_{i,j}$, with $q$ replaced by $q^{-1}$. Equivalently,

  * (R1′)

$\partial_{k,j}\partial_{k,i}=q\partial_{k,i}\partial_{k,j}$,
$\partial_{j,k}\partial_{i,k}=q\partial_{i,k}\partial_{j,k}$ for $i<j$.

  * (R2′)

$\partial_{k,j}\partial_{i,l}=\partial_{i,l}\partial_{k,j},\partial_{k,l}\partial_{i,j}-\partial_{i,j}\partial_{k,l}=(q-q^{-1})\partial_{k,j}\partial_{i,l}$
for $i<k$ and $j<l$.

######  Definition 3.4.1.

Let $\mathscr{P}_{n\times n}$ denote the subalgebra of
$U_{q}(\mathfrak{gl}_{n})^{\circ}$ generated by the $t_{i,j}$, for $1\leq
i,j\leq n$. Similarly, let $\mathscr{D}_{n\times n}$ denote the subalgebra of
$U_{q}(\mathfrak{gl}_{n})^{\circ}$ generated by the $\partial_{i,j}$, for
$1\leq i,j\leq n$.

It is well known (for example see [Ta92]) that the relations (R1)–(R2) yield a
presentation of $\mathscr{P}_{n\times n}$ by generators and relations. Since
$\mathscr{D}_{n\times n}\cong\mathscr{P}_{n\times n}^{\mathrm{op}}$, a similar
statement holds for $\mathscr{D}_{n\times n}$. From Section 2.2 it follows
that both $\mathscr{P}_{n\times n}$ and $\mathscr{D}_{n\times n}$ are
bialgebras with the coproducts satisfying

$t_{i,j}\mapsto\sum_{k}t_{i,k}\otimes
t_{k,j}\quad\text{and}\quad\partial_{i,j}\mapsto\sum_{k}\partial_{i,k}\otimes\partial_{k,j},$

and the counits satisfying $t_{i,j},\partial_{i,j}\mapsto\llbracket
i,j\rrbracket$. Henceforth we denote the coproducts of $\mathscr{P}_{n\times
n}$ and $\mathscr{D}_{n\times n}$ by $\Delta_{\mathscr{P}}$ and
$\Delta_{\mathscr{D}}$, respectively. In the proof of Lemma 3.4.2 we use the
relations

$t_{i,j}(K_{\varepsilon_{k}})=\llbracket i,j\rrbracket q^{-\llbracket
i,k\rrbracket}\quad,\quad t_{i,j}(E_{k})=\llbracket i,k+1\rrbracket\llbracket
j,k\rrbracket\quad,\quad t_{i,j}(F_{k})=\llbracket i,k\rrbracket\llbracket
j,k+1\rrbracket.$

######  Lemma 3.4.2.

$t_{i,j}^{\natural}=t_{j,i}$ and $\partial_{i,j}^{\natural}=\partial_{j,i}$,
where $t_{i,j}^{\natural}$ and $\partial_{i,j}^{\natural}$ are defined by
(50).

###### Proof.

We only give the proof for the $t_{i,j}$, as the argument for the
$\partial_{i,j}$ is similar. The assertion follows if we verify that

(51) $\langle t_{i,j},x^{\natural}\rangle=\langle
t_{j,i},x\rangle\quad\text{for $x\in U_{q}(\mathfrak{gl}_{n})$.}$

It suffices to check (51) when $x$ is a generator of
$U_{q}(\mathfrak{gl}_{n})$, because if (51) holds for $x,y\in
U_{q}(\mathfrak{gl}_{n})$ then

$\langle t_{i,j},(xy)^{\natural}\rangle=\langle
t_{i,j},y^{\natural}x^{\natural}\rangle=\sum_{a}\langle
t_{i,a},y^{\natural}\rangle\langle t_{a,j},x^{\natural}\rangle=\sum_{a}\langle
t_{a,i},y\rangle\langle t_{j,a},x\rangle=\langle t_{j,i},xy\rangle,$

hence (51) also holds for $xy$. When $x$ is one of the standard generators of
$U_{q}(\mathfrak{gl}_{n})$, checking (51) is a direct calculation. For example
for $x=E_{k}$ we have

$\langle t_{i,j},E_{k}^{\natural}\rangle=q\sum_{a}\langle
t_{i,a},K_{k}\rangle\langle t_{a,j},F_{k}\rangle.$

The right hand side vanishes unless $j=k+1$ and $i=k$, in which case we have
$\langle t_{i,j},E_{k}^{\natural}\rangle=1$. It follows immediately that
$\langle t_{i,j},E_{k}^{\natural}\rangle=\langle t_{j,i},E_{k}\rangle$ for all
$i,j,k$. ∎

According to Remark 2.2.1, the canonical $U_{q}(\mathfrak{gl}_{n})\otimes
U_{q}(\mathfrak{gl}_{n})$-module structure of $U_{q}(\mathfrak{gl}_{n})$ by
left and right translation equips both $\mathscr{P}_{n\times n}$ and
$\mathscr{D}_{n\times n}$ with $U_{q}(\mathfrak{gl}_{n})\otimes
U_{q}(\mathfrak{gl}_{n})$-module algebra structures. Our next goal is to
describe the latter actions explicitly (all of the actions are from the left
side).

Let $\mathcal{R}_{\mathscr{D}}$ be the action of $U_{q}(\mathfrak{gl}_{n})$ on
$\mathscr{D}_{n\times n}$ by right translation, as in Remark 2.2.1. We have

$\mathcal{R}_{\mathscr{D}}(x)u=\sum\langle u_{2},x\rangle u_{1}\quad\text{for
}x\in U_{q}(\mathfrak{gl}_{n}),\ u\in\mathscr{D}_{n\times n},$

where as usual $\Delta(u)=\sum u_{1}\otimes u_{2}$. Similarly, let
$\mathcal{L}_{\mathscr{D}}$ be the action of $U_{q}(\mathfrak{gl}_{n})$ on
$\mathscr{D}_{n\times n}$ by left translation. Thus

$\mathcal{L}_{\mathscr{D}}(x)u=\sum\langle u_{1},x^{\natural}\rangle
u_{2}\quad\text{for }x\in U_{q}(\mathfrak{gl}_{n}),\ u\in\mathscr{D}_{n\times
n}.$

By Remark 2.2.1 both $\mathcal{L}_{\mathscr{D}}$ and
$\mathcal{R}_{\mathscr{D}}$ equip $\mathscr{D}_{n\times n}$ with
$U_{q}(\mathfrak{gl}_{n})$-module algebra structures.

Next we define the left and right $U_{q}(\mathfrak{gl}_{n})$-actions on
$\mathscr{P}_{n\times n}$. For $c\in\Bbbk$ let $\xi_{c}$ denote the unique
automorphism of $U_{q}(\mathfrak{gl}_{n})$ defined by

$\xi_{c}(E_{i}):=cE_{i}\quad,\quad\xi_{c}(F_{i}):=c^{-1}F_{i}\quad,\quad\xi_{c}(K_{\varepsilon_{i}})=K_{\varepsilon_{i}}.$

######  Lemma 3.4.3.

$t_{i,j}\circ\xi_{c}=c^{i-j}t_{i,j}$ and
$\partial_{i,j}\circ\xi_{c}=c^{j-i}\partial_{i,j}$.

###### Proof.

We only give the proof of the assertion for the $t_{i,j}$. In this case we
need to verify the equality

(52) $t_{i,j}(\xi_{c}(x))=c^{i-j}t_{i,j}(x)$

for $x\in U_{q}(\mathfrak{gl}_{n})$. This is a straightforward calculation in
the special case where $x$ is one of the standard generators of
$U_{q}(\mathfrak{gl}_{n})$. To complete the proof of (52) note that if (52)
holds for $x$ and $x^{\prime}$, then it also holds for $xx^{\prime}$ because

$t_{i,j}(\xi_{c}(xx^{\prime}))=\sum_{a=1}^{n}t_{i,a}(\xi_{c}(x))t_{a,j}(\xi_{c}(x^{\prime}))=\sum_{a=1}^{n}c^{(i-a)+(a-j)}t_{i,a}(x)t_{a,j}(x^{\prime})=c^{i-j}t_{i,j}(xx^{\prime}).\qed$

The map

$\Xi:U_{q}(\mathfrak{gl}_{n})\to U_{q}(\mathfrak{gl}_{n})^{\mathrm{op,cop}}\
,\ x\mapsto\xi_{-1/q}(S(x))$

is an isomorphism of Hopf algebras. Thus, the pullback of $\Xi$ induces an
isomorphism of Hopf algebras
$U_{q}(\mathfrak{gl}_{n})^{\circ}\to\left(U_{q}(\mathfrak{gl}_{n})^{\circ}\right)^{\mathrm{op,cop}}$,
given by $u\mapsto u\circ\Xi$. Set

(53) $\iota(u):=u\circ\Xi\quad\text{ for }u\in
U_{q}(\mathfrak{gl}_{n})^{\circ}.$

######  Lemma 3.4.4.

We have

(54) $\iota(t_{i,j})=\partial_{j,i}\quad\text{for }1\leq i,j\leq n.$

In particular, the restriction of $\iota$ to $\mathscr{P}_{n\times n}$ is an
isomorphism of bialgebras $\iota:\mathscr{P}_{n\times
n}\to\mathscr{D}_{n\times n}^{\mathrm{op,cop}}$.

###### Proof.

We need to verify $t_{i,j}(\Xi(x))=\partial_{j,i}(x)$ for $x\in
U_{q}(\mathfrak{gl}_{n})$. It suffices to check the latter relation for the
standard generators of $U_{q}(\mathfrak{gl}_{n})$, and this special case
follows from a direct calculation. ∎

Next note that the map

$\underline{\iota}:\mathscr{P}_{n\times n}\to\mathscr{D}_{n\times
n}^{\mathrm{op}}\ ,\ \underline{\iota}(u):=\iota(u)^{\natural}$

is an isomorphism of bialgebras (indeed
$\underline{\iota}(t_{i,j})=\partial_{i,j}$). Let
$\mathcal{R}_{\mathscr{D},\tau}$ and $\mathcal{L}_{\mathscr{D},\tau}$ denote
the $\tau$-twists of $\mathcal{R}_{\mathscr{D}}$ and
$\mathcal{L}_{\mathscr{D}}$ (see Remark 2.2.1), where we set
$\tau(x):=S^{-1}(x)^{\natural}$ for $x\in U_{q}(\mathfrak{gl}_{n})$. For
$u\in\mathscr{P}_{n\times n}$ and $x\in U_{q}(\mathfrak{gl}_{n})$ set

$\mathcal{R}_{\mathscr{P}}(x)u:=\underline{\iota}^{-1}\left(\mathcal{R}_{\mathscr{D},\tau}(x)\underline{\iota}(u)\right)\quad\text{
and
}\quad\mathcal{L}_{\mathscr{P}}(x)u:=\underline{\iota}^{-1}\left(\mathcal{L}_{\mathscr{D},\tau}(x)\underline{\iota}(u)\right).$

By Remark 2.2.1(ii), $\mathcal{R}_{\mathscr{D},\tau}$ and
$\mathcal{L}_{\mathscr{D},\tau}$ equip $\mathscr{D}_{n\times n}$ with
$U_{q}(\mathfrak{gl}_{n})^{\mathrm{cop}}$-module algebra structures. It
follows that $\mathcal{R}_{\mathscr{P}}$ and $\mathcal{L}_{\mathscr{P}}$ equip
$\mathscr{P}_{n\times n}$ with $U_{q}(\mathfrak{gl}_{n})$-module structures.
By a direct calculation

$\mathcal{R}_{\mathscr{P}}(x)u=\sum\langle\iota(u_{2}),S^{-1}(x)\rangle
u_{1}\quad\text{and}\quad\mathcal{L}_{\mathscr{P}}(x)u=\sum\langle\iota(u_{1}),S^{-1}(x)^{\natural}\rangle
u_{2},$

for $x\in U_{q}(\mathfrak{gl}_{n})$ and $u\in\mathscr{P}_{n\times n}$. Using
(53), Lemma 3.3.1 and the relation $S^{2}=\xi_{q^{2}}$ we obtain

$\mathcal{R}_{\mathscr{P}}(x)u=\sum\langle u_{2}\circ\xi_{-1/q},x\rangle
u_{1}\quad\text{and}\quad\mathcal{L}_{\mathscr{P}}(x)u=\sum\langle(u_{1}\circ\xi_{-q})^{\natural},x\rangle
u_{2}.$

###  3.5. The algebras $\mathscr{P}$ and $\mathscr{D}$

Our next goal is to extend the constructions of Subsection 3.4 to the $m\times
n$ case.

######  Definition 3.5.1.

Let $m$ and $n$ be positive integers and set $N:=\max\\{m,n\\}$. We define the
algebra $\mathscr{P}:=\mathscr{P}_{m\times n}$ (respectively,
$\mathscr{D}:=\mathscr{D}_{m\times n}$) to be the subalgebra of
$\mathscr{P}_{N\times N}$ (respectively, $\mathscr{D}_{N\times N}$) that is
generated by the $t_{i,j}$ (respectively, the $\partial_{i,j}$) where $1\leq
i\leq m$ and $1\leq j\leq n$.

Note that by restricting the $U_{q}(\mathfrak{gl}_{N})\otimes
U_{q}(\mathfrak{gl}_{N})$-module algebra structures on $\mathscr{P}_{N\times
N}$ and $\mathscr{D}_{N\times N}$ we obtain $U_{LR}$-module algebra structures
on $\mathscr{P}$ and $\mathscr{D}$. Let us describe these $U_{LR}$-module
algebras more precisely. For convenience we first assume that $m\leq n$. Then
the subalgebra of $U_{q}(\mathfrak{gl}_{n})$ generated by $E_{i},F_{i},K^{\pm
1}_{\varepsilon_{j}}$ for $1\leq i\leq m-1$ and $1\leq j\leq m$ is isomorphic
to $U_{q}(\mathfrak{gl}_{m})\cong U_{L}$. With this identification of $U_{L}$
with a subalgebra of $U_{q}(\mathfrak{gl}_{n})$ we have the following lemma.

######  Lemma 3.5.2.

Suppose that $m\leq n$ and we identify $U_{L}$ with a subalgebra of
$U_{q}(\mathfrak{gl}_{n})$ as above. Then
$\mathcal{L}_{\mathscr{D}}(x)\mathscr{D}\subseteq\mathscr{D}$ and
$\mathcal{L}_{\mathscr{P}}(x)\mathscr{P}\subseteq\mathscr{P}$ for $x\in
U_{L}$.

###### Proof.

We only prove this for $\mathcal{L}_{\mathscr{D}}$ (for
$\mathcal{L}_{\mathscr{P}}$ the proof is similar). Since $\mathscr{D}_{n\times
n}$ is a $U_{L}$-module algebra, it suffices to check that if $i\leq m$ and
$x$ is one of the standard generators of $U_{L}$ then
$\mathcal{L}_{\mathscr{D}}(x)\partial_{i,j}\in\mathscr{D}$. If $x=E_{k}$ for
$1\leq k\leq m-1$ then

(55) $\displaystyle\mathcal{L}_{\mathscr{D}}(E_{k})\partial_{i,j}$
$\displaystyle=\sum_{a=1}^{n}\langle\partial_{i,a},E_{k}^{\natural}\rangle\partial_{a,j}=\sum_{a=1}^{n}\langle\partial_{i,a},qK_{k}F_{k}\rangle\partial_{a,j}=q\sum_{a=1}^{n}\sum_{b=1}^{n}\langle\partial_{i,b},K_{k}\rangle\langle\partial_{b,a},F_{k}\rangle\partial_{a,j}.$

We have $\langle\partial_{b,a},F_{k}\rangle=\langle
e_{b}^{*},\mathsf{E}_{k+1,k}e_{a}\rangle=\llbracket a,k\rrbracket\llbracket
b,k+1\rrbracket$. In particular, $\langle\partial_{b,a},F_{k}\rangle=0$ unless
$a\leq m-1$. It follows that the right hand side of (55) is a linear
combination of the $\partial_{a,j}$ where $a\leq m-1$, hence it lies in
$\mathscr{D}$. The calculations for the cases $x=F_{k}$ for $1\leq k\leq m-1$
and $x=K_{\varepsilon_{k}}^{\pm 1}$ for $1\leq k\leq m$ are similar. ∎

Lemma 3.5.2 implies that $\mathscr{P}$ and $\mathscr{D}$ are $U_{LR}$-stable
subspaces of $\mathscr{P}_{n\times n}$ and $\mathscr{D}_{n\times n}$, where we
consider

$U_{LR}=U_{q}(\mathfrak{gl}_{m})\otimes U_{q}(\mathfrak{gl}_{n})\cong
U_{L}\otimes U_{R}$

as a subalgebra of $U_{q}(\mathfrak{gl}_{n})\otimes U_{q}(\mathfrak{gl}_{n})$
via the aforementioned embedding $U_{L}\hookrightarrow
U_{q}(\mathfrak{gl}_{n})$. Thus, $\mathscr{P}$ and $\mathscr{D}$ inherit
$U_{LR}$-module algebra structures from $\mathscr{P}_{n\times n}$ and
$\mathscr{D}_{n\times n}$.

Henceforth we mostly drop the symbols $\mathcal{L}_{\mathscr{P}}$,
$\mathcal{L}_{\mathscr{D}}$, $\mathcal{R}_{\mathscr{P}}$ and
$\mathcal{R}_{\mathscr{D}}$ from our notation. Instead, we use the notation

$(x\otimes y)\cdot u$

to denote the action of $x\otimes y\in U_{LR}$ on $u\in\mathscr{P}$ (or
$u\in\mathscr{D}$). The actions of $x\otimes y\in U_{L}\otimes U_{R}$ on
$u\in\mathscr{P}$ and on $v\in\mathscr{D}$ are given explicitly by the
formulas

(56) $(x\otimes y)\cdot
u=\sum\langle\iota(u_{1}),S^{-1}(x)^{\natural}\rangle\langle\iota(u_{3}),S^{-1}(y)\rangle
u_{2}=\sum\langle(u_{1}\circ\xi_{-q})^{\natural},x\rangle\langle
u_{3}\circ\xi_{-1/q},y\rangle u_{2}$

and

(57) $(x\otimes y)\cdot v=\sum\langle v_{1},x^{\natural}\rangle\langle
v_{3},y\rangle v_{2}=\sum\langle v_{1}^{\natural},x\rangle\langle
v_{3},y\rangle v_{2},$

with $(\Delta_{\mathscr{P}}\otimes 1)\circ\Delta_{\mathscr{P}}(u)=\sum
u_{1}\otimes u_{2}\otimes u_{3}$ and $(\Delta_{\mathscr{D}}\otimes
1)\circ\Delta_{\mathscr{D}}(v)=\sum v_{1}\otimes v_{2}\otimes v_{3}$ in
Sweedler notation, where $\Delta_{\mathscr{P}}$ and $\Delta_{\mathscr{D}}$
denote the coproducts of $\mathscr{P}_{n\times n}$ and $\mathscr{D}_{n\times
n}$, respectively.

######  Remark 3.5.3.

In the case $m>n$ the construction of the $U_{LR}$-action is the same, except
that we embed $U_{q}(\mathfrak{gl}_{m})\otimes U_{q}(\mathfrak{gl}_{n})$ in
$U_{q}(\mathfrak{gl}_{m})\otimes U_{q}(\mathfrak{gl}_{m})$. However, formulas
(56) and (57) remain the same.

######  Definition 3.5.4.

The map $\phi_{U}:U_{LR}\to\mathrm{End}_{\Bbbk}(\mathscr{P})$ is the
homomorphism of algebras induced by the action (56).

###  3.6. $U_{LR}$-module decomposition of $\mathscr{P}$ and $\mathscr{D}$

For any integer partition $\lambda$ satisfying $\ell(\lambda)\leq n$, where
$\ell(\lambda)$ denotes the length of $\lambda$, let $V_{\lambda}$ denote the
irreducible finite dimensional $U_{R}$-module of type $(1,\ldots,1)$ with
highest weight $q^{\sum_{i}\lambda_{i}\varepsilon_{i}}$ (with respect to
$\Delta_{n}^{+}$). If $\lambda$ satisfies $\ell(\lambda)\leq m$ we use the
same notation $V_{\lambda}$ to denote the analogously defined module of
$U_{L}$.

The algebras $\mathscr{P}$ and $\mathscr{D}$ are naturally graded by degree of
monomials. For $d\geq 0$ let $\mathscr{P}^{(d)}$ (respectively,
$\mathscr{D}^{(d)}$) denote the graded component of degree $d$ of
$\mathscr{P}$ (respectively, $\mathscr{D}$). Furthermore, let $\Lambda_{d,r}$
be the set of integer partitions $\lambda$ such that $\ell(\lambda)\leq d$ and
$|\lambda|=r$, where $|\lambda|$ denotes the size of $\lambda$. The following
proposition is well known and its proof can be found for example in [NYM93,
Ta92, Zh02].

######  Proposition 3.6.1.

Set $d:=\min\\{m,n\\}$. We have isomorphisms of $U_{LR}$-modules

$\mathscr{P}^{(r)}\cong\bigoplus_{\lambda\in\Lambda_{d,r}}V_{\lambda}^{*}\otimes
V_{\lambda}^{*}\quad\text{and}\quad\mathscr{D}^{(r)}\cong\bigoplus_{\lambda\in\Lambda_{d,r}}V_{\lambda}\otimes
V_{\lambda}.$

######  Remark 3.6.2.

The action of $U_{L}\otimes U_{R}$ on the generators of $\mathscr{P}$ and
$\mathscr{D}$ can be computed explicitly. For the subalgebra $U_{R}\cong
1\otimes U_{R}$ of $U_{LR}$, the action is given by

$\begin{array}[]{ccccc}E_{k}\cdot\partial_{i,j}=\llbracket
k+1,j\rrbracket\partial_{i,k}&,&F_{k}\cdot\partial_{i,j}=\llbracket
k,j\rrbracket\partial_{i,k+1}&,&K_{\varepsilon_{k}}\cdot\partial_{i,j}=q^{\llbracket
k,j\rrbracket}\partial_{i,j},\\\ E_{k}\cdot t_{i,j}=-\llbracket k,j\rrbracket
q^{-1}t_{i,k+1}&,&F_{k}\cdot t_{i,j}=-\llbracket k+1,j\rrbracket
qt_{i,k}&,&K_{\varepsilon_{k}}\cdot t_{i,j}=q^{-\llbracket
k,j\rrbracket}t_{i,j},\end{array}$

where $1\leq k\leq n-1$, $1\leq i\leq m$ and $1\leq j\leq n$. For $U_{L}\cong
U_{L}\otimes 1$ the formulas are similar but the action occurs in the first
index (thus, they are obtained by replacing $\partial_{i,j}$ by
$\partial_{j,i}$ and $t_{i,j}$ by $t_{j,i}$).

###  3.7. The algebras ${\mathscr{PD}^{\mathrm{gr}}}$ and $\mathscr{PD}$

For $n\geq 1$ let $\mathcal{R}^{(n)}$ and $\underline{\mathcal{R}}^{(n)}$ be
as in Definition 3.2.1.

######  Definition 3.7.1.

We set $\mathcal{C}_{L}:=\mathcal{C}^{(m)}$ and
$\mathcal{C}_{R}:=\mathcal{C}^{(n)}$. Furthermore, we set

${\mathcal{R}}_{L}:={\mathcal{R}}^{(m)}\quad,\quad{\underline{\mathcal{R}}}_{L}:={\underline{\mathcal{R}}}^{(m)}\quad,\quad{\mathcal{R}}_{R}:={\mathcal{R}}^{(n)}\quad,\quad{\underline{\mathcal{R}}}_{R}:={\underline{\mathcal{R}}}^{(n)}.$

We define braidings $\check{\mathcal{R}}_{L}$ and
$\check{\underline{\mathcal{R}}}_{L}$ on $\mathcal{C}_{L}$, and
$\check{\mathcal{R}}_{R}$ and $\check{\underline{\mathcal{R}}}_{L}$ on
$\mathcal{C}_{R}$ as in (45).

From Proposition 3.2.2 it follows that
$(U_{L},\mathcal{C}_{L},\check{\underline{\mathcal{R}}}_{L})$ and
$(U_{R},\mathcal{C}_{R},\check{\underline{\mathcal{R}}}_{R})$ are locally
finite braided triples. Thus the product (in the sense of Definition 2.5.3) of
$(U_{L},\mathcal{C}_{L},\check{\underline{\mathcal{R}}}_{L})$ and
$(U_{R},\mathcal{C}_{R},\check{\underline{\mathcal{R}}}_{R})$ is also a
locally finite braided triple of the form
$(U_{LR},\mathcal{C}_{LR},\check{\underline{\mathcal{R}}}_{LR})$, where
$\check{\underline{\mathcal{R}}}_{LR}$ is defined as in (28). Furthermore,
Proposition 3.6.1 implies that
$\mathscr{P},\mathscr{D}\in\mathrm{Obj}(\mathcal{C}_{LR})$.

Recall that $\mathscr{D}^{(1)}$ and $\mathscr{P}^{(1)}$ are $U_{LR}$-modules.
Let $\psi_{\circ}:\mathscr{D}^{(1)}\times\mathscr{P}^{(1)}\to\Bbbk$ be the
$U_{LR}$-invariant $\Bbbk$-bilinear form, in the sense of (19), that is
defined by

$\psi_{\circ}(\partial_{i,j},t_{k,l}):=\llbracket i,k\rrbracket\llbracket
j,l\rrbracket\quad\text{for }1\leq i,k\leq m,\text{ and }1\leq j,l\leq n.$

######  Definition 3.7.2.

We define the algebras $\mathscr{PD}^{\mathrm{gr}}$ and $\mathscr{PD}$ by

(58)
$\mathscr{PD}^{\mathrm{gr}}:=\mathscr{P}\otimes_{\check{\mathcal{R}}}\mathscr{D}\quad\text{and}\quad\mathscr{PD}:=\mathscr{P}\otimes_{\check{\mathcal{R}},\psi_{\circ}}\mathscr{D},$

according to Definition 2.4.2 and Definition 2.4.9, with $A:=\mathscr{P}$,
$E_{A}:=\mathscr{P}^{(1)}$, $B:=\mathscr{D}$, $E_{B}:=\mathscr{D}^{(1)}$,
$\psi:=\psi_{\circ}$, and
$\check{\mathcal{R}}:=\check{\underline{\mathcal{R}}}_{LR}$.

It turns out that there is an equivalent description of $\mathscr{PD}$ and
$\mathscr{PD}^{\mathrm{gr}}$ by generators and relations. Recall the notation
$\llbracket a,b\rrbracket$ that was defined in (40). We set

$\llbracket a,b\rrbracket_{q}:=\begin{cases}q&\text{ if }a=b,\\\
q-q^{-1}&\text{if }a\neq b.\end{cases}$

######  Definition 3.7.3.

The algebra $\mathscr{PD}$ is generated by $2mn$ generators $t_{i,j}$ and
$\partial_{i,j}$ for $1\leq i\leq m$ and $1\leq j\leq n$, modulo the relations
(R1), (R2), (R1′), (R2′) of Subsection 3.4 and the relations

(59)
$\displaystyle\partial_{\overline{a}_{1},\overline{a}_{2}}t_{a_{1},a_{2}}=\llbracket
a_{1},\overline{a}_{1}\rrbracket\llbracket
a_{2},\overline{a}_{2}\rrbracket+\sum_{b_{1}\geq
a_{1}}\sum_{\overline{b}_{1}\geq\overline{a}_{1}}\sum_{b_{2}\geq
a_{2}}\sum_{\overline{b}_{2}\geq\overline{a}_{2}}\left(\diamondsuit_{1}+\square_{1}\right)\left(\diamondsuit_{2}+\square_{2}\right)t_{b_{1},b_{2}}\partial_{\overline{b}_{1},\overline{b}_{2}},$

where

$\diamondsuit_{i}:=\llbracket a_{i},\overline{a}_{i}\rrbracket\llbracket
b_{i},\overline{b}_{i}\rrbracket\llbracket
a_{i},b_{i}\rrbracket_{q}\quad\text{ and }\quad\square_{i}:=\left(1-\llbracket
a_{i},\overline{a}_{i}\rrbracket\right)\llbracket
a_{i},b_{i}\rrbracket\llbracket\overline{a}_{i},\overline{b}_{i}\rrbracket.$

The algebra $\mathscr{PD}^{\mathrm{gr}}$ is also generated by $2mn$ generators
$t_{i,j}$ and $\partial_{i,j}$ for $1\leq i\leq m$ and $1\leq j\leq n$ modulo
the same relations, except that $\llbracket
a_{1},\overline{a}_{1}\rrbracket\llbracket a_{2},\overline{a}_{2}\rrbracket$
does not occur on the right hand side of (59).

######  Remark 3.7.4.

The relation (59) of $\mathscr{PD}$ can be written more explicitly as the
relations (R3)–(R6) below:

  * (R3)

$\partial_{c,b}t_{d,a}=t_{d,a}\partial_{c,b}$ if $b\neq a$ and $c\neq d$.

  * (R4)

$\displaystyle\partial_{c,b}t_{c,a}=qt_{c,a}\partial_{c,b}+\sum_{c^{\prime}>c}(q-q^{-1})t_{c^{\prime},a}\partial_{c^{\prime},b}$
if $b\neq a$.

  * (R5)

$\displaystyle\partial_{c,a}t_{d,a}=qt_{d,a}\partial_{c,a}+\sum_{a^{\prime}>a}(q-q^{-1})t_{d,a^{\prime}}\partial_{c,a^{\prime}}$
if $c\neq d$.

  * (R6)

$\displaystyle\partial_{c,d}t_{c,d}=1+\sum_{c^{\prime}\geq
c}\,\sum_{d^{\prime}\geq d}q^{\llbracket c^{\prime},c\rrbracket+\llbracket
d^{\prime},d\rrbracket}(q-q^{-1})^{2-\llbracket
c^{\prime},c\rrbracket-\llbracket
d^{\prime},d\rrbracket}t_{c^{\prime},a^{\prime}}\partial_{c^{\prime},a^{\prime}}$.

For $\mathscr{PD}^{\mathrm{gr}}$ the relation (59) has the same explicit form,
except that (R6) should be replaced by

  * (R6′)

$\displaystyle\partial_{c,d}t_{c,d}=\sum_{c^{\prime}\geq
c}\,\sum_{d^{\prime}\geq d}q^{\llbracket c^{\prime},c\rrbracket+\llbracket
d^{\prime},d\rrbracket}(q-q^{-1})^{2-\llbracket
c^{\prime},c\rrbracket-\llbracket
d^{\prime},d\rrbracket}t_{c^{\prime},d^{\prime}}\partial_{c^{\prime},d^{\prime}}$.

######  Proposition 3.7.5.

Definition 3.7.2 and Definition 3.7.3 are equivalent.

###### Proof.

We just need to explain how to compute the mixed relations (17) and (20). As a
$U_{LR}$-module,

$\mathscr{D}^{(1)}\cong\breve{V}^{(m)}\otimes\breve{V}^{(n)}\quad\text{and}\quad\mathscr{P}^{(1)}\cong
V^{(m)}\otimes V^{(n)},$

where the isomorphisms are $\partial_{i,j}\mapsto e_{i}\otimes e_{j}$ and
$t_{i,j}\mapsto e_{i}\otimes e_{j}$. By a direct calculation using Definition
3.2.1 we obtain

$\left(\underline{\mathcal{R}}_{L}\right)_{\breve{V}^{(m)},V^{(m)}}=q\sum_{1\leq
i\leq m}\mathsf{E}_{i,i}\otimes\mathsf{E}_{i,i}+\sum_{1\leq i\neq j\leq
m}\mathsf{E}_{i,i}\otimes\mathsf{E}_{j,j}+(q-q^{-1})\sum_{1\leq i<j\leq
m}\mathsf{E}_{j,i}\otimes\mathsf{E}_{j,i}.$

The formula for
$\left(\underline{\mathcal{R}}_{R}\right)_{\breve{V}^{(n)},V^{(n)}}$ is
similar, with $m$ replaced by $n$. The mixed relations (20) of $\mathscr{PD}$
and (17) of $\mathscr{PD}^{\mathrm{gr}}$ can now be computed explicitly based
on Definition 2.4.2 and Definition 2.4.9. ∎

###  3.8. Bases of monomials for $\mathscr{PD}$ and
$\mathscr{PD}^{\mathrm{gr}}$

Consider the monomials

(60) $t_{1,1}^{a_{1,1}}\cdots t_{1,n}^{a_{1,n}}\cdots t_{m,1}^{a_{m,1}}\cdots
t_{m,n}^{a_{m,n}}\partial_{m,n}^{b_{m,n}}\cdots\partial_{m,1}^{b_{m,1}}\cdots\partial_{1,n}^{b_{1,n}}\cdots\partial_{1,1}^{b_{1,1}},\quad
a_{i,j},b_{i,j}\in\mathbb{Z}^{\geq 0}.$

The expression (60) makes sense both as an element of $\mathscr{PD}$ and an
element of $\mathscr{PD}^{\mathrm{gr}}$.

######  Proposition 3.8.1.

The monomials (60) form a $\Bbbk$-basis of $\mathscr{PD}$.

###### Proof.

By a standard straightening argument we can show that by using the relations
(R1), (R1′), (R2), (R2′) and (R3)–(R6) any product of the $t_{i,j}$ and the
$\partial_{i,j}$ can be expressed as a linear combination of the monomials
(60). The fact that the latter monomials are indeed linearly independent
follows from Bergman’s Diamond Lemma and some straightforward (although
tedious) computations. This was also pointed out in [SSV04, Sec. 10]. In
[LSS22a] we give a more conceptual proof of this assertion using the theory of
PBW deformations of quadratic algebras. ∎

######  Proposition 3.8.2.

The algebra $\mathscr{P}$ has a basis consisting of monomials (60) where
$b_{i,j}=0$ for all $i,j$. The algebra $\mathscr{D}$ has a basis consisting of
monomials (60) where $a_{i,j}=0$ for all $i,j$.

###### Proof.

This follows from Proposition 3.8.1. It is also proved for example in [NYM93,
Thm 1.4]. ∎

By Proposition 2.4.8 the algebra $\mathscr{PD}^{\mathrm{gr}}$ is a quotient of
the free algebra on $2mn$ generators $t_{i,j}$ and $\partial_{i,j}$. Note that
by a slight abuse of notation we use the same notation for generators of
$\mathscr{PD}$ and $\mathscr{PD}^{\mathrm{gr}}$. In the next proposition we
describe a basis for this quotient.

######  Proposition 3.8.3.

The monomials (60) form a basis of $\mathscr{PD}^{\mathrm{gr}}$.

###### Proof.

This follows from the vector space decomposition
$\mathscr{PD}^{\mathrm{gr}}=\mathscr{P}\otimes\mathscr{D}$ and Proposition
3.8.2. ∎

######  Remark 3.8.4.

From the results of this subsection it follows that
$\mathscr{PD}^{\mathrm{gr}}$ has two realizations:

  * (i)

According to Definition 3.7.2 we have
$\mathscr{PD}^{\mathrm{gr}}\cong\mathscr{P}\otimes\mathscr{D}$ as a
$\Bbbk$-vector space. Thus, $\mathscr{PD}^{\mathrm{gr}}$ is generated by $2mn$
generators $t_{i,j}\otimes 1$ and $1\otimes\partial_{i,j}$.

  * (ii)

By Proposition 2.4.8 we can realize $\mathscr{PD}^{\mathrm{gr}}$ as a quotient
of the free $\Bbbk$-algebra generated by $2mn$ generators: the $t_{i,j}$ and
the $\partial_{i,j}$.

######  Definition 3.8.5.

Define a total order $\prec$ on the set of pairs $(i,j)$ with $1\leq i\leq m$
and $1\leq j\leq n$ as follows: we set $(i,j)\prec(i^{\prime},j^{\prime})$ if
either $i+j<i^{\prime}+j^{\prime}$, or $i+j=i^{\prime}+j^{\prime}$ and
$i<i^{\prime}$.

######  Remark 3.8.6.

The algebra $\mathscr{PD}$ has another basis consisting of monomials of the
form

(61)
$\left(\prod_{i,j}t_{i,j}^{a_{i,j}}\right)\left(\prod_{i,j}\partial_{i,j}^{b_{i,j}}\right),$

where the $\partial_{i,j}$ (respectively, the $t_{i,j}$) occur in ascending
(respectively, descending) order relative to the total order $\prec$. This can
be deduced from Proposition 3.8.1. Indeed by an elementary argument one can
show that any monomial of the $t_{i,j}$ of total degree $d$ that is sorted in
the order given in Proposition 3.8.1 can be expressed as a linear combination
of monomials of the $t_{i,j}$ of total degree $d$ that are sorted in the order
given in (61). A similar assertion holds for monomials in the
$\partial_{i,j}$. Thus the monomials of the form (61) span $\mathscr{PD}$. A
dimension counting argument implies that the latter monomials also form a
basis.

By an analogous reasoning we can also show that $\mathscr{PD}$ has a basis
that consists of the monomials

(62) $t_{m,n}^{a_{m,n}}\cdots t_{m,1}^{a_{m,1}}\cdots t_{1,n}^{a_{1,1}}\cdots
t_{1,1}^{a_{1,1}}\partial_{1,1}^{b_{1,1}}\cdots\partial_{1,n}^{b_{1,n}}\cdots\partial_{m,1}^{b_{m,1}}\cdots\partial_{m,n}^{b_{m,n}},\quad
a_{i,j},b_{i,j}\in\mathbb{Z}^{\geq 0}.$

Here the $\partial_{i,j}$ (respectively, the $t_{i,j}$) are sorted according
to the lexicographic order (respectively, the reverse lexicographic order) on
indices.

###  3.9. The algebras $\mathscr{A}_{k,l,n}$ and
$\mathscr{A}_{k,l,n}^{\mathrm{gr}}$

In this subsection we consider two families of algebras, the
$\mathscr{A}_{k,l,n}$ and the $\mathscr{A}_{k,l,n}^{\mathrm{gr}}$, that
slightly generalize $\mathscr{PD}$ and $\mathscr{PD}^{\mathrm{gr}}$.

######  Definition 3.9.1.

Fix integers $k,l,n\geq 1$ and set $m:=\max\\{k,l\\}$. Let $\tilde{t}_{i,j}$
and $\tilde{\partial}_{i,j}$ be as in (5) where $a=m$ and $b=n$. We define
$\mathscr{A}_{k,l,n}$ (respectively, $\mathscr{A}_{k,l,n}^{\mathrm{gr}}$) to
be the subalgebra of $\mathscr{PD}=\mathscr{PD}_{m\times n}$ (respectively,
$\mathscr{PD}^{\mathrm{gr}}\cong\mathscr{P}\otimes\mathscr{D}$) that is
generated by the $\tilde{t}_{i,j}$ and the $\tilde{\partial}_{i^{\prime},j}$
(respectively, the $\tilde{t}_{i,j}\otimes 1$ and the
$1\otimes\tilde{\partial}_{i^{\prime},j}$) where

(63) $1\leq i\leq k\quad,\quad 1\leq i^{\prime}\leq l\quad\text{and}\quad
1\leq j\leq n.$

######  Proposition 3.9.2.

The algebras $\mathscr{A}_{k,l,n}$ and $\mathscr{A}_{k,l,n}^{\mathrm{gr}}$
have the following presentations:

  * (i)

$\mathscr{A}_{k,l,n}$ is isomorphic to the quotient of the free
$\Bbbk$-algebra generated by the symbols $t_{i,j}$ and
$\partial_{i^{\prime},j}$ with $i$, $i^{\prime}$, $j$ satisfying (63), modulo
the relations (R1), (R2), (R1′), (R2′) of Subsection 3.4 and the relations
(R3)–(R6) of Remark 3.7.4.

  * (ii)

$\mathscr{A}^{\mathrm{gr}}_{k,l,n}$ is isomorphic to the quotient of the free
$\Bbbk$-algebra generated by the symbols $t_{i,j}$ and
$\partial_{i^{\prime},j}$ with $i$, $i^{\prime}$, $j$ satisfying (63), modulo
the relations (R1), (R2), (R1′), (R2′) of Subsection 3.4 and the relations
(R3)–(R5) and (R6′) of Remark 3.7.4.

###### Proof.

(i) Denote the quotient of the free algebra by $\mathscr{F}_{k,l,n}$. Since
$\mathscr{A}_{k,l,n}$ is a subalgebra of $\mathscr{PD}=\mathscr{PD}_{m\times
n}$ for $m:=\max\\{k,l\\}$, from the explicit description of the relations of
$\mathscr{PD}$ it follows that there exists a natural epimorphism
$\mathsf{f}_{k,l,n}:{\mathscr{F}}_{k,l,n}\to\mathscr{A}_{k,l,n}$ that is
uniquely defined by the assignments $t_{i,j}\mapsto t_{m-k+i,j}$ and
$\partial_{i^{\prime},j}\mapsto\partial_{m-l+i^{\prime},j}$. A standard
straightening argument proves that every element of $\mathscr{F}_{k,l,n}$ is a
linear combination of monomials of the form (60). Proposition 3.8.1 implies
that $\mathsf{f}_{k,l,n}$ maps the latter monomials to a linearly independent
set of elements of $\mathscr{A}_{k,l,n}$. Thus $\mathsf{f}_{k,l,n}$ is an
isomorphism.

(ii) Similar to the proof of (i), with Proposition 3.8.1 replaced by
Proposition 3.8.3. ∎

######  Definition 3.9.3.

For any $1\leq r\leq n$ we can identify $U_{q}(\mathfrak{gl}_{r})$ with a Hopf
subalgebra of $U_{q}(\mathfrak{gl}_{n})$ via the monomorphism of associative
algebras

(64) $\boldsymbol{\kappa}_{r,n}:U_{q}(\mathfrak{gl}_{r})\to
U_{q}(\mathfrak{gl}_{n}),$

defined by $\boldsymbol{\kappa}_{r,n}(E_{i}):=E_{i+n-r}$,
$\boldsymbol{\kappa}_{r,n}(F_{i}):=F_{i+n-r}$, and
$\boldsymbol{\kappa}_{r,n}(K_{\varepsilon_{i}}^{\pm
1}):=K_{\varepsilon_{i+n-r}}^{\pm 1}$.

In the next proposition we establish the existence of the map (4). Recall from
Remark 3.8.4(ii) that we consider both $\mathscr{PD}$ and
$\mathscr{PD}^{\mathrm{gr}}$ as algebras generated by $2mn$ generators
$t_{i,j}$ and $\partial_{i,j}$.

######  Proposition 3.9.4.

Fix $1\leq m^{\prime}\leq m$ and $1\leq n^{\prime}\leq n$. Let
$\tilde{t}_{i,j},\tilde{\partial}_{i,j}\in\mathscr{PD}_{m^{\prime}\times
n^{\prime}}$ (respectively,
$\tilde{t}_{i,j},\tilde{\partial}_{i,j}\in\mathscr{PD}^{\mathrm{gr}}_{m^{\prime}\times
n^{\prime}}$) for $1\leq i\leq m^{\prime}$ and $1\leq j\leq n^{\prime}$ be as
in (5) for $a:=m^{\prime}$ and $b:=n^{\prime}$, that is

$\tilde{t}_{i,j}:=t_{m^{\prime}+1-i,n^{\prime}+1-j}\quad\text{and}\quad\tilde{\partial}_{i,j}:=\partial_{m^{\prime}+1-i,n^{\prime}+1-j}.$

Also, let $\tilde{t}_{i,j},\tilde{\partial}_{i,j}\in\mathscr{PD}_{m\times n}$
(respectively,
$\tilde{t}_{i,j},\tilde{\partial}_{i,j}\in\mathscr{PD}^{\mathrm{gr}}_{m\times
n}$) for $1\leq i\leq m$ and $1\leq j\leq n$ be as in (5) for $a:=m$ and
$b:=n$, that is

$\tilde{t}_{i,j}:=t_{m+1-i,n+1-j}\quad\text{and}\quad\tilde{\partial}_{i,j}:=\partial_{m+1-i,n+1-j}.$

Then the following assertions hold.

  * (i)

The assignments $\tilde{t}_{i,j}\mapsto\tilde{t}_{i,j}$ and
$\tilde{\partial}_{i,j}\mapsto\tilde{\partial}_{i,j}$ for $1\leq i\leq
m^{\prime}$ and $1\leq j\leq n^{\prime}$ define unique embeddings of algebras

$\boldsymbol{e}:=\boldsymbol{e}_{m^{\prime}\times n^{\prime}}^{m\times
n}:\mathscr{PD}_{m^{\prime}\times n^{\prime}}\to\mathscr{PD}_{m\times
n}\quad\text{and}\quad\boldsymbol{e}^{\mathrm{gr}}:=(\boldsymbol{e}^{\mathrm{gr}})_{m^{\prime}\times
n^{\prime}}^{m\times n}:\mathscr{PD}_{m^{\prime}\times
n^{\prime}}^{\mathrm{gr}}\to\mathscr{PD}_{m\times n}^{\mathrm{gr}}.$

  * (ii)

If we identify $U_{q}(\mathfrak{gl}_{m^{\prime}})\otimes
U_{q}(\mathfrak{gl}_{n^{\prime}})$ with a subalgebra of
$U_{q}(\mathfrak{gl}_{m})\otimes U_{q}(\mathfrak{gl}_{n})$ via
$\boldsymbol{\kappa}_{m^{\prime},m}\otimes\boldsymbol{\kappa}_{n^{\prime},n}$
then the maps $\boldsymbol{e}$ and $\boldsymbol{e}^{\mathrm{gr}}$ are
$U_{q}(\mathfrak{gl}_{m^{\prime}})\otimes
U_{q}(\mathfrak{gl}_{n^{\prime}})$-equivariant.

###### Proof.

We only give the details of the proofs of these assertions for $\mathscr{PD}$.
The arguments for $\mathscr{PD}^{\mathrm{gr}}$ are analogous.

(i) From Definition 3.7.3 and Remark 3.7.4 it follows that the generators
$\tilde{t}_{i,j}$ and $\tilde{\partial}_{i,j}$ of
$\mathscr{PD}_{m^{\prime}\times n^{\prime}}$ and $\mathscr{PD}_{m\times n}$
satisfy identical relations. It follows that there exists a homomorphism of
algebras $\mathscr{PD}_{m^{\prime}\times n^{\prime}}\to\mathscr{PD}_{m\times
n}$. By Proposition 3.8.1 the latter map takes a basis of
$\mathscr{PD}_{m^{\prime}\times n^{\prime}}$ to a basis of
$\mathscr{PD}_{m\times n}$, hence it is an injection. The uniqueness assertion
is trivial.

(ii) We give the proof for $\boldsymbol{e}$ only, since the proof for
$\boldsymbol{e}^{\mathrm{gr}}$ is similar. Since
$\mathscr{PD}_{m^{\prime}\times n^{\prime}}$ and $\mathscr{PD}_{m\times n}$
are module algebras it suffices to verify equivariance for standard generators
of $U_{q}(\mathfrak{gl}_{m^{\prime}})$ and $U_{q}(\mathfrak{gl}_{n^{\prime}})$
on the $\tilde{t}_{i,j}$ and the $\tilde{\partial}_{i,j}$. This can be done
using the explicit formulas of Remark 3.6.2. ∎

###  3.10. The action of $\mathscr{PD}$ on $\mathscr{P}$ and the map
$\phi_{PD}$

Recall that by Proposition 2.4.10, $\mathscr{PD}$ is a $U_{LR}$-module
algebra. We denote the action of $x\in U_{LR}$ on $D\in\mathscr{PD}$ by
$x\cdot D$.

Let $\mathscr{I}$ denote the left ideal of $\mathscr{PD}$ that is generated by
$\mathscr{D}^{(1)}$. By Proposition 3.8.1 we have a $U_{LR}$-invariant
decomposition

$\mathscr{PD}\cong\mathscr{I}\oplus\mathscr{P}.$

This decomposition equips $\mathscr{P}\cong\mathscr{PD}/\mathscr{I}$ with a
$\mathscr{PD}$-module structure given by

(65) $\mathscr{PD}\otimes\mathscr{P}\to\mathscr{P}\ ,\
D\otimes(f+\mathscr{I})\mapsto D\cdot f\quad\text{for }D\in\mathscr{PD}\text{
and }f\in\mathscr{P},$

where $D\cdot f:=(Df)+\mathscr{I}$.

######  Definition 3.10.1.

The map $\phi_{PD}:\mathscr{PD}\to\mathrm{End}_{\Bbbk}(\mathscr{P})$ is the
homomorphism of algebras induced by (65).

To simplify our notation, henceforth for $X\in\mathscr{PD}$ and
$f\in\mathscr{P}$ we write $X\cdot f$ instead of $\phi_{PD}(X)f$.

######  Lemma 3.10.2.

The map (65) is a $U_{LR}$-module homomorphism.

###### Proof.

This is a consequence of the following general fact: let $H$ be a Hopf
algebra, $A$ be an $H$-module algebra, and $I\subseteq A$ be an $H$-stable
left ideal of $A$. Then the canonical $A$-module structure map $A\otimes
A/I\to A/I$ is an $H$-module homomorphism. ∎

###  3.11. $\mathscr{P}$ is a faithful $\mathscr{PD}$-module

The goal of this subsection is to provide a purely algebraic proof of
Proposition 3.11.4. This proposition is also proved in [SSV04, Thm 2.6] using
analytic tools.

######  Lemma 3.11.1.

Let $\mathscr{P}^{(\leq k)}:=\bigoplus_{i=0}^{k}\mathscr{P}^{(i)}$ for $k\geq
0$. Then $\partial_{i,j}\cdot\mathscr{P}^{(\leq k)}\subseteq\mathscr{P}^{(\leq
k-1)}$.

###### Proof.

Follows by induction on $k$ and the mixed relations (R3)–(R6) in Subsection
3.7. ∎

Recall the total order $\prec$ on the set of pairs $(i,j)$ with $1\leq i\leq
m$ and $1\leq j\leq n$ from Definition 3.8.5.

######  Lemma 3.11.2.

Assume that $(i_{r},j_{r})\prec(i,j)$ for $1\leq r\leq k$. Then
$\partial_{i,j}\cdot(t_{i_{1},j_{1}}\cdots t_{i_{k},j_{k}})=0$.

###### Proof.

We use induction on $k$. From $(i_{1},j_{1})\prec(i,j)$ it follows that either
$i>i_{1}$ or $j>j_{1}$. If $i>i_{1}$ then by the mixed relations (R3) or (R5)
we have

$\partial_{i,j}t_{i_{1},j_{1}}\cdots
t_{i_{k},j_{k}}=c_{1}t_{i_{1},j_{1}}\partial_{i,j}t_{i_{2},j_{2}}\cdots
t_{i_{k},j_{k}}+\delta_{j,j_{1}}c_{2}\sum_{j^{\prime}>j}t_{i_{1},j^{\prime}}\partial_{i,j^{\prime}}t_{i_{2},j_{2}}\cdots
t_{i_{k},j_{k}},$

for some $c_{1},c_{2}\in\Bbbk$. The claim now follows from the induction
hypothesis, because $i+j^{\prime}>i+j$ and therefore
$(i,j)\prec(i,j^{\prime})$. When $j>j_{1}$ the argument is similar. ∎

For $a\in\mathbb{Z}$ we set

(66) $\mathbf{c}(a):=\begin{cases}\sum_{i=0}^{a}q^{2i}&\text{ if }a\geq 0,\\\
0&\text{otherwise.}\end{cases}$

######  Lemma 3.11.3.

Assume that $(i_{r},j_{r})\prec(i,j)$ for $1\leq r\leq k$. Then

$\partial_{i,j}\cdot(t^{a}_{i,j}t_{i_{1},j_{1}}\cdots
t_{i_{k},j_{k}})=\mathbf{c}(a-1)t_{i,j}^{a-1}t_{i_{1},j_{1}}\cdots
t_{i_{k},j_{k}}\quad\text{for $a\geq 1$}.$

###### Proof.

The mixed relation (R6) implies

$\displaystyle\partial_{i,j}t_{i,j}^{a}t_{i_{1},j_{1}}$ $\displaystyle\cdots
t_{i_{k},j_{k}}=t_{i,j}^{a-1}t_{i_{1},j_{1}}\cdots
t_{i_{k},j_{k}}+q^{2}t_{i,j}\partial_{i,j}t_{i,j}^{a-1}t_{i_{1},j_{1}}\cdots
t_{i_{k},j_{k}}$
$\displaystyle+(q^{2}-1)\sum_{i^{\prime}>i}t_{i^{\prime},j}\partial_{i^{\prime},j}t_{i,j}^{a-1}t_{i_{1},j_{1}}\cdots
t_{i_{k},j_{k}}+(q^{2}-1)\sum_{j^{\prime}>j}t_{i,j^{\prime}}\partial_{i,j^{\prime}}t_{i,j}^{a-1}t_{i_{1},j_{1}}\cdots
t_{i_{k},j_{k}}$
$\displaystyle+(q-q^{-1})^{2}\sum_{i^{\prime}>i,j^{\prime}>j}t_{i^{\prime},j^{\prime}}\partial_{i^{\prime},j^{\prime}}t_{i,j}^{a-1}t_{i_{1},j_{1}}\cdots
t_{i_{k},j_{k}}.$

Since $\min\\{i^{\prime}+j,i+j^{\prime},i^{\prime}+j^{\prime}\\}>i+j$, by
Lemma 3.11.2 the sums on the second and the third line lie in the ideal
$\mathscr{I}$. The assertion follows by induction on $a$. ∎

######  Proposition 3.11.4.

$\mathscr{P}$ is a faithful $\mathscr{PD}$-module.

###### Proof.

Let $D\in\mathscr{PD}$ and assume that $D\neq 0$. Then $D=\sum_{d\geq 0}D_{d}$
where each $D_{d}$ is a linear combination of monomials of the form (61) with
$\sum_{i,j}b_{i,j}=d$. Set $d_{\circ}:=\min\\{d:D_{d}\neq 0\\}$. By Lemma
3.11.1, for $f\in\mathscr{P}^{(d_{\circ})}$ we have $D\cdot
f=D_{d_{\circ}}\cdot f$. Let $\mathcal{T}$ denote the set of all the
$mn$-tuples $\mathsf{b}:=(b_{i,j})$ for which a monomial of the form (61)
occurs in $D_{d_{\circ}}$ with a nonzero coefficient. We sort the components
of the $\mathsf{b}:=(b_{i,j})$ according to $\prec$ on the pairs $(i,j)$. In
other words, we assume that
$\mathsf{b}:=(b_{1,1},b_{1,2},b_{2,1},\ldots,b_{m-1,n},b_{m,n-1},b_{m,n})$.
Let $\tilde{\mathsf{b}}:=(\tilde{b}_{i,j})$ be the minimum of $\mathcal{T}$ in
the reverse lexicographic order. Thus, we have

$\tilde{b}_{m,n}=\min\\{b_{m,n}\,:\,(b_{i,j})\in\mathcal{T}\\},$

then also
$\tilde{b}_{m-1,n}=\min\\{b_{m-1,n}\,:\,(b_{i,j})\in\mathcal{T}\text{ and
}b_{m,n}=\tilde{b}_{m,n}\\}$, and so on. From Lemma 3.11.2 and Lemma 3.11.3 it
follows that $D_{d_{\circ}}\cdot\prod_{i,j}t^{\tilde{b}_{i,j}}\neq 0$. ∎

###  3.12. Two $U_{LR}$-actions on $\mathscr{PD}$ are identical

By Proposition 3.11.4 the map $\phi_{PD}$ is an injection and consequently we
can consider $\mathscr{PD}$ as a subalgebra of
$\mathrm{End}_{\Bbbk}(\mathscr{P})$. Thus according to Lemma 2.3.1 there
exists another action of $U_{LR}$ on elements of $\mathscr{PD}$. We
temporarily denote this action by $x\bullet D$ for $x\in U_{LR}$ and
$D\in\mathscr{PD}$. In the following proposition, we show that the latter
action is identical to the action that is defined in the beginning of
Subsection 3.10.

######  Proposition 3.12.1.

$x\bullet D=x\cdot D$ for $x\in U_{LR}$ and $D\in\mathscr{PD}$.

###### Proof.

By Lemma 3.10.2, for $f\in\mathscr{P}$ we have

$\displaystyle(x\bullet D)\cdot f=\sum x_{1}\cdot(D\cdot$
$\displaystyle(S(x_{2})\cdot f))=\sum(x_{1}\cdot
D)\cdot(x_{2}\cdot(S(x_{3})\cdot f))$ $\displaystyle=\sum(x_{1}\cdot
D)\cdot(\epsilon(x_{2})f)=\sum(x_{1}\epsilon(x_{2})\cdot D)\cdot f=(x\cdot
D)\cdot f.$

Since $\mathscr{P}$ is a faithful module over
$\mathrm{End}_{\Bbbk}(\mathscr{P})$, it follows that $x\bullet D=x\cdot D$. ∎

Henceforth we only use the notation $x\cdot D$ to denote the $U_{LR}$-action
on $\mathscr{PD}$.

###  3.13. The maps $\mathsf{P}_{k,l,n}$

Recall that $\mathscr{PD}^{\mathrm{gr}}\cong\mathscr{P}\otimes\mathscr{D}$ as
a vector space. Let

(67) $\mathsf{P}:\mathscr{PD}^{\mathrm{gr}}\to\mathscr{PD}$

be the linear map uniquely defined by $\mathsf{P}(a\otimes b):=ab$ for
$a\in\mathscr{P}$ and $b\in\mathscr{D}$.

######  Proposition 3.13.1.

The map $\mathsf{P}$ is an isomorphism of $U_{LR}$-modules.

###### Proof.

The map $\mathsf{P}$ is identical to the map (26) when $A:=\mathscr{P}$,
$B:=\mathscr{D}$ and $H:=U_{LR}$. Hence by Remark 2.4.11 it is a homomorphism
of $U_{LR}$-modules. From Proposition 3.8.1 and Proposition 3.8.3 it follows
that $\mathsf{P}$ maps a basis of $\mathscr{PD}^{\mathrm{gr}}$ to a basis of
$\mathscr{PD}$, hence it is indeed an isomorphism of $U_{LR}$-modules. ∎

######  Definition 3.13.2.

Given $k,l,n\geq 1$, we set $m:=\max\\{k,l\\}$ and define the map

(68)
$\mathsf{P}_{k,l,n}:\mathscr{A}_{k,l,n}^{\mathrm{gr}}\to\mathscr{A}_{k,l,n}\
,\ D\mapsto\mathsf{P}(D),$

where $\mathsf{P}:\mathscr{PD}^{\mathrm{gr}}\to\mathscr{PD}$ is as in (67).

For $r,s\in\mathbb{Z}^{\geq 0}$ we set

(69)
$\mathscr{PD}^{\mathrm{gr},(r,s)}:=\mathscr{P}^{(r)}\otimes\mathscr{D}^{(s)}\quad\text{
and
}\quad\mathscr{A}_{k,l,n}^{\mathrm{gr},(r,s)}:=\mathscr{A}_{k,l,n}^{\mathrm{gr}}\cap\mathscr{PD}^{\mathrm{gr},(r,s)},$

so that

$\mathscr{PD}^{\mathrm{gr}}=\bigoplus_{r,s\geq
0}\mathscr{PD}^{\mathrm{gr},(r,s)}\quad\text{and}\quad\mathscr{A}_{k,l,n}^{\mathrm{gr}}:=\bigoplus_{r,s\geq
0}\mathscr{A}_{k,l,n}^{\mathrm{gr},(r,s)}.$

By Proposition 3.6.1 we obtain an isomorphism of
$U_{q}(\mathfrak{gl}_{k})\otimes U_{q}(\mathfrak{gl}_{l})\otimes
U_{q}(\mathfrak{gl}_{n})\otimes U_{q}(\mathfrak{gl}_{n})$-modules

(70)
$\mathscr{A}_{k,l,n}^{\mathrm{gr},(r,s)}\cong\bigoplus_{\footnotesize\begin{array}[]{c}\lambda\in\Lambda_{\underline{k},r}\\\\[-2.84526pt]
\mu\in\Lambda_{\underline{l},s}\end{array}}(V_{\lambda}^{*}\otimes
V_{\mu})\otimes(V_{\lambda}^{*}\otimes V_{\mu}),$

where $\underline{k}:=\min\\{k,n\\}$ and $\underline{l}:=\min\\{l,n\\}$. Here
we consider the left copy of $V_{\lambda}^{*}\otimes V_{\mu}$ as a module for
$U_{q}(\mathfrak{gl}_{k})\otimes U_{q}(\mathfrak{gl}_{l})$ and the right copy
of $V_{\lambda}^{*}\otimes V_{\mu}$ as a module for $U_{R}\otimes U_{R}\cong
U_{q}(\mathfrak{gl}_{n})\otimes U_{q}(\mathfrak{gl}_{n})$. Of course by
restriction along the coproduct map $U_{R}\to U_{R}\otimes U_{R}$ we can also
consider the right copy as a $U_{R}$-module.

######  Proposition 3.13.3.

For $a\otimes
a^{\prime}\in\mathscr{P}^{(r)}\otimes_{\tilde{\mathcal{R}}}\mathscr{D}^{(r^{\prime})}$
and $b\otimes
b^{\prime}\in\mathscr{P}^{(s)}\otimes_{\tilde{\mathcal{R}}}\mathscr{D}^{(s^{\prime})}$
we have

(71) $\mathsf{P}(a\otimes a^{\prime})\mathsf{P}(b\otimes
b^{\prime})-\mathsf{P}\big{(}(a\otimes a^{\prime})(b\otimes
b^{\prime})\big{)}\in\bigoplus_{i=1}^{\min\\{u,u^{\prime}\\}}\mathscr{PD}^{(u-i,u^{\prime}-i)},$

where $u:=r+s$ and $u^{\prime}:=r^{\prime}+s^{\prime}$.

###### Proof.

We have $\mathsf{P}(a\otimes a^{\prime})\mathsf{P}(b\otimes
b^{\prime})=aa^{\prime}bb^{\prime}$. Using the explicit relations of
$\mathscr{PD}$ (see Remark 3.7.4) we can move the $\partial_{i,j}$ past the
$t_{i,j}$ to express $a^{\prime}b$ as $a^{\prime}b=\sum
a^{\prime\prime}b^{\prime\prime}$ where $a^{\prime\prime}\in\mathscr{P}$ and
$b^{\prime\prime}\in\mathscr{D}$. Thus

$\mathsf{P}(a\otimes a^{\prime})\mathsf{P}(b\otimes b^{\prime})=\sum
aa^{\prime\prime}b^{\prime\prime}b^{\prime}.$

Similarly, using the relations of $\mathscr{PD}^{\mathrm{gr}}$ we can move the
$1\otimes\partial_{i,j}$ past the $t_{i,j}\otimes 1$ and as a result we obtain
$(1\otimes a^{\prime})(b\otimes
1)=\sum\underline{a}^{\prime\prime}\otimes\underline{b}^{\prime\prime}$ where
$\underline{a}^{\prime\prime}\in\mathscr{P}$ and
$\underline{b}^{\prime\prime}\in\mathscr{D}$. It follows that

$\mathsf{P}\big{(}(a\otimes a^{\prime})(b\otimes b^{\prime})\big{)}=\sum
a\underline{a}^{\prime\prime}\underline{b}^{\prime\prime}b^{\prime}.$

The only difference between the relations of $\mathscr{PD}$ and
$\mathscr{PD}^{\mathrm{gr}}$ is (R6) vs. (R6′). Since (R6′) is the homogenized
form of (R6′), it follows that

$\sum
a^{\prime\prime}b^{\prime\prime}-\sum\underline{a}^{\prime\prime}\underline{b}^{\prime\prime}\in\bigoplus_{i=1}^{\min\\{r^{\prime},s\\}}\mathscr{PD}^{(s-i,r^{\prime}-i)}.$

From the latter inclusion (71) follows immediately. ∎

Recall that $\mathrm{gr}(\mathscr{A}_{k,l,n})$ denotes the associated graded
algebra corresponding to the degree filtration on $\mathscr{A}_{k,l,n}$, i.e.,
the filtration obtained by setting $\deg(t_{i,j})=\deg(\partial_{i,j})=1$.
Note that we have a canonical isomorphism
$\mathscr{A}_{k,l,n}^{\mathrm{gr}}\cong\mathrm{gr}(\mathscr{A}_{k,l,n}^{\mathrm{gr}})$
since $\mathscr{A}_{k,l,n}^{\mathrm{gr}}$ is graded.

The maps
$\mathsf{P}_{k,l,n}:\mathscr{A}_{k,l,n}^{\mathrm{gr}}\to\mathscr{A}_{k,l,n}$
do _not_ induce isomorphisms of associative algebras. However, the following
statement holds.

######  Corollary 3.13.4.

The associated graded map $\mathrm{gr}(\mathsf{P}_{k,l,n})$ induces a
$U_{R}$-equivariant isomorphism between
$\mathscr{A}_{k,l,n}^{\mathrm{gr}}\cong\mathrm{gr}(\mathscr{A}_{k,l,n}^{\mathrm{gr}})$
and $\mathrm{gr}(\mathscr{A}_{k,l,n})$. When $k=l=m$, the latter map is
$U_{LR}$-equivariant.

###### Proof.

From Proposition 3.13.3 it follows that the associated graded map
$\mathrm{gr}(\mathsf{P}_{k,l,n})$ is an isomorphism of associative algebras
from
$\mathrm{gr}(\mathscr{A}_{k,l,n}^{\mathrm{gr}})\cong\mathscr{A}_{k,l,n}^{\mathrm{gr}}$
onto $\mathrm{gr}(\mathscr{A}_{k,l,n})$. The equivariance statements follow
from Proposition 3.13.1. ∎

###  3.14. The algebras $\mathring{U}_{L}$, $\mathring{U}_{R}$ and
$\mathring{U}_{LR}$

We set

$\mathring{U}_{LR}:=\phi_{U}^{-1}(\mathscr{PD}):=\\{x\in
U_{LR}\,:\,\phi_{U}(x)\in\mathscr{PD}\\}.$

Also, we set

$\mathring{U}_{L}:=\\{x\in U_{L}\,:\,x\otimes
1\in\mathring{U}_{LR}\\}\quad\text{and}\quad\mathring{U}_{R}:=\\{x\in
U_{R}\,:\,1\otimes x\in\mathring{U}_{LR}\\}.$

Recall that the adjoint action of $U_{LR}$ is $\mathrm{ad}_{y}(x):=\sum
y_{1}xS(y_{2})$ for $x,y\in U_{LR}$. Recall that we equip
$\mathrm{End}_{\Bbbk}(\mathscr{P})$ with the $U_{LR}$-module structure of
Lemma 2.3.1 and we denote the latter action by $x\cdot T$ for $x\in U_{LR}$
and $T\in\mathrm{End}_{\Bbbk}(\mathscr{P})$. Then it is straightforward from
the definition of $\phi_{U}$ that

(72) $\phi_{U}({\mathrm{ad}_{y}(x)})=y\cdot\phi_{U}(x)\quad\text{ for $x,y\in
U_{LR}$. }$

######  Lemma 3.14.1.

$\mathrm{ad}_{x}(\mathring{U}_{LR})\subseteq\mathring{U}_{LR}$ for $x\in
U_{LR}$.

###### Proof.

Follows immediately from (72) and Proposition 3.12.1. ∎

######  Remark 3.14.2.

When $m\leq n$, Proposition 3.6.1 implies that the map
$\mathcal{L}_{\mathscr{P}}:U_{L}\to\mathrm{End}_{\Bbbk}(\mathscr{P})$ is an
injection. The argument is similar to the proof of [KS97, Thm 7.1.5.13].

Let $\mathscr{K}_{n}$ denote the kernel of
$\mathcal{R}_{\mathscr{P}}:U_{R}\to\mathrm{End}_{\Bbbk}(\mathscr{P})$. For the
next proposition recall the notation $\mathscr{F}(H,I)$ defined in (11).

######  Proposition 3.14.3.

Assume that $m\leq n$. Then $\mathring{U}_{L}\subseteq\mathscr{F}(U_{L})$ and
$\mathring{U}_{R}\subseteq\mathscr{F}(U_{R},\mathscr{K}_{n})$.

###### Proof.

Since the actions of $U_{LR}$ on $\mathscr{P}$ and $\mathscr{D}$ are degree
preserving, it follows that $\mathscr{PD}$ is a locally finite
$U_{LR}$-module. The assertions of the proposition follow from the fact that
the maps $\mathring{U}_{L}\xrightarrow{\phi_{U}}\mathscr{PD}$ and
$\mathring{U}_{R}/(\mathring{U}_{R}\cap\mathscr{K}_{n})\xrightarrow{\phi_{U}}\mathscr{PD}$
are injective and $U_{LR}$-equivariant (this is equivalent to (72)). ∎

The following proposition is a “no-go theorem” that provides evidence that the
commutative diagram (1) cannot be fully quantized.

######  Proposition 3.14.4.

There does not exist a $\Bbbk$-algebra $\widetilde{\mathscr{PD}}$ with the
following properties:

  * (i)

$\widetilde{\mathscr{PD}}$ is a locally finite $U_{LR}$-module.

  * (ii)

$\mathscr{P}$ is a $\widetilde{\mathscr{PD}}$-module and the action map
$\widetilde{\mathscr{PD}}\otimes\mathscr{P}\to\mathscr{P}$ is a homomorphism
of $U_{LR}$-module.

  * (iii)

There exists a homomorphism of algebras
$\widetilde{\phi}:U_{LR}\to\widetilde{\mathscr{PD}}$ such that the diagram

$\textstyle{U_{LR}\otimes\mathscr{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\otimes