1.1 The Hopf algebra $U_q(\mathfrak {gl}(N,\mathbb {C}))$ and its real form $U_q(\mathfrak {u}(N))$

For the following, we refer to any standard reference work on quantum groups, for example, [Reference Klimyk and SchmüdgenKS97]. We adapt results to fit our conventions.

Definition 1.1. We define $U_q(\mathfrak {n}(N))$ as the universal unital $\mathbb {C}$ -algebra generated by elements $E_1,\ldots , E_{N-1}$ such that $E_iE_j = E_jE_i$ for all $1\leq i,j< N$ with $|i-j|\geq 2$ , and such that for all $1\leq i,j< N$ with $|i-j|= 1$ it holds that

(1.1) $$ \begin{align} E_i^2E_j - (q+q^{-1})E_iE_jE_i + E_jE_i^2 =0, \end{align} $$

We define $U_q(\mathfrak {n}^-(N))$ as an independent copy of $U_q(\mathfrak {n}(N))$ , with generators written as $F_i$ for $1\leq i < N$ .

We define $U_q(\mathfrak {h}(N))$ as the $\mathbb {C}$ -algebra $\mathbb {C}[K_1^{\pm 1},\ldots , K_N^{\pm 1}]$ , and write $\hat {K}_i = K_iK_{i+1}^{-1}$ for $1\leq i < N$ .

We define $U_q(\mathfrak {gl}(N,\mathbb {C}))$ as the algebra generated by $U_q(\mathfrak {n}(N)),U_q(\mathfrak {h}(N)),U_q(\mathfrak {n}^-(N))$ with extra relations

(1.2) $$ \begin{align} K_iE_j =q^{\delta_{ij}-\delta_{i,j+1}}E_j K_i,\qquad 1 \leq i\leq N,1\leq j< N, \end{align} $$

(1.3) $$ \begin{align} K_iF_j =q^{\delta_{i,j+1}-\delta_{ij}}F_j K_i,\qquad 1 \leq i\leq N,1\leq j< N, \end{align} $$

(1.4) $$ \begin{align} E_i F_j - F_jE_i= \delta_{ij} \frac{\hat{K}_i - \hat{K}_i^{-1}}{q-q^{-1}},\qquad 1\leq i,j<N. \end{align} $$

Note that $U_q(\mathfrak {h}(N))$ is just the group algebra of $\mathbb {Z}^N$ , and in particular does not depend on q. The notation $U_q(\mathfrak {h}(N))$ is purely meant to guide classical intuition as to the role that this algebra plays in the theory.

By, for example, the proof of [Reference Klimyk and SchmüdgenKS97, Theorem 6.14], the following multiplication maps are bijective:

(1.5) $$ \begin{align} U_q(\mathfrak{n}(N)) \otimes U_q(\mathfrak{h}(N)) \otimes U_q(\mathfrak{n}^-(N)) \rightarrow U_q(\mathfrak{gl}(N,\mathbb{C})), \end{align} $$

(1.6) $$ \begin{align} U_q(\mathfrak{n}^-(N)) \otimes U_q(\mathfrak{h}(N)) \otimes U_q(\mathfrak{n}(N)) \rightarrow U_q(\mathfrak{gl}(N,\mathbb{C})). \end{align} $$

By (1.5), one has that $U_q(\mathfrak {b}(N))$ , resp. $U_q(\mathfrak {b}^-(N))$ , is isomorphic to the universal algebra generated by $U_q(\mathfrak {h}(N))$ and $U_q(\mathfrak {n}(N))$ , resp. $U_q(\mathfrak {h}(N))$ and $U_q(\mathfrak {n}^-(N))$ , satisfying the commutation relations (1.2), resp. (1.3).

Definition 1.3. We endow $U_q(\mathfrak {gl}(N,\mathbb {C}))$ with the Hopf algebra structure

(1.7) $$ \begin{align} \Delta(K_i) = K_i \otimes K_i,\qquad \Delta(E_i) = E_i \otimes 1 + \hat{K}_i\otimes E_i,\qquad \Delta(F_i) = F_i \otimes \hat{K}_i^{-1} + 1 \otimes F_i, \end{align} $$

where counit and antipode are determined by

(1.8) $$ \begin{align} \varepsilon(E_i) = \varepsilon(F_i)= 0,\quad \varepsilon(K_i) = 1,\qquad S(E_i) = -\hat{K}_i^{-1}E_i,\quad S(F_i) = -F_i\hat{K}_i,\quad S(K_i) = K_i^{-1}. \end{align} $$

Note that $U_q(\mathfrak {b}(N))$ and $U_q(\mathfrak {b}^-(N))$ are then Hopf subalgebras.

Definition 1.4. The Hopf $*$ -algebra $U_q(\mathfrak {u}(N))$ has underlying Hopf algebra $U_q(\mathfrak {gl}(N,\mathbb {C}))$ and $*$ -structure

(1.9) $$ \begin{align} K_i^* = K_i,\qquad E_i^* = q^{-1}F_i\hat{K}_i,\qquad F_i^* = q\hat{K}_i^{-1}E_i. \end{align} $$

A weight vector in a $U_q(\mathfrak {gl}(N,\mathbb {C}))$ -module is a joint (nonzero) eigenvector for all $K_i$ . A weight vector v is said to be

• ○ a highest weight vector if $E_i v = 0$ for all i,

• ○ an admissible weight vector if the associated eigenvalues of the $K_i$ are positive. The unique $\lambda \in \mathbb {R}^N$ such that $K_iv = q^{\lambda _i}v$ is then called the weight Footnote ² $\mathrm {wt}(v)$ of v,

• ○ an integral weight vector if v is admissible and $\mathrm {wt}(v)\in \mathbb {Z}^N$ , and

• ○ a positive integral weight vector if v is integral and $\mathrm {wt}(v) \in \mathbb {Z}_{\geq 0}^{N}$ .

A $U_q(\mathfrak {gl}(N,\mathbb {C}))$ -module is admissible, resp. integral, resp. positive integral if it has a basis of admissible, resp. integral, resp. positive integral weight vectors.

Denote the set of weakly integral, integral and positive integral weights respectively as

$$\begin{align*}P_{{\mathrm{w.int}}} = \{\lambda = (\lambda_1,\ldots,\lambda_N) \in \mathbb{R}^N \mid \lambda_i - \lambda_j \in \mathbb{Z} \textrm{ for all }i,j\},\qquad P_{{\mathrm{int}}} = \mathbb{Z}^N,\qquad P_{{\mathrm{pos.int}}} = \mathbb{Z}_{\geq0}^{N}. \end{align*}$$

Call a vector $\lambda \in \mathbb {R}^N$ dominant if $\lambda _i \geq \lambda _{j}$ whenever $i\leq j$ . The dominant vectors in $P_{\bullet }$ are denoted as $P_{\bullet }^{{\mathrm {dom}}}$ . We also denote the standard bilinear form on $\mathbb {R}^N$ as

$$\begin{align*}(\omega,\chi) = \sum_{i=1}^N \omega_i \chi_i. \end{align*}$$

We write $Q \subseteq P_{{\mathrm {int}}}$ for the root lattice consisting of integral weights $\lambda $ with $\sum \lambda _i = 0$ , and $Q^+ \subseteq Q$ for the $\mathbb {Z}_{\geq 0}$ -span of the $\mathbf {a}_i = e_i -e_{i+1}$ for $1\leq i<N$ . For $\omega ,\omega '\in P_{{\mathrm {int}}}$ we write

$$\begin{align*}\omega \preceq \omega' \quad \Leftrightarrow \quad \omega' - \omega \in Q^+ \qquad \Leftrightarrow \qquad \sum_{k=1}^N \omega_k = \sum_{k=1}^N \omega_k'\textrm{ and } \sum_{k=1}^l \omega_k \leq \sum_{k=1}^l \omega_k' \textrm{ for all }1\leq l \leq N. \end{align*}$$

Note that the order (0.15) can be related to the partial order on $P_{{\mathrm {int}}}$ as follows: with

we have

$$\begin{align*}I \preceq J \qquad \textrm{if and only if} \qquad \varpi_J \preceq \varpi_I. \end{align*}$$

In the following, we will simply identify a $\{0,1\}$ -valued weight $\lambda $ with its associated set,

(1.10) $$ \begin{align} \lambda = \varpi_I = I = \{i \mid \lambda_i= 1\}. \end{align} $$

The following is well-known, see, for example, [Reference Klimyk and SchmüdgenKS97, Theorem 7.23] for the case of integral weights. Recall here that for an algebra A and an A-module M, a vector $m\in M$ is cyclic if $Am =M$ .

We briefly comment below on why the $V_{\lambda }$ with $\lambda \in P_{{\mathrm {pos.int}}}^{{\mathrm {dom}}}$ are positive integral modules.

Using the notation (1.10), consider the irreducible positive integral modules for $0\leq k \leq N$ . Here is the trivial module, given by the counit, while we can identify as the vector $U_q(\mathfrak {gl}(N,\mathbb {C}))$ -module, defined by

(1.11) $$ \begin{align} \Theta: U_q(\mathfrak{gl}(N,\mathbb{C})) \rightarrow {\mathrm{End}}(\mathbb{C}^N),\qquad \Theta(K_i)e_j = q^{\delta_{ij}}e_j,\quad \Theta(E_i) e_j = \delta_{i,j-1}e_{j-1},\quad \Theta(F_i) e_j = \delta_{ij} e_{j+1}. \end{align} $$

Note that $\Theta $ defines in fact a representation of $U_q(\mathfrak {u}(N))$ , with the standard basis of $\mathbb {C}^N$ as orthonormal basis.

To construct the other modules , we follow [Reference Parshall and WangPW91, Chapter 8]. Consider the quadratic algebra $\wedge _q(\mathbb {C}^N)$ generated by vectors $\{e_i\mid 1\leq i\leq N\}$ with defining relations given by $e_i \wedge _q e_i = 0$ and

$$\begin{align*}e_j \wedge_q e_i = -q e_i \wedge_q e_j,\qquad 1\leq i<j \leq N. \end{align*}$$

In other words, we are dividing out the tensor algebra of $\mathbb {C}^N$ by the image of

$$\begin{align*}\hat{R}+q: \mathbb{C}^N \otimes \mathbb{C}^N \rightarrow \mathbb{C}^N \otimes \mathbb{C}^N. \end{align*}$$

Noting that $\hat {R}$ is an intertwiner for $\Theta ^{*2} := (\Theta \otimes \Theta )\circ \Delta $ , we get that $\wedge _q(\mathbb {C}^N)$ is a $U_q(\mathfrak {gl}(N,\mathbb {C}))$ -module algebra:

(1.12) $$ \begin{align} \rhd: U_q(\mathfrak{gl}(N,\mathbb{C}))\otimes \Lambda_q(\mathbb{C}^N) \rightarrow \Lambda_q(\mathbb{C}^N),\qquad x\rhd (ab) = (x_{(1)}\rhd a)(x_{(2)}\rhd b), \quad x \rhd 1 = \varepsilon(x)1, \end{align} $$

where we use the (sumless) Sweedler notation

$$\begin{align*}\Delta(x) = x_{(1)}\otimes x_{(2)}. \end{align*}$$

Denoting

(1.13)

we obtain that $\wedge _q^k(\mathbb {C}^N)$ is a model for . We can also represent $\wedge _q^k(\mathbb {C}^N)$ directly as

$$\begin{align*}\wedge_q^k(\mathbb{C}^N) = (\mathbb{C}^N)^{\otimes k} \Big/ \sum_{i=1}^{k-1} \mathrm{Im}(\hat{R}_{i,i+1}+q), \end{align*}$$

and the $e_I$ will give a basis for $\wedge _q^k(\mathbb {C}^N)$ . With the $e_I$ orthonormal, we obtain a representation of $U_q(\mathfrak {u}(N))$ .

If we now consider a general $V_{\lambda }$ with $\lambda \in P_{{\mathrm {pos.int}}}^{{\mathrm {dom}}}$ , we see that $V_{\lambda }$ is embedded in some , sending the highest weight vector of $V_{\lambda }$ to the tensor product of the highest weight vectors in the range. Since the above tensor product is positive integral, the same is true of $V_{\lambda }$ .

1.2 The quantum matrix algebra $\mathcal {O}_q(M_N(\mathbb {C}))$ and its localisation $\mathcal {O}_q(GL(N,\mathbb {C}))$

Below, we continue to use the identification (0.4).

Definition 1.6 [Reference Faddeev, Reshetikhin and TakhtajanFRT88]

We define the FRT (= Faddeev-Reshetikhin-Takhtajan) algebra $\mathcal {O}_q(M_N(\mathbb {C}))$ as the universal unital $\mathbb {C}$ -algebra generated by the matrix entries of $X = \sum _{ij} e_{ij} \otimes x_{ij}$ with universal relations

(1.14) $$ \begin{align} \hat{R}_{12} X_{13}X_{23} = X_{13}X_{23} \hat{R}_{12}. \end{align} $$

It is a bialgebra by the coproduct and counit

$$\begin{align*}({\mathrm{id}}\otimes \Delta)X = X_{12}X_{13},\qquad ({\mathrm{id}}\otimes \varepsilon)(X) = I_N. \end{align*}$$

Writing out (1.14), we find the following commutation relations between the $x_{ij}$ ,

(1.15) $$ \begin{align} \left\{\begin{array}{lll} x_{ij} x_{kj} = q x_{kj}x_{ij} & \textrm{for} & i<k,\\ x_{ij} x_{ik} = qx_{ik}x_{ij} &\textrm{for}&j<k,\\ x_{ij} x_{kl} = x_{kl} x_{ij}&\textrm{for}& i< k \textrm{ and } j>l,\\ x_{ij}x_{kl} =x_{kl}x_{ij} + (q-q^{-1})x_{il}x_{kj} & \textrm{for}&i<k\textrm{ and } j<l. \end{array}\right. \end{align} $$

As $\hat {R}$ is an intertwiner for the tensor product of the vector module of $U_q(\mathfrak {gl}(N,\mathbb {C}))$ with itself, it follows that there is a pairing of bialgebras between $\mathcal {O}_q(M_N(\mathbb {C}))$ and $U_q(\mathfrak {gl}(N,\mathbb {C}))$ , given by

(1.16) $$ \begin{align} \tau(x_{ij},x) = \Theta(x)_{ij},\qquad x \in U_q(\mathfrak{gl}(N,\mathbb{C})), \end{align} $$

with $\Theta $ as in (1.11), and with matrix coefficients taken with respect to the standard basis of $\mathbb {C}^N$ .

Stating that this is a pairing simply means that $\tau $ is a bilinear map

$$\begin{align*}\tau: \mathcal{O}_q(M_N(\mathbb{C}))\times U_q(\mathfrak{gl}(N,\mathbb{C})) \rightarrow \mathbb{C} \end{align*}$$

with

$$\begin{align*}\tau(a,xy) = \tau(a_{(1)},x)\tau(a_{(2)},y),\quad \tau(a,1) = \varepsilon(a),\quad \tau(ab,x)= \tau(a,x_{(1)})\tau(b,x_{(2)}),\quad \tau(1,x) = \varepsilon(x). \end{align*}$$

This duality is nondegenerate [Reference Klimyk and SchmüdgenKS97, Chapter 11, Corollary 22],Footnote ³ so $\tau $ gives injective homomorphisms into the respective vector space duals,

$$\begin{align*}U_q(\mathfrak{gl}(N,\mathbb{C}))\rightarrow \mathrm{Lin}(\mathcal{O}_q(M_N(\mathbb{C})),\mathbb{C}),\qquad \mathcal{O}_q(M_N(\mathbb{C})) \rightarrow \mathrm{Lin}(U_q(\mathfrak{gl}(N,\mathbb{C})),\mathbb{C}), \end{align*}$$

where the right hand sides are endowed with the convolution algebra structure.

By the universal property, we also see that the comodule structure

$$\begin{align*}\delta: \mathbb{C}^N \rightarrow \mathbb{C}^N\otimes \mathcal{O}_q(M_N(\mathbb{C})),\qquad e_j \mapsto \sum_i e_i \otimes x_{ij} \end{align*}$$

lifts to a comodule algebra map

(1.17) $$ \begin{align} \wedge \delta: \wedge_q(\mathbb{C}^N) \rightarrow \wedge_q(\mathbb{C}^N)\otimes \mathcal{O}_q(M_N(\mathbb{C})), \end{align} $$

which in turn restricts to comodule structures

$$\begin{align*}\wedge^k\delta: \wedge_q^k(\mathbb{C}^N) \rightarrow \wedge_q^k(\mathbb{C}^N) \otimes \mathcal{O}_q(M_N(\mathbb{C})). \end{align*}$$

It follows that the $U_q(\mathfrak {gl}(N,\mathbb {C}))$ -modules can be ‘integrated’ to $\mathcal {O}_q(M_N(\mathbb {C}))$ -comodules, that is,

Definition 1.7. For we define the quantum minors

$$\begin{align*}x_{IJ} \in \mathcal{O}_q(M_N(\mathbb{C})) \end{align*}$$

as the matrix coefficients of $\wedge ^k\delta $ with respect to the canonical basis, so

$$\begin{align*}\wedge^k\delta(e_J) = \sum_J e_I \otimes x_{IJ}. \end{align*}$$

We then put

and we call the k-th exterior power of X.

For brevity, we write

$$\begin{align*}x_I = x_{I,I}. \end{align*}$$

In particular, we denote

(1.18)

It can be shown that ${\mathrm {Det}}_q(X)$ is central [Reference Parshall and WangPW91, Theorem 4.6.1]. More generally, we have

Definition 1.8. We define $\mathcal {O}_q(GL(N,\mathbb {C}))$ as the localisation

$$\begin{align*}\mathcal{O}_q(GL(N,\mathbb{C})) = \mathcal{O}_q(M_N(\mathbb{C}))[{\mathrm{Det}}_q(X)^{-1}]. \end{align*}$$

As $\mathcal {O}_q(M_N(\mathbb {C}))$ has no zero divisors [Reference Parshall and WangPW91, Theorem 3.5.1], it embeds in its localisation. Moreover, X becomes invertible in $\mathcal {O}_q(GL(N,\mathbb {C}))$ . This implies that $\mathcal {O}_q(GL(N,\mathbb {C}))$ becomes a Hopf algebra.

The pairing (1.16) extends uniquely to a nondegenerate pairing of Hopf algebras

(1.19) $$ \begin{align} \tau: \mathcal{O}_q(GL(N,\mathbb{C})) \otimes U_q(\mathfrak{gl}(N,\mathbb{C})) \rightarrow \mathbb{C}. \end{align} $$

It follows that $\mathcal {O}_q(GL(N,\mathbb {C}))$ is co-semisimple, and that an $U_q(\mathfrak {gl}(N,\mathbb {C}))$ -module lifts to a $\mathcal {O}_q(GL(N,\mathbb {C}))$ -comodule through the pairing (1.19) if and only if it is integral. In particular, we can identify the coalgebras

$$\begin{align*}\mathcal{O}_q(GL(N,\mathbb{C})) \cong \oplus_{\lambda \in P_{{\mathrm{int}}}^{{\mathrm{dom}}}} \mathrm{Lin}({\mathrm{End}}(V_{\lambda}),\mathbb{C}), \end{align*}$$

and hence have a concrete model for the convolution algebra of $\mathcal {O}_q(GL(N,\mathbb {C}))$ ,

(1.20) $$ \begin{align} \mathscr{U}_q(\mathfrak{gl}(N,\mathbb{C})) := \mathrm{Lin}(\mathcal{O}_q(GL(N,\mathbb{C})),\mathbb{C}) \cong \prod_{\lambda \in P_{{\mathrm{int}}}^{{\mathrm{dom}}}} {\mathrm{End}}(V_{\lambda}). \end{align} $$

Note in particular the following corollary.

Similar statements hold for tensor products. For example, any tensor product of finite-dimensional integral $U_q(\mathfrak {gl}(N,\mathbb {C}))$ -modules uniquely extends to a module of

$$\begin{align*}\mathscr{U}_q(\mathfrak{gl}(N,\mathbb{C})) \widehat{\otimes}\mathscr{U}_q(\mathfrak{gl}(N,\mathbb{C})) := \mathrm{Lin}(\mathcal{O}_q(GL(N,\mathbb{C}))\otimes \mathcal{O}_q(GL(N,\mathbb{C})),\mathbb{C}), \end{align*}$$

and we have an identification

(1.21) $$ \begin{align} \mathscr{U}_q(\mathfrak{gl}(N,\mathbb{C})) \widehat{\otimes}\mathscr{U}_q(\mathfrak{gl}(N,\mathbb{C})) \cong \prod_{\lambda,\lambda' \in P_{{\mathrm{int}}}^{{\mathrm{dom}}}} {\mathrm{End}}(V_{\lambda})\otimes {\mathrm{End}}(V_{\lambda'}). \end{align} $$

Going back to $\mathcal {O}_q(M_N(\mathbb {C}))$ , also the following is well-known. Let us give a brief proof.

We can hence denote the dual convolution algebra of $\mathcal {O}_q(M_N(\mathbb {C}))$ as

(1.22) $$ \begin{align} \mathscr{U}_q^{{\mathrm{pos}}}(\mathfrak{gl}(N,\mathbb{C})) := \mathrm{Lin}(\mathcal{O}_q(M_N(\mathbb{C})),\mathbb{C}) \cong \prod_{\lambda \in P_{{\mathrm{pos.int}}}^{{\mathrm{dom}}}} {\mathrm{End}}(V_{\lambda}). \end{align} $$

Since the positive integral representations of $U_q(\mathfrak {gl}(N,\mathbb {C}))$ are stable under tensor products, $\mathscr {U}_q^{{\mathrm {pos}}}(\mathfrak {gl}(N,\mathbb {C}))$ is still endowed with a coproduct

$$\begin{align*}\Delta^{{\mathrm{pos}}}: \mathscr{U}_q^{{\mathrm{pos}}}(\mathfrak{gl}(N,\mathbb{C})) \rightarrow \mathscr{U}_q^{{\mathrm{pos}}}(\mathfrak{gl}(N,\mathbb{C})) \widehat{\otimes} \mathscr{U}_q^{{\mathrm{pos}}}(\mathfrak{gl}(N,\mathbb{C})). \end{align*}$$

1.3 The W $^*$ -category $\mathbf {U}_q(N)$

The Hopf algebra $\mathcal {O}_q(GL(N,\mathbb {C}))$ gives a quantization of $GL(N,\mathbb {C})$ as a complex group variety. To obtain its real form $U(N)$ , we endow $\mathcal {O}_q(GL(N,\mathbb {C}))$ with the following $*$ -structure.

Definition 1.11. The Hopf $*$ -algebra $\mathcal {O}_q(U(N)) = (\mathcal {O}_q(GL(N,\mathbb {C})),*)$ is determined uniquely by

$$\begin{align*}X^* = X^{-1}. \end{align*}$$

One can note that this is well-defined as the composition $S(-)^*$ is well-defined, given that ${S(x_{ij})^* = x_{ji}}$ . This $*$ -structure is compatible with the real form $U_q(\mathfrak {u}(N))$ under the pairing $\tau $ of (1.19), in the sense that

(1.23) $$ \begin{align} \tau(f,x^*) = \overline{\tau(S(f)^*,x)},\qquad \tau(f^*,x) = \overline{\tau(f,S(x)^*)},\qquad f\in \mathcal{O}_q(U(N)),x \in U_q(\mathfrak{u}(N)). \end{align} $$

When considering $\mathcal {O}_q(U(N))$ , we will denote the generating matrix as U rather than X.

Equivalently, we may see this as the full W $^*$ -subcategory of $\mathbf {G}\mathbf {L}_q(N,\mathbb {C})$ (defined above (0.12)) given by couples $(\mathcal {H},U)$ with U unitary. This is a tensor W $^*$ -subcategory, and the tensor product coincides with the one on $\mathcal {O}_q(U(N))$ -representations given by the coproduct,

(1.24) $$ \begin{align} \pi*\pi' := (\pi\otimes \pi')\Delta. \end{align} $$

It follows by the Tannaka–Krein-reconstruction theorem [Reference WoronowiczWor88], together with the fact that $\mathcal {O}_q(U(N))$ is nondegenerately paired with $U_q(\mathfrak {u}(N))$ , that $\mathcal {O}_q(U(N))$ is C $^*$ -faithful. This implies that the Hopf $*$ -algebra

$$\begin{align*}U_q(N) := (\mathcal{O}_q(U(N)),\Delta) \end{align*}$$

defines a compact quantum group [Reference WoronowiczWor87, Reference Levendorskii and SoibelmanLS91, Reference Dijkhuizen and KoornwinderDK94]: there exists on $\mathcal {O}_q(U(N))$ a (necessarily faithful, necessarily unique) invariant state

$$\begin{align*}\int_{U_q(N)}: \mathcal{O}_q(U(N)) \rightarrow \mathbb{C}, \end{align*}$$

called the Haar integral. The above conditions mean that, for all $f\in \mathcal {O}_q(U(N))$ ,

$$\begin{align*}\int_{U_q(N)} f^*f \geq 0,\qquad \int_{U_q(N)}1 = 1,\qquad \left(\int_{U_q(N)}\otimes {\mathrm{id}}\right)\Delta(f) = \int_{U_q(N)}f = \left({\mathrm{id}}\otimes \int_{U_q(N)}\right)\Delta(f). \end{align*}$$

We can then consider the universal C $^*$ -envelope $C_q(U(N))$ of $\mathcal {O}_q(U(N))$ . On the other hand, one can also consider the associated reduced C $^*$ -envelope, given by completing $\mathcal {O}_q(U(N))$ in its natural representation $\lambda _{{\mathrm {st}}}$ on the GNS-space $L^2_q(U(N))$ with respect to $\int _{U_q(N)}$ .

Let us look at the finer structure of $\mathbf {U}_q(N)$ .

First, recall that the Hopf $*$ -algebra $\mathcal {O}_q(SU(2))$ is the universal unital Hopf $*$ -algebra generated by $a,c$ with the requirement that $U=\begin {pmatrix} a & -qc^* \\ c& a^*\end {pmatrix}$ is a unitary corepresentation matrix. Any irreducible representation of $\mathcal {O}_q(SU(2))$ is equivalent to one of the following list of mutually inequivalent irreducible representations: the family of $*$ -characters $\chi _{\theta }$ and the family of representations $\mathbf {s}*\chi _{\theta }$ , where $\chi _{\theta }$ is the $*$ -character

$$\begin{align*}\chi_{\theta}(U) = \begin{pmatrix} e^{2\pi i\theta} & 0 \\0 & e^{-2\pi i\theta}\end{pmatrix},\qquad \theta \in \mathbb{R}/\mathbb{Z}, \end{align*}$$

and where $\mathbf {s}$ is the representation on $l^2(\mathbb {N})$ given by

$$\begin{align*}\mathbf{s}(a)e_n = (1-q^{2n})^{1/2}e_{n-1},\qquad \mathbf{s}(c)e_n = q^{n} e_{n}. \end{align*}$$

Now $\mathcal {O}_q(SU(2))$ is dual to $U_q(\mathfrak {su}(2))$ , defined as the $*$ -subalgebra of $U_q(\mathfrak {u}(2))$ generated by $E,F,\hat {K}^{\pm 1}$ . Since $U_q(\mathfrak {su}(2))$ has for each $1\leq i < N$ a copy in $U_q(\mathfrak {u}(N))$ , generated by $E_i, F_i$ and $\hat {K}_i$ , we obtain a corresponding restriction map

$$\begin{align*}\mathrm{res}_i: \mathcal{O}_q(U(N)) \rightarrow \mathcal{O}_q(SU(2)),\qquad 1\leq i <N. \end{align*}$$

We denote $\mathbf {s}_i = \mathbf {s} \circ \mathrm {res}_i$ . On the other hand, we also have for $\theta \in (\mathbb {R}/\mathbb {Z})^N$ the $*$ -character

$$\begin{align*}\chi_{\theta}: \mathcal{O}_q(U(N)) \rightarrow \mathbb{C},\quad U_{kl} \mapsto \delta_{kl} e^{2\pi i \theta_k}. \end{align*}$$

In particular, $\mathbf {s}_w$ only depends on the reduced decomposition of w up to unitary equivalence.

We also have the following theorem which is a direct consequence of [Reference Reshetikhin and YakimovRY01, Theorem 5.2]. We call a representation of $\mathcal {O}_q(U(N))$ normal if it lifts to a normal representation of the von Neumann algebra

$$\begin{align*}L^{\infty}_q(U(N)) := \lambda_{{\mathrm{st}}}(\mathcal{O}_q(U(N)))". \end{align*}$$

Theorem 1.15. Let $M=\binom {N}{2}$ , and let $s_{i_1}\ldots s_{i_{M}}$ be a fixed reduced expression for the longest word $w_0$ in . Let $\mathbf {s}_{w_0} = \mathbf {s}_{i_1}*\ldots *\mathbf {s}_{i_{M}}$ . Then

$$\begin{align*}\lambda_0 = \int_{(\mathbb{R}/\mathbb{Z})^N} \mathbf{s}_{w_0}*\chi_{\theta}{\mathrm{d}\!} \theta \end{align*}$$

is a faithful normal representation of $\mathcal {O}_q(U(N))$ . Moreover,

$$\begin{align*}\lambda_{{\mathrm{st}}} \cong \int_{(\mathbb{R}/\mathbb{Z})^N}\mathbf{s}_{w_0}*\mathbf{s}_{w_0}*\chi_{\theta}{\mathrm{d}\!} \theta \cong \oplus_{\mathbb{N}} \lambda_0. \end{align*}$$

1.4 The W $^*$ -category $\mathbf {T}_q(N)$

Definition 1.16. We define $\mathcal {O}_q(B(N))$ as the quotient algebra of $\mathcal {O}_q(GL(N,\mathbb {C}))$ by putting $x_{ij} =0$ for $i>j$ . We denote by

$$\begin{align*}\pi_B: \mathcal{O}_q(GL(N,\mathbb{C})) \rightarrow \mathcal{O}_q(B(N)) \end{align*}$$

the quotient map. We write the resulting images of $x_{ij}$ as $t_{ij}$ (when not zero).

In the following, we put $t_{ii} = t_i$ . Then $\mathcal {O}_q(B(N))$ is the universal algebra generated by invertible elements $t_{i}$ for $1\leq i\leq N$ and elements $t_{ij}$ with $1\leq i<j\leq N$ such that the following relations hold:

$$\begin{align*}t_i t_{kl} = q^{\delta_{ik}-\delta_{il}}t_{kl}t_i,\qquad k\leq l, \end{align*}$$

(1.25) $$ \begin{align} \left\{\begin{array}{lll} t_{ij} t_{kj} = q t_{kj}t_{ij} & \textrm{for} & i<k<j,\\ t_{ij} t_{ik} = qt_{ik}t_{ij} &\textrm{for}&i<j<k,\\ t_{ij} t_{kl} = t_{kl} t_{ij}&\textrm{for}& i< k \textrm{ and } j>l,\\ t_{ij}t_{kl} =t_{kl}t_{ij} + (q-q^{-1})t_{il}t_{kj} & \textrm{for}&i<k\textrm{ and } j<l. \end{array}\right. \end{align} $$

The pairing (1.19) restricts to a nondegenerate pairing of Hopf algebras

(1.26) $$ \begin{align} \tau: \mathcal{O}_q(B(N))\times U_q(\mathfrak{b}(N))\rightarrow \mathbb{C}. \end{align} $$

On the other hand, we have the following result which is for example contained in [Reference Klimyk and SchmüdgenKS97, Theorem 8.33].

Lemma 1.17. There is an isomorphism of Hopf algebras

(1.27) $$ \begin{align} \kappa: U_q(\mathfrak{b}^-(N))^{\mathrm{cop}}\rightarrow \mathcal{O}_q(B(N)),\qquad K_i \mapsto t_{i}^{-1},\quad F_i \mapsto (q^{-1}-q)^{-1} t_{i,i+1}t_{i+1}^{-1}. \end{align} $$

We denote the inverse as

(1.28) $$ \begin{align} \iota: \mathcal{O}_q(B(N))\rightarrow U_q(\mathfrak{b}^-(N))^{\mathrm{cop}}, \end{align} $$

so in particular

$$\begin{align*}\iota(t_{i}) = K_i^{-1},\qquad \iota(t_{i,i+1}) = (q^{-1}-q)F_iK_{i+1}^{-1}. \end{align*}$$

The inverse map $\iota $ has a more conceptual description as follows. With $\Sigma $ the flip map, consider the Yang-Baxter matrix

$$\begin{align*}R = \Sigma \circ \hat{R} = \sum_{i,j} q^{-\delta_{ij}}e_{ii}\otimes e_{jj} + (q^{-1} -q)\sum_{i<j} e_{ij} \otimes e_{ji}. \end{align*}$$

From R one obtains the skew bicharacter $\mathbf {r}: \mathcal {O}_q(M_N(\mathbb {C}))\otimes \mathcal {O}_q(M_N(\mathbb {C})) \rightarrow \mathbb {C}$ , uniquely determined by

(1.29) $$ \begin{align} ({\mathrm{id}}\otimes {\mathrm{id}}\otimes \mathbf{r})(X_{13}X_{24}) = R. \end{align} $$

This follows from the relations of $\mathcal {O}_q(M_n(\mathbb {C}))$ , together with the defining properties of a skew bicharacter:

(1.30) $$ \begin{align} \mathbf{r}(ab,c) = \mathbf{r}(a,c_{(1)})\mathbf{r}(b,c_{(2)}),\quad \mathbf{r}(a,bc) = \mathbf{r}(a_{(2)},b)\mathbf{r}(a_{(1)},c),\qquad \mathbf{r}(a,1) = \varepsilon(a),\quad \mathbf{r}(1,b) = \varepsilon(b). \end{align} $$

This skew bicharacter automatically extends to $\mathcal {O}_q(GL(N,\mathbb {C}))$ .

Through (1.21), we can view $\mathbf {r}$ as being induced by an element

$$\begin{align*}\mathscr{R} \in \mathscr{U}_q(\mathfrak{gl}(N,\mathbb{C})) \widehat{\otimes} \mathscr{U}_q(\mathfrak{gl}(N,\mathbb{C})) = \prod_{\lambda,\kappa\in P_{{\mathrm{int}}}^{{\mathrm{dom}}}} {\mathrm{End}}(V_{\lambda}) \otimes {\mathrm{End}}(V_{\kappa}), \end{align*}$$

the universal R-matrix, which then satisfies

(1.31) $$ \begin{align} (\Delta\otimes {\mathrm{id}})\mathscr{R} = \mathscr{R}_{13}\mathscr{R}_{23},\quad ({\mathrm{id}}\otimes \Delta)\mathscr{R} = \mathscr{R}_{13}\mathscr{R}_{12}, \end{align} $$

(1.32) $$ \begin{align} \mathscr{R} \Delta(-) = \Delta^{\mathrm{op}}(-)\mathscr{R}. \end{align} $$

These identities can be interpreted within the dual vector space of $\otimes _{i=1}^n \mathcal {O}_q(GL(N,\mathbb {C}))$ for appropriate n. We further have that

(1.33) $$ \begin{align} \mathscr{R}(\xi\otimes \eta) = q^{-(\mathrm{wt}(\xi),\mathrm{wt}(\eta))}\xi\otimes \eta \end{align} $$

for a highest weight vector $\xi $ and lowest weight vector $\eta $ . Together with (1.32) this uniquely determines $\mathscr {R}$ .

We will need the following information on $\mathscr {R}$ . For $\alpha \in Q^+ \subseteq \mathbb {R}^N$ , consider the finite-dimensional spaces

(1.34) $$ \begin{align} U_q(\mathfrak{n}(N))_{\alpha} = \{X \in U_q(\mathfrak{n}(N))\mid K_i XK_i^{-1} = q^{\alpha_i}X\}, \end{align} $$

(1.35) $$ \begin{align} U_q(\mathfrak{n}^-(N))_{-\alpha} = \{X \in U_q(\mathfrak{n}^-(N))\mid K_i XK_i^{-1} = q^{-\alpha_i}X\}. \end{align} $$

Then one can factorize

(1.36) $$ \begin{align} \mathscr{R} = \widetilde{\mathscr{R}} \mathscr{Q}, \end{align} $$

where for general weight vectors $\xi ,\eta $

(1.37) $$ \begin{align} \mathscr{Q}(\xi\otimes \eta) = q^{-(\mathrm{wt}(\xi),\mathrm{wt}(\eta))}\xi\otimes \eta \end{align} $$

and

(1.38) $$ \begin{align} \widetilde{\mathscr{R}} = \sum_{\alpha \in Q^+} \widetilde{\mathscr{R}}_{\alpha},\qquad \widetilde{\mathscr{R}}_{\alpha} \in U_q(\mathfrak{n}(N))_{\alpha} \otimes U_q(\mathfrak{n}^-(N))_{-\alpha}. \end{align} $$

The latter sum converges weakly as a functional on $\mathcal {O}_q(GL(N,\mathbb {C}))^{\otimes 2}$ : for any $f,g\in \mathcal {O}_q(GL(N,\mathbb {C}))$ , the sum

$$\begin{align*}\sum_{\alpha} \tau(f\otimes g,\widetilde{\mathscr{R}}_{\alpha}) \end{align*}$$

contains only finitely many nonzero terms. We have for example, for $\mathbf {a}_r = e_r - e_{r+1}$ ,

(1.39) $$ \begin{align} \widetilde{\mathscr{R}}_{0} = 1\otimes 1,\qquad \widetilde{\mathscr{R}}_{\mathbf{a}_r} = (q^{-1}-q) E_r \otimes F_r. \end{align} $$

It can now be checked that the map

(1.40) $$ \begin{align} \mathcal{O}_q(GL(N,\mathbb{C})) \rightarrow \mathscr{U}_q(\mathfrak{gl}(N,\mathbb{C})),\qquad f\mapsto (\tau(f,-)\otimes {\mathrm{id}})\mathscr{R} \end{align} $$

factors through a map $\mathcal {O}_q(B(N)) \rightarrow U_q(\mathfrak {b}^-(N))$ , which is precisely the map $\iota $ :

(1.41) $$ \begin{align} \iota: \mathcal{O}_q(B(N)) \rightarrow U_q(\mathfrak{b}^-(N))^{\mathrm{cop}},\quad f\mapsto (\tau(f,-)\otimes {\mathrm{id}})\mathscr{R}. \end{align} $$

The above gives a quantization of the group of invertible upper triangular matrices as a complex algebraic group. We now want to quantize the underlying real space.

If A is a complex algebra, we denote by $A^{*}$ its formal antilinear, anti-isomorphic copy: it is the unique structure of complex algebra on the set of formal symbols $\{a^*\mid a\in A\}$ such that

$$\begin{align*}A \rightarrow A^*,\qquad a \mapsto a^* \end{align*}$$

is an antilinear, antimultiplicative map.

Definition 1.18. The unital $*$ -algebra $\mathcal {O}_q^{\mathbb {R}}(B(N))$ is generated by $\mathcal {O}_q(B(N))$ and $\mathcal {O}_q(B(N))^*$ , with relations

(1.42) $$ \begin{align} T_{23}\hat{R}_{12}T^*_{23} = T^*_{13}\hat{R}_{12}T_{13}. \end{align} $$

We define $\mathcal {O}_q(T(N))$ as the quotient of $\mathcal {O}_q^{\mathbb {R}}(B(N))$ by putting $t_{i}^* = t_i$ for all i.

We have that $\mathcal {O}_q(T(N))$ is the universal unital $*$ -algebra generated by invertible self-adjoint elements $t_i=t_{ii}$ for $1\leq i \leq N$ and elements $t_{ij}$ with $1\leq i<j\leq N$ such that (1.25) holds, as well as

(1.43) $$ \begin{align} \left\{\begin{array}{lllll} t_{kj}t_{li}^{*} - t_{li}^{*}t_{kj}&=&0,&& i\neq j, k\neq l,\\ t_{kj}t_{lj}^{*} - qt_{lj}^{*}t_{kj}&=& -(1-q^2)\sum_{i<j} t_{ki}t_{li}^{*} ,&&k\neq l,\\ qt_{kj}t_{ki}^{*}- t_{ki}^{*}t_{kj} &=& (1-q^2)\sum_{k<l\leq\mathrm{min}\{i,j\}} t_{li}^{*}t_{lj},&& i\neq j, \\ t_{kj}t_{kj}^{*}- t_{kj}^{*}t_{kj} &=& (1-q^2)\left(\sum_{k<l\leq j} t_{lj}^{*}t_{lj} - \sum_{k\leq l<j} t_{kl}t_{kl}^{*}\right). && \end{array}\right. \end{align} $$

Then $\mathcal {O}_q(T(N))$ is a Hopf $*$ -algebra for the unique coproduct such that

(1.44) $$ \begin{align} ({\mathrm{id}}\otimes \Delta)(T) = T_{12}T_{13}. \end{align} $$

The Hopf algebra isomorphism (1.27) extends uniquely to a Hopf $*$ -algebra isomorphism

(1.45) $$ \begin{align} \kappa: U_q(\mathfrak{u}(N))^{\mathrm{cop}} \rightarrow \mathcal{O}_q(T(N)). \end{align} $$

whose inverse we again denote by

(1.46) $$ \begin{align} \iota: \mathcal{O}_q(T(N)) \rightarrow U_q(\mathfrak{u}(N))^{\mathrm{cop}}. \end{align} $$

Consider now the W $^*$ -category $\mathbf {T}_q(N)$ of representations $\lambda $ of $\mathcal {O}_q(T(N))$ for which the $\lambda (t_{i})$ are positive operators. Equivalently, we can view $\mathbf {T}_q(N)$ as the monoidal W $^*$ -subcategory of $\mathbf {G}\mathbf {L}_q(N,\mathbb {C})$ consisting of all $(\mathcal {H},T)$ for which T is upper triangular, with positive, invertible operators on the diagonal. Again, the monoidal structure is the same as the one induced by the coproduct of $\mathcal {O}_q(T(N))$ .

By the $*$ -isomorphism (1.46), a finite-dimensional admissible $U_q(\mathfrak {u}(N))$ -representation becomes a $\mathcal {O}_q(T(N))$ -representation in $\mathbf {T}_q(N)$ , and we can transfer the unitary irreducible admissible representations $V_{\lambda }$ for $\lambda \in P_{{\mathrm {w.int}}}^{{\mathrm {dom}}}$ to representations of $\mathcal {O}_q(T(N))$ . We then have the following.

1.5 The W $^*$ -category $\mathbf {G}\mathbf {L}_q(N,\mathbb {C})$

Definition 1.20. We define the FRT double $\mathcal {O}_{q}^{(2)}(M_N(\mathbb {C}))$ as the universal $\mathbb {C}$ -algebra generated by the matrix entries of $X = \sum _{ij} e_{ij} \otimes x_{ij}$ and $Y = \sum _{ij} e_{ij} \otimes y_{ij}$ with universal relations

(1.47) $$ \begin{align} \hat{R}_{12}X_{13}X_{23} = X_{13}X_{23}\hat{R}_{12},\qquad \hat{R}_{12}Y_{23}Y_{13} = Y_ {23}Y_{13}\hat{R}_{12},\qquad X_{23}\hat{R}_{12}Y_{23} = Y_{13}\hat{R}_{12}X_{13}. \end{align} $$

We define $\mathcal {O}_{q}^{\mathbb {R}}(M_N(\mathbb {C}))$ as $\mathcal {O}_{q}^{(2)}(M_N(\mathbb {C}))$ endowed with the $*$ -structure

$$\begin{align*}X^{*} = Y,\qquad Y^{*} = X. \end{align*}$$

The commutation relations can be written out as follows: apart from the relations in (1.15), we also have

(1.48) $$ \begin{align} \left\{\begin{array}{lllll} x_{kj} x_{li}^{*} - x_{li}^{*}x_{kj}&=&0,&& i\neq j, k\neq l,\\ x_{kj}x_{lj}^{*} - qx_{lj}^{*}x_{kj}&=& -(1-q^2)\sum_{i<j} x_{ki}x_{li}^{*} ,&&k\neq l,\\ qx_{kj}x_{ki}^{*}- x_{ki}^{*}x_{kj} &=& (1-q^2)\sum_{k<l} x_{li}^{*}x_{lj},&& i\neq j, \\ x_{kj}x_{kj}^{*}- x_{kj}^{*}x_{kj} &=& (1-q^2)\left(\sum_{k<l} x_{lj}^{*}x_{lj} - \sum_{l<j} x_{kl}x_{kl}^{*}\right). && \end{array}\right. \end{align} $$

The element ${\mathrm {Det}}_q(X)$ is central in $\mathcal {O}_{q}^{\mathbb {R}}(M_N(\mathbb {C}))$ [Reference De Commer and FloréDCF19, Lemma 3.5], so with ${\mathrm {Det}}_q(X^{*}) := {\mathrm {Det}}_q(X)^{*}$ , we can define the Hopf $*$ -algebra $\mathcal {O}_q^{\mathbb {R}}(GL(N,\mathbb {C}))$ as the localisation

$$\begin{align*}\mathcal{O}_{q}^{\mathbb{R}}(GL(N,\mathbb{C})) = \mathcal{O}_{q}^{\mathbb{R}}(M_N(\mathbb{C}))[{\mathrm{Det}}_q(X)^{-1},{\mathrm{Det}}_q(X^{*})^{-1}]. \end{align*}$$

The W $^*$ -category $\mathbf {G}\mathbf {L}_q(N,\mathbb {C})$ , defined below (0.11), can be identified with the representation category of $\mathcal {O}_q^{\mathbb {R}}(GL(N,\mathbb {C}))$ : we only need to observe that since ${\mathrm {Det}}_q(X)$ arises as a direct summand of $X^{\otimes n}$ , any representation of $\mathcal {O}_q^{\mathbb {R}}(M_N(\mathbb {C}))$ for which the image of X is invertible, extends to a $\mathcal {O}_q^{\mathbb {R}}(\mathrm {GL}(N,\mathbb {C}))$ -representation.

We have natural quotient maps of Hopf $*$ -algebras

$$\begin{align*}\pi_U: \mathcal{O}_q^{\mathbb{R}}(GL(N,\mathbb{C})) \rightarrow \mathcal{O}_q(U(N)),\qquad X \mapsto U, \end{align*}$$

$$\begin{align*}\pi_T: \mathcal{O}_q^{\mathbb{R}}(GL(N,\mathbb{C}))\rightarrow \mathcal{O}_q(T(N)),\quad X\mapsto T, \end{align*}$$

leading to W $^*$ -functors

The following is a categorical analogue of the Gauss decomposition.

Theorem 1.21. For any irreducible representation $\lambda $ in $\mathbf {G}\mathbf {L}_q(N,\mathbb {C})$ , there exist unique (up to unitary equivalence) irreducible representations $\lambda _U,\lambda _U'$ in $\mathbf {U}_q(N)$ and unique (up to unitary equivalence) irreducible representations $\lambda _T,\lambda _T'$ in $\mathbf {T}_q(N)$ such that

More generally, [Reference De Commer and FloréDCF19, Proposition 4.18] implies that any representation of $\mathcal {O}_q^{\mathbb {R}}(GL(N,\mathbb {C}))$ is weakly contained in a representation of the form for $\lambda \in \mathbf {U}_q(N)$ and $\lambda '\in \mathbf {T}_q(N)$ .

2.1 Quantum hermitian matrices

Consider $\mathcal {O}_q(H(N))$ as defined via (0.5). Writing $\tau = q^{-1}-q$ , these relations can be written out entrywise as

(2.1) $$ \begin{align} q^{-\delta_{ik} -\delta_{jk}} & z_{ij} z_{kl} + \delta_{k<i}\tau q^{-\delta_{ij}} z_{kj}z_{il} +\delta_{jk} \tau q^{-\delta_{ij}}\sum_{p<j} z_{ip} z_{pl} + \delta_{ij} \delta_{k<i}\tau^2 \sum_{p<i} z_{kp}z_{pl} \nonumber\\ & = q^{-\delta_{il}-\delta_{jl}} z_{kl} z_{ij} + \delta_{l<j}\tau q^{-\delta_{ij}} z_{kj}z_{il} + \delta_{il}\tau q^{-\delta_{ij}} \sum_{p<i} z_{kp}z_{pj} +\delta_{ij}\delta_{l<j} \tau \sum_{p<j} z_{kp}z_{pl}. \end{align} $$

In [Reference MudrovMud02, Theorem 2.2], using just the above relations, Mudrov was able to classify all characters of the reflection equation algebra. From this, we can easily obtain also a classification of all $*$ -characters Footnote ⁴

$$\begin{align*}\chi: \mathcal{O}_q(H(N)) \rightarrow \mathbb{C}. \end{align*}$$

Theorem 2.1. Each $*$ -character $\chi $ of $\mathcal {O}_q(H(N))$ is of the form $\chi = \chi _{k,l,a,c,y}$ where

• ○ $k,l\in \mathbb {Z}_{\geq 0}$ with $k+l \leq N-l$ ,

• ○ $a>0$ and $c\in \mathbb {R}^{\times }$ ,

• ○ $y = (y_{0},y_{1},\ldots ,y_{l-1})$ with $y_i\in \mathbb {C}$ and $|y_i| = 1$ ,

such that

$$\begin{align*}\chi(Z) = c\left(a \sum_{i = k+l+1}^{N} e_{ii} - a^{-1} \sum_{i=N-l+1}^N e_{ii} + \sum_{i=0}^{l-1} (y_{i} e_{k+i+1,N-i} + \overline{y_{i}} e_{N-i,k+i+1})\right). \end{align*}$$

For example, for $N= 4$ we have the following types of solutions, where empty entries signify $0$ :

$$\begin{align*}\begin{pmatrix} &&& \\ &&& \\ &&& \\ &&& \end{pmatrix}\,,\quad \begin{pmatrix} &&& \\ &&& \\ &&& \\ &&& c \end{pmatrix}\,,\quad \begin{pmatrix} &&& \\ &&& \\ && c& \\ &&& c \end{pmatrix}\,, \quad \begin{pmatrix} &&& \\ & c&& \\ && c& \\ &&& c \end{pmatrix}\,,\quad \begin{pmatrix} c &&& \\ & c&&\\ && c& \\ &&& c \end{pmatrix}\,, \end{align*}$$

$$\begin{align*}c \begin{pmatrix} &&& \\ &&& \\ &&& y_0 \\ && \overline{y_0} & a-a^{-1}\end{pmatrix}\,,\quad c\begin{pmatrix} &&& \\ &&& y_0 \\ && a& \\ &\overline{y_0} && a-a^{-1}\end{pmatrix}\,,\quad c\begin{pmatrix} &&& y_0 \\ & a && \\ && a & \\ \overline{y_0} &&& a-a^{-1}\end{pmatrix}\,,\end{align*}$$

$$\begin{align*}c\begin{pmatrix} &&& y_0 \\ &&y_1 & \\ &\overline{y_1} & a-a^{-1} & \\ \overline{y_0} &&&a-a^{-1}\end{pmatrix}\,. \end{align*}$$

2.2 Classification of irreducible representations for $N=2$

For $\mathcal {O}_q(H(2))$ , one can work directly from the defining relations to obtain a classification of the irreducible representations.

Fix in the following $N=2$ , and write

$$\begin{align*}Z = \begin{pmatrix} z & w \\ v & u\end{pmatrix} \in M_2(\mathcal{O}_q(H(2))). \end{align*}$$

Define the quantum trace T and quantum determinant D to be

$$\begin{align*}T = {\mathrm{Tr}}_q(Z) = qz + q^{-1}u,\qquad D = {\mathrm{Det}}_q(Z) = uz -q^{-2}vw. \end{align*}$$

We find by a tedious but direct comparison with (2.1) that the universal relations of $\mathcal {O}_q(H(2))$ are centrality and self-adjointness of $T,D$ together with the relations

$$\begin{align*}z^* = z,\quad w^* = v,\quad zw = q^2wz,\qquad vz = q^2zv \end{align*}$$

and

$$\begin{align*}q^{-2}vw = -D+qTz - q^{2}z^2,\qquad q^{-2}wv = -D+q^{-1}Tz -q^{-2}z^2. \end{align*}$$

For A a unital commutative $*$ -algebra, write ${\mathrm {Spec}}_*(A)$ for the space of unital $*$ -characters. For A a general $*$ -algebra, write $\mathscr {Z}(A)$ for the centre of A.

Exploiting the fact that z either commutes or $q^{\pm 2}$ -commutes with all generators, one sees that $\pi (z)$ must be diagonalizable for any irreducible representation $\pi $ , and that moreover $\pi (v)$ must vanish on some eigenvector of $\pi (z)$ if $\pi (z)$ is not identically zero. The reader will then have no difficulty in providing the details for the following proposition.

Proposition 2.2. We have $\mathscr {Z}(\mathcal {O}_q(H(2))) \cong \mathbb {C}[D,T]$ . Identifying ${\mathrm {Spec}}_*(\mathbb {C}[D,T])\cong \mathbb {R}^2/{\mathrm {Sym}}(2)$ via

$$\begin{align*}\chi \mapsto \{\textrm{roots of }\lambda^2 -\chi(T)\lambda + \chi(D)\}, \end{align*}$$

we have that there exists a (nonzero) irreducible representation of $\mathcal {O}_q(H(2))$ fibering over $\lambda \in \mathbb {R}^2/{\mathrm {Sym}}(2)$ if and only if

$$\begin{align*}\lambda \in \Lambda_q = \Lambda_- \cup \Lambda_0 \cup \Lambda_+ \end{align*}$$

with

$$ \begin{align*} \Lambda_0 &= \{\{0,\lambda\}\mid \lambda \in \mathbb{R}\},\\ \Lambda_+ &= \{\{cq^{n+1},cq^{-n-1}\}\mid c\in \mathbb{R}^{\times}, n\in \mathbb{Z}_{\geq0}\}, \\ \Lambda_- &= \{\{\lambda_+,\lambda_-\}\mid \lambda_+ \in \mathbb{R}_{>0},\lambda_- \in \mathbb{R}_{<0}\}. \end{align*} $$

More precisely:

1. (1) There is a unique irreducible representation fibering over $\{cq^{n+1},cq^{-n-1}\}\in \Lambda _+$ , namely

   $$\begin{align*}\mathcal{S}_{c^2,n}\cong \mathbb{C}^{n+1},\qquad ze_k= cq^{-n+2k}e_k,\quad ve_0 = 0. \end{align*}$$

   We say that it has rank $2$ and shape $\begin {pmatrix} \mathrm {sgn}(c) & 0 \\ 0 & \mathrm {sgn}(c) \end {pmatrix}$ .

2. (2) For $\{0,\lambda \} \in \Lambda _0$ we have the $*$ -character $\chi (z) = \chi (v) = \chi (w) =0$ . When $\lambda =0$ this is the only irreducible representation, and we say that it has rank $0$ . If $\lambda \neq 0$ the only other irreducible representation is given by

   $$\begin{align*}\mathcal{S}_{0,\lambda} \cong l^2(\mathbb{N}),\quad ze_n = q^{2n+1}\lambda e_n,\quad ve_0 = 0, \end{align*}$$
   and we say that it has rank $1$ and shape $\begin {pmatrix} 1 & 0 \\ 0 & 0 \end {pmatrix}$ .

3. (3) For $\{ca,-ca^{-1}\} \in \Lambda _-$ with $a,c>0$ , we have as complete list of nonequivalent irreducible representations the following.

   1. (a) The $*$ -characters $\chi _{\theta }$ for $\theta \in [0,1)$ , with $\chi _{\theta }(z)= 0$ and $\chi _{\theta }(v) = qce^{2\pi i\theta }$ . We say that they have rank $2$ and shape $\begin {pmatrix} 0 & e^{-2\pi i\theta } \\ e^{2\pi i\theta } & 0 \end {pmatrix}$ .

   2. (b) The representations

      $$\begin{align*}\mathcal{S}^{\pm}_{-c^2,a} \cong l^2(\mathbb{N}),\qquad ze_n = \pm ca^{\pm 1} q^{2n+1}e_n,\qquad ve_0 =0. \end{align*}$$

      We say that they have rank $2$ and shape $\begin {pmatrix} \pm 1 & 0 \\ 0 & \mp 1\end {pmatrix}$ .

A general definition of the rank of a representation will be given in Section 2.7. The terminology ‘shape’ will be further motivated and generalized in [Reference De Commer and MooreDCMo25].

2.3 Covariantisation

For the following, see, for example, [Reference MajidMaj95], [Reference Klimyk and SchmüdgenKS97, Chapter 8-10] or [Reference Donin, Kulish and MudrovDKM03].

Recall the skew bicharacter $\mathbf {r}$ from (1.29). This defines a coquasitriangular structure on $\mathcal {O}_q(GL(N,\mathbb {C}))$ , in the sense that we also have the ‘braided commutativity’ property

(2.2) $$ \begin{align} \mathbf{r}(a_{(1)},b_{(1)})a_{(2)}b_{(2)} = b_{(1)}a_{(1)} \mathbf{r}(a_{(2)},b_{(2)}). \end{align} $$

Since R is invertible, the functional $\mathbf {r}: A\otimes A \rightarrow \mathbb {C}$ has a convolution inverse $\mathbf {r}^{-1}$ , where we endow $A\otimes A$ with the tensor product coalgebra structure. Then (2.2) can alternatively be written as

(2.3) $$ \begin{align} \mathbf{r}^{-1}(b_{(1)},a_{(1)})a_{(2)}b_{(2)} = b_{(1)}a_{(1)} \mathbf{r}^{-1}(b_{(2)},a_{(2)}), \end{align} $$

or as

(2.4) $$ \begin{align} ab = \mathbf{r}^{-1}(a_{(1)},b_{(1)})b_{(2)}a_{(2)} \mathbf{r}(a_{(3)},b_{(3)}) = \mathbf{r}(b_{(1)},a_{(1)})b_{(2)}a_{(2)} \mathbf{r}^{-1}(b_{(3)},a_{(3)}). \end{align} $$

However, $\mathbf {r}$ also has a convolution inverse $\mathbf {r}'$ when considered as a functional on $(A,\Delta )\otimes (A,\Delta ^{\mathrm {op}})$ . Then $\mathbf {r}'$ is uniquely determined by

$$\begin{align*}\mathbf{r}'(ab,c) = \mathbf{r}'(a,c_{(2)})\mathbf{r}'(b,c_{(1)}),\quad\!\! \mathbf{r}'(a,bc) = \mathbf{r}'(a_{(1)},b)\mathbf{r}'(a_{(2)},c),\quad\!\! \mathbf{r}'(a,1) = \varepsilon(a),\quad\!\! \mathbf{r}'(1,b) = \varepsilon(b) \end{align*}$$

and, with t the ordinary transpose of a matrix,

$$\begin{align*}({\mathrm{id}}\otimes {\mathrm{id}}\otimes \mathbf{r}')(X_{13}X_{24}) = R' := ((R^{{\mathrm{id}}\otimes t})^{-1})^{{\mathrm{id}}\otimes t} = \sum_{i,j} q^{\delta_{ij}}e_{ii}\otimes e_{jj} + (q-q^{-1})\sum_{i<j} q^{2(j-i)} e_{ij}\otimes e_{ji}. \end{align*}$$

These properties of $\mathbf {r}$ allow to endow $\mathcal {O}_q(M_N(\mathbb {C}))$ with the new product

(2.5) $$ \begin{align} f*g = \mathbf{r}(f_{(1)},g_{(2)})f_{(2)}g_{(3)} \mathbf{r}'(f_{(3)},g_{(1)}), \end{align} $$

which can be checked to be associative and unital (with the same unit). Inverting (2.5), we obtain that the original product can be obtained from the deformed product via

(2.6) $$ \begin{align} fg = \mathbf{r}^{-1}(f_{(1)},g_{(1)})f_{(2)}*g_{(3)} \mathbf{r}(f_{(3)},g_{(2)}). \end{align} $$

One has the following result, see, for example, [Reference MajidMaj95, Section 7.4] or [Reference Klimyk and SchmüdgenKS97, Example 10.18].

Theorem 2.4. There exists a (unique) isomorphism of algebras

(2.7) $$ \begin{align} \Phi: \mathcal{O}_q(H(N)) \rightarrow (\mathcal{O}_q(M_N(\mathbb{C})),*),\qquad Z \mapsto X. \end{align} $$

The $*$ -structure of $\mathcal {O}_q(H(N))$ can be transported along $\Phi $ to a $*$ -structure on $(\mathcal {O}_q(M_N(\mathbb {C})),*)$ , which we denote by $f \mapsto f^{\#}$ . Concretely, we have

(2.8) $$ \begin{align} x_{ij}^{\#} = x_{ji}. \end{align} $$

The above isomorphism allows to define quantum minors inside $\mathcal {O}_q(H(N))$ .

Definition 2.5. For , we define the quantum minors

(2.9) $$ \begin{align} z_{IJ} = \Phi^{-1}(x_{IJ}). \end{align} $$

More generally, if V is a $\mathcal {O}_q(M_N(\mathbb {C}))$ -comodule, we denote by

(2.10) $$ \begin{align} X_V(\xi,\eta) \in \mathcal{O}_q(M_N(\mathbb{C})),\qquad \xi,\eta\in V \end{align} $$

the corresponding matrix units, and we similarly write

$$\begin{align*}Z_V(\xi,\eta) = \Phi^{-1}(X_V(\xi,\eta)) \in \mathcal{O}_q(H(N)). \end{align*}$$

Then the quantum minors $z_{IJ}$ arise as matrix coefficients of

(2.11)

Lemma 2.6. The $*$ -structure on $\mathcal {O}_q(H(N))$ satisfies

(2.12) $$ \begin{align} z_{IJ}^* = z_{JI},\qquad Z_V(\xi,\eta)^* = Z_V(\eta,\xi), \end{align} $$

Proof. First note that $\#$ , as defined by (2.8), extends uniquely to an antilinear automorphism on $\mathcal {O}_q(M_N(\mathbb {C}))$ with the original product. This automorphism will then be anticomultiplicative:

$$\begin{align*}\Delta(f^{\#}) =f_{(2)}^{\;\#}\otimes f_{(1)}^{\;\#},\qquad f\in \mathcal{O}_q(M_N(\mathbb{C})). \end{align*}$$

As then

(2.13) $$ \begin{align} \mathbf{r}(f^{\#},g^{\#}) = \overline{\mathbf{r}(g,f)},\qquad f,g\in \{x_{ij}\mid 1\leq i,j\leq N\}, \end{align} $$

it follows by (1.30) that (2.13) holds for all $f,g\in \mathcal {O}_q(M_N(\mathbb {C}))$ . The same condition must then hold for $\mathbf {r}'$ . But this is easily seen to imply (using (2.5) and (2.2)) that

$$\begin{align*}(f*g)^{\#} = g^{\#}*f^{\#},\qquad f,g\in \mathcal{O}_q(M_N(\mathbb{C})). \end{align*}$$

It follows that $z \mapsto \Phi ^{-1}(\Phi (z)^{\#})$ must coincide with the original $*$ -structure on $\mathcal {O}_q(H(N))$ , as it coincides on the generators $z_{ij}$ .

The conclusion now follows by the easy verification that

$$\begin{align*}X_V(\xi,\eta)^{\#} = X_V(\eta,\xi), \end{align*}$$

using that $\#$ is multiplicative on $\mathcal {O}_q(M_N(\mathbb {C}))$ with the original product.

2.4 Commutation relations and Laplace expansions

For and , denote

(2.14) $$ \begin{align} \hat{R}^{IJ}_{I'J'} := \mathbf{r}(x_{JI},x_{I'J'}) \end{align} $$

and

By the braided commutativity (2.2), one gets that

(2.15)

Lemma 2.7. We have

(2.16) $$ \begin{align} \hat{R}^{IJ}_{I'J'} \neq 0 \quad \textrm{only if}\quad J\preceq I,J'\preceq I'\quad \textrm{and}\quad J \setminus I = J'\setminus I',\;I\setminus J = I'\setminus J'. \end{align} $$

Moreover,

$$\begin{align*}\hat{R}^{II}_{I'I'} = q^{- |I\cap I'|}. \end{align*}$$

We also write and

The $(\hat {R}^{-1})_{I'J'}^{IJ}$ satisfy the same conditions (2.16) except that now $(\hat {R}^{-1})^{II}_{I'I'} = q^{|I\cap I'|}$ .

We have the following q-analogue of the Laplace expansion [Reference Parshall and WangPW91, Theorem 4.4.3], which can be obtained by using the comodule algebra $\wedge _q(\mathbb {C}^N)$ . We use the notation as introduced in (0.16).

Proposition 2.8. For and the following identities hold in $\mathcal {O}_q(M_N(\mathbb {C}))$ ,

(2.17)

(2.18)

Using Theorem 2.4 and (2.15), we see that the quantum minors from (2.11) satisfy

(2.19)

By (2.19) we obtain the following commutation relations between the $z_{IJ}$ : for all $I,J,I',J'$

(2.20) $$ \begin{align} \sum_{K,L,L'} \left(\sum_{P'}\hat{R}_{JK}^{P'I'}\hat{R}_{P'L'}^{IL}\right) z_{KL} z_{L'J'} = \sum_{K,L,L'} \left(\sum_{P'}\hat{R}_{JK}^{P'L'}\hat{R}_{P'J'}^{IL}\right) z_{I'L'} z_{KL}. \end{align} $$

Putting and using (2.16), we obtain from (2.20) that the leading quantum minors

are mutually commuting self-adjoint elements which satisfy q-commutation relations with the $z_{IJ}$ , namely

(2.21)

In particular, we can consider the localisation

(2.22)

Also the Laplace expansion has an analogue in $\mathcal {O}_q(H(N))$ .

Proposition 2.9. For and with $m\leq k \leq N$ we have

(2.23)

and

(2.24)

Proof. Using (2.6), we see that

(2.25) $$ \begin{align} x_{IJ}x_{I'J'} = \sum_{A,B,C,D} (\hat{R}^{-1})_{I',B}^{A,I} \hat{R}_{B,D}^{J,C} x_{AC}*x_{DJ'}. \end{align} $$

Applying (2.25) to (2.17) and invoking Theorem 2.4 and the concrete formulas (2.5), we obtain (2.23). Applying $*$ we obtain (2.24).

2.5 Actions by quantum groups

The right action (0.13) of $\mathbf {G}\mathbf {L}_q(N,\mathbb {C})$ on $\mathbf {H}_q(N)$ is implemented by a coaction (= comodule $*$ -algebra structure)

(2.26) $$ \begin{align} {\mathrm{Ad}}^{GL}_q: \mathcal{O}_q(H(N)) \rightarrow \mathcal{O}_q(H(N)) \otimes \mathcal{O}_q^{\mathbb{R}}(GL(N,\mathbb{C})),\qquad Z \mapsto X_{13}^{*}Z_{12}X_{13}. \end{align} $$

The well-definedness of ${\mathrm {Ad}}^{GL}_q$ is immediate from the defining relations.

Similarly, the restrictions of the above action to actions of $\mathbf {U}_q(N)$ and $\mathbf {T}_q(N)$ on $\mathbf {H}_q(N)$ , arise from coactions

(2.27) $$ \begin{align} {\mathrm{Ad}}^{U}_q: \mathcal{O}_q(H(N)) \rightarrow \mathcal{O}_q(H(N)) \otimes \mathcal{O}_q(U(N)),\qquad Z \mapsto U_{13}^{*}Z_{12}U_{13}. \end{align} $$

(2.28) $$ \begin{align} {\mathrm{Ad}}^{T}_q: \mathcal{O}_q(H(N)) \rightarrow \mathcal{O}_q(H(N)) \otimes \mathcal{O}_q(T(N)),\qquad Z \mapsto T_{13}^{*}Z_{12}T_{13}. \end{align} $$

Using the pairing (1.19) between $\mathcal {O}_q(U(N))$ and $U_q(\mathfrak {u}(N))$ , the $*$ -algebra $\mathcal {O}_q(H(N))$ becomes a left $U_q(\mathfrak {u}(N))$ -module $*$ -algebra by considering the infinitesimal action dual to ${\mathrm {Ad}}^U_q$ ,

(2.29) $$ \begin{align} x\rhd z = ({\mathrm{id}}\otimes \tau(-,x)){\mathrm{Ad}}^U_q(z),\qquad x\in U_q(\mathfrak{u}(N)),z\in\mathcal{O}_q(H(N)). \end{align} $$

One has the global formula

(2.30) $$ \begin{align} x\rhd Z_{V}(\xi,\eta)= Z_{V}(S(x_{(1)})^*\xi,x_{(2)}\eta). \end{align} $$

Indeed, one notes that the right hand side also defines a module $*$ -algebra structure on $\mathcal {O}_q(H(N))$ , so that the identity only needs to be checked on the generating matrix, where it is obvious.

Classically, the $U(N)$ -orbit of the identity matrix in $H(N)$ is of course trivial under the adjoint action. On the other hand, by the Cholesky decomposition one has that the $T(N)$ -orbit of the identity matrix under the adjoint action gives the space of all positive-definite matrices. As the latter are Zariski dense in $H(N)$ , the following proposition is not surprising. We will use in the statement the map $\iota $ from (1.40), the infinitesimal adjoint action (2.29) as well as the right adjoint action $\blacktriangleleft $ of $\mathcal {O}_q(T(N))$ on $\mathcal {O}_{q}(T(N))$ , given by

(2.31) $$ \begin{align} x \blacktriangleleft g = S(g_{(1)})xg_{(2)},\qquad x,g\in \mathcal{O}_q(T(N)), \end{align} $$

with S denoting the antipode.

Proposition 2.10. The $*$ -homomorphism

$$\begin{align*}i_T:\mathcal{O}_q(H(N)) \rightarrow \mathcal{O}_q(T(N)),\qquad Z \mapsto T^*T \end{align*}$$

is injective, and

$$\begin{align*}i_{T}(\iota(S(g))\rhd f) = i_{T}(f) \blacktriangleleft g,\qquad f\in \mathcal{O}_q(H(N)), g\in \mathcal{O}_q(T(N)). \end{align*}$$

More generally, we have for any positive integral $U_q(\mathfrak {u}(N))$ -representation V that

(2.32) $$ \begin{align} ({\mathrm{id}}\otimes i_T) Z_V = T_V^*T_V, \end{align} $$

writing $T_V$ for the image of $X_V$ in ${\mathrm {End}}(V)\otimes \mathcal {O}_q(T(N))$ . See the proof of Lemma 4.6 for a more general statement. In particular,

(2.33)

2.6 Inclusions and quotients between the $\mathcal {O}_q(H(N))$

One can map $\mathcal {O}_q(H(K))$ into $\mathcal {O}_q(H(N))$ by placement in the upper left corner:

(2.34) $$ \begin{align} \mathcal{O}_q(H(K)) \rightarrow \mathcal{O}_q(H(N)),\qquad z_{ij}\mapsto z_{ij},\qquad 1\leq i,j\leq K. \end{align} $$

Similarly, we can also project down to the right bottom corner:

Lemma 2.11. Let $J_M$ be the 2-sided $*$ -ideal generated by the $z_{ij}$ with $1\leq i \leq M$ and $1\leq j \leq N$ . Then

$$\begin{align*}\mathcal{O}_q(H(N))/J_M \cong \mathcal{O}_q(H(N-M)),\qquad (z_{ij})_{1\leq i,j\leq N} \mapsto \begin{pmatrix} 0_M & 0_{M,N-M} \\ 0_{N-M,M} & (z_{ij})_{1\leq i,j\leq N-M}\end{pmatrix} \end{align*}$$

is a $*$ -homomorphism.

We will now show that, upon a small modification, the corner (2.34) can also be embedded in a different way. Namely, let $1\leq k \leq l \leq N$ , and put $M = l-k+1$ . Let $U_q(\mathfrak {u}(M))_{k,l} \subseteq U_q(\mathfrak {u}(N))$ be generated by the $E_i,F_i$ with $k\leq i <l$ and the $K_i^{\pm 1}$ with $k \leq i \leq l$ . Then we can consider the restriction of the infinitesimal adjoint action $\rhd $ to $U_q(\mathfrak {u}(M))$ via $U_q(\mathfrak {u}(M)) \cong U_q(\mathfrak {u}(M))_{k,l}$ .

Proposition 2.12. There is a unique $U_q(\mathfrak {u}(M))$ -equivariant $*$ -homomorphism

(2.35) $$ \begin{align} \rho_{k,l}: \mathcal{O}_q(H(M)) \rightarrow \mathcal{O}_q(H(N)) \end{align} $$

such that . Moreover, we then have .

Proof. By (4.3) and (1.44), it follows immediately that we have a Hopf $*$ -algebra homomorphism

$$\begin{align*}\kappa_{k,l}: \mathcal{O}_q(T(M)) \rightarrow \mathcal{O}_q(T(N)),\qquad t_{ij} \mapsto t_{i+k-1,j+k-1}. \end{align*}$$

Then as the adjoint action $\blacktriangleleft $ , defined in (2.31), is inner, we have

$$\begin{align*}\kappa_{k,l}(x \blacktriangleleft y) = \kappa_{k,l}(x) \blacktriangleleft \kappa_{k,l}(y),\qquad x,y\in \mathcal{O}_q(T(M)). \end{align*}$$

Consider now the modified $*$ -homomorphism

$$\begin{align*}\widetilde{\kappa}_{k,l}: \mathcal{O}_q(T(M)) \rightarrow \mathcal{O}_q(T(N)),\quad t_{ij} \mapsto t_1\ldots t_{k-1} \kappa_{k,l}(t_{ij}), \end{align*}$$

using that $t_i$ for $i<k$ commutes with all elements of $\kappa _{k,l}(\mathcal {O}_q(T(M)))$ . Then we clearly still have

$$\begin{align*}\widetilde{\kappa}_{k,l}(x \blacktriangleleft y) = \widetilde{\kappa}_{k,l}(x) \blacktriangleleft \kappa_{k,l}(y), \qquad x,y\in \mathcal{O}_q(T(M)). \end{align*}$$

Pulling $\widetilde {\kappa }_{k,l}$ back through the respective embeddings $i_T$ , we hence obtain from Proposition 2.10 a $U_q(\mathfrak {u}(M))$ -equivariant $*$ -homomorphism (2.35), which must be unique since $z_{11}$ is a generator for $\mathcal {O}_q(H(M))$ as a module $*$ -algebra. It is also immediate from the above construction and (2.33) that .

2.7 Rank of a representation

Definition 2.13. Let $1\leq M \leq N$ . We define $I_M$ to be the 2-sided $*$ -ideal of $\mathcal {O}_q(H(N))$ generated by all $z_{IJ}$ with for $k\geq M$ . We define

$$\begin{align*}\mathcal{O}_q^{\leq M}(H(N)) = \mathcal{O}_q(H(N))/I_{M+1}. \end{align*}$$

By convention, we put $I_{N+1} =\{0\}$ .

3.1 Spectral weight

Let $\mathscr {Z}(\mathcal {O}_q(H(N)))$ be the centre of $\mathcal {O}_q(H(N))$ , and let $\mathcal {O}_q(H(N))^{{\mathrm {Ad}}^U_q}$ be the $*$ -algebra of coinvariants for the coaction (2.27),

$$\begin{align*}\mathcal{O}_q(H(N))^{{\mathrm{Ad}}^U_q} = \{z\in \mathcal{O}_q(H(N)) \mid {\mathrm{Ad}}^U_q(z) = z\otimes 1\}. \end{align*}$$

The following result is well-known. For the convenience of a direct reference for the $GL$ -setting, we refer to [Reference De Commer and FloréDCF19, Lemma 3.30].

Theorem 3.1. An element of $\mathcal {O}_q(H(N))$ is central if and only if it is ${\mathrm {Ad}}^U_q$ -coinvariant, that is,

$$\begin{align*}\mathscr{Z}(\mathcal{O}_q(H(N))) = \mathcal{O}_q(H(N))^{{\mathrm{Ad}}^U_q}. \end{align*}$$

The centre of $\mathcal {O}_q(H(N))$ can further be described explicitly, see [Reference Jordan and WhiteJW20, Theorem 1.3] or also [Reference FloréFlo20, (1.12)].

Theorem 3.2. We have $\mathscr {Z}(\mathcal {O}_q(H(N))) \cong \mathbb {C}[\sigma _1,\ldots ,\sigma _N]$ , where the $\sigma _k$ are self-adjoint, algebraically independent elements given byFootnote ⁵

where ${\mathrm {Sym}}(I)$ is the group of permutations of leaving the complement of $I = \{i_1<\ldots < i_k\}$ pointwise fixed, where $\mathrm {wt}(I)$ is as in (0.17), and where $a(\sigma ) = |\{l\mid \sigma (l)<l\}|$ (called the antiexceedance).

Moreover, the following quantum Cayley-Hamilton identity holds: with $\sigma _0 = 1$ , we have

(3.1) $$ \begin{align} \sum_{k=0}^N (-1)^k \sigma_k Z ^{N-k} = 0. \end{align} $$

In particular, we define the quantum trace and quantum determinant

$$\begin{align*}{\mathrm{Tr}}_q(Z) = q^{-(N-1)}\sigma_1 = q^{N+1}\sum_{i=1}^N q^{-2i}z_{ii}, \end{align*}$$

It can be shown [Reference Jordan and WhiteJW20] that

As we have a unital $*$ -homomorphism $\mathcal {O}_q(H(K)) \rightarrow \mathcal {O}_q(H(N))$ by $Z \mapsto (z_{ij})_{1\leq i,j\leq K}$ , we obtain

(3.2)

Applying $*$ , we also obtain the alternative version

(3.3)

If $\pi $ is a factor representation of $\mathcal {O}_q(H(N))$ , then $\pi $ restricts to a $*$ -character on $\mathscr {Z}(\mathcal {O}_q(H(N)))$ .

Definition 3.3. If $\pi : \mathcal {O}_q(H(N)) \rightarrow B(\mathcal {H})$ is a factor representation, we denote

$$\begin{align*}\chi_{\pi}^{\mathscr{Z}}: \mathscr{Z}(\mathcal{O}_q(H(N))) \rightarrow \mathbb{C},\qquad \sigma_k \mapsto \sigma_k^{\pi} \end{align*}$$

for the associated $*$ -character on $\mathscr {Z}(\mathcal {O}_q(H(N)))$ , and call it the central $*$ -character of $\pi $ .

We call $s\in \mathbb {R}^N$ centrally admissible if there exists a factor representation $\pi $ such that

$$\begin{align*}\sigma_k^{\pi} = s_k,\qquad 1\leq k \leq N. \end{align*}$$

We call $\lambda \in \mathbb {R}^N/{\mathrm {Sym}}(N)$ a spectral weight if there exists a centrally admissible $s \in \mathbb {R}^N$ for which $\lambda $ is the multiset of roots of the polynomial

$$\begin{align*}P_s(x) = \sum_k (-1)^k s_k x^{k}, \end{align*}$$

and we denote

$$\begin{align*}\Lambda_q = \{ \textrm{spectral weights} \}. \end{align*}$$

For $\pi $ a factor representation of $\mathcal {O}_q(H(N))$ , we denote by $\lambda _{\pi }$ the spectral weight of $\pi $ , given as the multiset of roots of $P_s$ for $s_k = \sigma _k^{\pi }$ .

Note that Theorem B concretely identifies the set $\Lambda _q$ . For now, we will content ourselves to show in the next section that $\Lambda _q$ parametrises $\mathbf {U}_q(N)$ -orbits.

3.2 Orbits under $\mathbf {U}_q(N)$

The results in this section will be a bit more general than needed for our immediate purposes. We start with the following well-known general fact.

Proposition 3.4. Let $\mathbb {G}$ be a compact quantum group with associated Hopf $*$ -algebra $H = \mathcal {O}(\mathbb {G})$ , and let A be a unital $*$ -algebra with coaction

$$\begin{align*}\alpha: A \rightarrow A\otimes H. \end{align*}$$

Assume that the algebra of coinvariants $A^{\alpha } = \{a \in A \mid \alpha (a) = a\otimes 1\}$ is trivial,

$$\begin{align*}A^{\alpha} = \mathbb{C}. \end{align*}$$

Then all representations of A on pre-Hilbert spaces are bounded, and A admits a universal C $^*$ -envelope.

To apply this to the representation theory of $\mathcal {O}_q(H(N))$ , note that, by Theorem 3.2, any $s\in \mathbb {R}^N$ determines a $*$ -character $\chi _s: \mathscr {Z}(\mathcal {O}_q(H(N))) \rightarrow \mathbb {C}$ such that $\chi _{s}(\sigma _k) = s_k$ for all $1\leq k\leq N$ .

Definition 3.5. For $s\in \mathbb {R}^N$ , we define the $*$ -algebra $\mathcal {O}_q(O_s)$ as the quotient of $\mathcal {O}_q(H(N))$ by the extra relations

$$\begin{align*}\sigma_k = s_k. \end{align*}$$

We call $\mathcal {O}_q(O_s)$ the quantum orbit $*$ -algebra at parameter s.

These quantum orbit algebras were studied from the viewpoint of deformation quantization in [Reference Donin and MudrovDM02b].

The following is clear by Theorem 3.1.

Proposition 3.6. The coaction ${\mathrm {Ad}}^U_q$ descends to a coaction of $\mathcal {O}_q(U(N))$ on $\mathcal {O}_q(O_s)$ , and

(3.4) $$ \begin{align} \mathcal{O}_q(O_s)^{{\mathrm{Ad}}^U_q} = \mathbb{C}. \end{align} $$

By Proposition 3.4, we now obtain:

Note that the canonical map $\mathcal {O}_q(O_s) \rightarrow C_q(O_s)$ need not be injective, even when s is centrally admissible.

Let us return now to a general compact quantum group $\mathbb {G}$ with associated Hopf $*$ -algebra $H = \mathcal {O}(\mathbb {G})$ . Let $L^{\infty }(\mathbb {G})$ be its associated von Neumann algebra, obtained as the double commutant of $\mathcal {O}(\mathbb {G})$ in its GNS-representation $\lambda _{{\mathrm {st}}}$ with respect to its invariant state $\int _{\mathbb {G}}$ .

Definition 3.8. Let A be a unital $*$ -algebra with a coaction

$$\begin{align*}\alpha: A \rightarrow A\otimes \mathcal{O}(\mathbb{G}). \end{align*}$$

We say that a representation $\pi $ of A is $\alpha $ -compatible if $\alpha $ descends to a coaction on $\pi (A)$ . We say that moreover $\pi $ is normal $\alpha $ -compatible if $\alpha $ extends to a normal $*$ -homomorphism

$$\begin{align*}\pi(A)" \rightarrow \pi(A)"\overline{\otimes} L^{\infty}(\mathbb{G}). \end{align*}$$

When $\pi _1,\pi _2$ are representations of a unital $*$ -algebra A, we say that $\pi _1$ and $\pi _2$ are stably unitarily equivalent if $\oplus _{i\in I} \pi _1$ is unitarily equivalent to $\oplus _{j\in J} \pi _2$ for some index sets $I,J$ .

Proof. Let V be the right regular unitary corepresentation

$$\begin{align*}L^2(\mathbb{G}) \otimes L^2(\mathbb{G}) \rightarrow L^2(\mathbb{G}) \otimes L^2(\mathbb{G}), \quad f\otimes g \mapsto \Delta(f)(1\otimes g),\qquad f,g\in\mathcal{O}(\mathbb{G}). \end{align*}$$

If then $\pi $ is a representation of A, we have that

$$\begin{align*}(\pi*\lambda_{{\mathrm{st}}}\otimes \lambda_{{\mathrm{st}}})(\alpha(x))= V_{23} (\pi*\lambda_{{\mathrm{st}}}(x)\otimes 1)V_{23}^*, \end{align*}$$

proving that $\pi *\lambda _{{\mathrm {st}}}$ is normal $\alpha $ -compatible. This proves the first half of (1).

Conversely, if $\pi $ is normal $\alpha $ -compatible, let $N = \pi (A)"$ and consider the normal coaction $\alpha _{\pi }: N \rightarrow N \overline {\otimes }L^{\infty }(\mathbb {G})$ . Let $E: N \rightarrow N^{\alpha }$ be the faithful normal conditional expectation

$$\begin{align*}E(n) = \left({\mathrm{id}}\otimes \int_{\mathbb{G}}\right)\alpha_{\pi}(n) \in N^{\alpha},\qquad n \in N. \end{align*}$$

Then on $L^2(N)$ , as obtained from $L^2(N^{\alpha })$ through induction by E, we obtain the unitary

$$\begin{align*}V_{\pi,\alpha}: L^2(N) \otimes L^2(\mathbb{G}) \rightarrow L^2(N) \otimes L^2(\mathbb{G}),\quad n\xi\otimes \eta \mapsto \alpha(n)(\xi\otimes \eta),\quad n\in N,\xi\in L^2(N^{\alpha}),\eta\in L^2(\mathbb{G}). \end{align*}$$

Then, with $\pi _N$ the GNS-representation of N,

$$\begin{align*}\pi_N*\lambda_{{\mathrm{st}}}(n) = V_{\pi,\alpha}(\pi_N(n)\otimes 1)V_{\pi,\alpha}^*. \end{align*}$$

As $\pi $ and $\pi _N$ are stably unitarily equivalent (being faithful normal representations of N), this finishes the proof of (1). Moreover, if $\mathbb {H}$ is a subgroup of $\mathbb {G}$ , consider

$$\begin{align*}V_{\mathbb{G},\mathbb{H}}: L^2(\mathbb{G}) \otimes L^2(\mathbb{H})\rightarrow L^2(\mathbb{G}) \otimes L^2(\mathbb{H}),\quad f\otimes h \mapsto ({\mathrm{id}}\otimes \pi_{\mathbb{G},\mathbb{H}})\Delta(f)(1\otimes h),\quad f\in \mathcal{O}(\mathbb{G}),h\in \mathcal{O}(\mathbb{H}). \end{align*}$$

Then by [Reference VaesVae01, Theorem 2.7], we obtain a unique normal coaction $\alpha _{\pi ,\mathbb {H}}$ of $L^{\infty }(\mathbb {H})$ on N such that

$$\begin{align*}(V_{\mathbb{G},\mathbb{H}})_{23}\alpha_{\pi}(n)_{12}(V_{\mathbb{G},\mathbb{H}})_{23}^* = (V_{\pi,\alpha})_{12}\alpha_{\pi,\mathbb{H}}(n)_{13}(V_{\pi,\alpha})_{12}^*. \end{align*}$$

It is then easily verified that

$$\begin{align*}\alpha_{\pi,\mathbb{H}}(\pi(a)) = (\pi \otimes \pi_{\mathbb{G},\mathbb{H}})\alpha(a),\qquad a\in A, \end{align*}$$

proving (2).

The first point of the previous proposition can be strengthened under the condition that A has trivial coinvariants.

Proof. Let $A_u$ be the universal C $^*$ -envelope of A, and let $\pi _u$ be some faithful representation of $A_u$ . Then $\alpha $ extends to a coaction $\alpha _u: A_u \rightarrow A_u \otimes C(\mathbb {G})$ which still has trivial coinvariants. In particular, we obtain on $A_u$ a unique state $\varphi $ determined by

$$\begin{align*}\varphi(x)1_{A_u} = \left({\mathrm{id}}\otimes \int_{\mathbb{G}}\right)\alpha_u(x). \end{align*}$$

Since $\int _{\mathbb {G}}$ is faithful on $C(\mathbb {G})$ , and $\alpha _u$ is faithful, it follows that $\varphi $ is faithful. Hence $A_u \cong C^*(\pi *\lambda _{{\mathrm {st}}})$ , and in particular $\pi ' \preccurlyeq \pi *\lambda _{{\mathrm {st}}}$ .

Returning now to $\mathbb {G} = U_q(N)$ , let us say that two factor representations $\pi ,\pi '$ of $\mathcal {O}_q(H(N))$ lie in the same $\mathbf {U}_q(N)$ -orbit if there exist $\lambda ,\lambda '\in \mathbf {U}_q(N)$ such that

$$\begin{align*}\pi \preccurlyeq \pi'*\lambda,\qquad \pi'\preccurlyeq \pi*\lambda. \end{align*}$$

The following theorem is a quantum analogue of the spectral theorem for finite-dimensional self-adjoint matrices.

Proof. Assume that $\pi ,\pi '$ have the same spectral weight. Then they factor through a representation of $\mathcal {O}_q(O_s)$ for some s, and so $\pi ,\pi '$ lie in the same $\mathbf {U}_q(N)$ -orbit by Corollary 3.12.

Conversely, assume $\pi ,\pi '$ lie in the same $\mathbf {U}_q(N)$ -orbit, and choose a representation $\rho \in \mathbf {U}_q(N)$ such that

$$\begin{align*}\pi' \preccurlyeq \pi*\rho. \end{align*}$$

Since $\mathcal {O}_q(H(N))^{{\mathrm {Ad}}_q} = \mathscr {Z}(\mathcal {O}_q(H(N)))$ , we find that

$$\begin{align*}(\pi*\rho)(\sigma_k) = \pi(\sigma_k)\otimes 1,\qquad 1\leq k \leq N \end{align*}$$

so that $\pi *\rho $ factors over $\mathcal {O}_q(O_s)$ . As $\pi ' \preccurlyeq \pi *\rho $ , the same must be true for $\pi '$ . This implies that the spectral weights of $\pi $ and $\pi '$ must coincide.

4.1 Big cell representations

To understand big cell representations, it is convenient to consider a deformation of $\mathcal {O}_q(T(N))$ . This is what we turn to next.

4.2 Deformed quantized function algebras $\mathcal {O}_q^{\epsilon }(T(N))$

For $\epsilon \in \mathbb {R}^N$ , let

(4.1)

Define, for $\epsilon \in \mathbb {R}^N$ , the $\epsilon $ -deformed braid operator by

$$\begin{align*}\hat{R}_{\epsilon} = \sum_{i,j} q^{-\delta_{ij}} e_{ji}\otimes e_{ij} +(q^{-1}-q)\sum_{i<j} \epsilon_{(i,j]} e_{jj}\otimes e_{ii}, \end{align*}$$

When $\epsilon _i = 1$ for all i, we write $\epsilon = +$ , so, for example, $\hat {R}_+ = \hat {R}$ .

Definition 4.2. We define $\mathcal {O}_{q,\epsilon }^{\mathbb {R}}(B(N))$ to be the algebra generated by $\mathcal {O}_q(B(N))$ and an independent copy of $\mathcal {O}_q(B(N))^*$ , together with the extra relations

(4.2) $$ \begin{align} T_{23}\hat{R}_{12}T^*_{23} = T^*_{13}\hat{R}_{\epsilon,12}T_{13}. \end{align} $$

We define $\mathcal {O}_q^{\epsilon }(T(N))$ as the quotient of $\mathcal {O}_{q,\epsilon }^{\mathbb {R}}(B(N))$ by the further relations $t_{i}^* = t_i$ for all i.

We then have that $\mathcal {O}_q^{\epsilon }(T(N))$ is the universal unital $*$ -algebra generated by invertible elements $t_i$ for $1\leq i \leq N$ and elements $t_{ij}$ with $1\leq i<j\leq N$ such that (1.25) hold, as well as

(4.3) $$ \begin{align} \left\{\begin{array}{lllll} t_{kj} t_{li}^{*} - t_{li}^{*}t_{kj}&=&0,&& i\neq j, k\neq l,\\ t_{kj}t_{lj}^{*} - qt_{lj}^{*}t_{kj}&=& -(1-q^2)\sum_{i<j} t_{ki}t_{li}^{*} ,&&k\neq l,\\ qt_{kj}t_{ki}^{*}- t_{ki}^{*}t_{kj} &=& (1-q^2)\sum_{k<l\leq\mathrm{min}\{i,j\}} \epsilon_{(k,l]}t_{li}^{*}t_{lj},&& i\neq j, \\ t_{kj}t_{kj}^{*}- t_{kj}^{*}t_{kj} &=& (1-q^2)\left(\sum_{k<l\leq j} \epsilon_{(k,l]} t_{lj}^{*}t_{lj} - \sum_{k\leq l<j} t_{kl}t_{kl}^{*}\right). && \end{array}\right. \end{align} $$

When $\epsilon = + = (+,+,\ldots ,+)$ , we find back $\mathcal {O}_q(T(N))$ .

In what follows we need a slightly more general class of $*$ -algebras $\mathcal {O}_q^{\epsilon }(T(N))$ parametrized by $\epsilon \in \sqcup _{M=0}^N \mathbb {R}^M$ .

Definition 4.3. Assume that $M \leq N$ , and let $\epsilon \in \mathbb {R}^M$ . We define $\mathcal {O}_q^{\leq M}(B(N))\subseteq \mathcal {O}_q(B(N))$ as the unital subalgebra generated by the $t_i^{\pm 1}$ and $t_{ij}$ with $i\leq M$ and $j\leq N$ . We define $\mathcal {O}_{q,\epsilon }^{\mathbb {R}}(B(N))$ to be the $*$ -subalgebra of $\mathcal {O}_{q,\widetilde {\epsilon }}^{\mathbb {R}}(B(N))$ generated by $\mathcal {O}_q^{\leq M}(B(N))$ , where

(4.4) $$ \begin{align} \widetilde{\epsilon} = (\epsilon,0,\ldots,0)\in \mathbb{R}^N. \end{align} $$

Similarly, we define $\mathcal {O}_q^{\epsilon }(T(N))$ as the $*$ -subalgebra of $\mathcal {O}_q^{\widetilde {\epsilon }}(T(N))$ generated by the $t_i^{\pm 1}$ and $t_{ij}$ with $i\leq M$ and $j\leq N$ .

Note that, for example, $\mathcal {O}_{q,\epsilon }^{\mathbb {R}}(B(N))$ implicitly depends on M, the dimension of the vector space in which the vector $\epsilon $ lives.

We denote

$$\begin{align*}\pi_T: \mathcal{O}_{q,\epsilon}^{\mathbb{R}}(B(N)) \rightarrow \mathcal{O}_q^{\epsilon}(T(N)) \end{align*}$$

for the natural quotient map.

Lemma 4.4. The multiplication maps

(4.5) $$ \begin{align} \mathcal{O}_q^{\leq M}(B(N))^* \otimes \mathcal{O}_q^{\leq M}(B(N)) \rightarrow \mathcal{O}_{q,\epsilon}^{\mathbb{R}}(B(N)),\quad x^*\otimes y\mapsto x^*y \end{align} $$

(4.6) $$ \begin{align} \mathcal{O}_q^{\leq M}(B(N)) \otimes \mathcal{O}_q^{\leq M}(B(N))^* \rightarrow \mathcal{O}_{q,\epsilon}^{\mathbb{R}}(B(N)),\quad x\otimes y^*\mapsto xy^* \end{align} $$

are bijective.

Here we note that, on the left hand side, we consider $\mathcal {O}_q^{\leq M}(B(N))^*$ to be a formal antilinear, anti-isomorphic copy of $\mathcal {O}_q^{\leq M}(B(N))$ .

Proof. Let us first prove the result for $\epsilon \in \mathbb {R}^N$ (so $M=N$ ). From the concrete commutation relations (4.3) and an easy induction argument, it is already clear that the maps will be surjective.

Let us show that (4.6) is injective, the proof for (4.5) is similar. Note first that there exists a unique skew bicharacter

(4.7) $$ \begin{align} \mathbf{r}_{\epsilon}: \mathcal{O}_q(GL(N,\mathbb{C})) \otimes \mathcal{O}_q(GL(N,\mathbb{C})) \rightarrow \mathbb{C} \end{align} $$

such that, with $R_{\epsilon } = \Sigma \circ \hat {R}_{\epsilon }$ ,

$$\begin{align*}({\mathrm{id}}\otimes {\mathrm{id}}\otimes \mathbf{r}_{\epsilon})X_{13}X_{24} = R_{\epsilon}. \end{align*}$$

In fact, $\mathbf {r}_{\epsilon }$ is obtained by pairing with $(\nu _{\epsilon }\otimes {\mathrm {id}})\mathscr {R} = ({\mathrm {id}}\otimes \nu _{\epsilon })\mathscr {R}$ , where $\nu _{\epsilon }$ is (the completion of) the Hopf algebra endomorphism of $U_q(\mathfrak {b}(N))$ , resp. $U_q(\mathfrak {b}^-(N))$ with

(4.8) $$ \begin{align} \nu_{\epsilon}(K_i) = K_i,\qquad \nu_{\epsilon}(E_i) = \epsilon_{i+1}E_i,\qquad \nu_{\epsilon}(F_i) = \epsilon_{i+1}F_i. \end{align} $$

This $\mathbf {r}_{\epsilon }$ descends to a skew bicharacter

$$\begin{align*}\mathbf{r}_{\epsilon}: \mathcal{O}_q(B(N)) \otimes \mathcal{O}_q(B^-(N)) \rightarrow \mathbb{C}. \end{align*}$$

We can then define on $\mathcal {O}_q^{\leq M}(B(N))^* \otimes \mathcal {O}_q^{\leq M}(B(N))$ a unique associative product such that

$$\begin{align*}(z^*\otimes x)(y^*\otimes w) := \mathbf{r}_{\epsilon}(S(x_{(1)}),y_{(1)}^*)z^*y_{(2)}^{*}\otimes x_{(2)}w \mathbf{r}(x_{(3)},y_{(3)}^*), \end{align*}$$

where we view $\mathcal {O}_q(B^-(N)) = \mathcal {O}_q(B(N))^*$ with respect to the compact real form on $\mathcal {O}_q(GL(N,\mathbb {C}))$ . The universal relations for $\mathcal {O}_{q,\epsilon }^{\mathbb {R}}(B(N))$ now allow to construct a splitting map

$$\begin{align*}\mathcal{O}_{q,\epsilon}^{\mathbb{R}}(B(N)) \rightarrow \mathcal{O}_q^{\leq M}(B(N)) \otimes \mathcal{O}_q^{\leq M}(B(N))^* \end{align*}$$

for (4.6), from which the injectivity of (4.6) directly follows.

Assume now in general that $\epsilon \in \mathbb {R}^M$ . Clearly, the multiplication maps (4.5) and (4.6) will then be injective by the first part of the proof. But they are also still surjective: all the right hand terms of (4.3) involving $t_{ij}$ for $i>M$ involve a factor $\widetilde {\epsilon }_i = 0$ , from which it follows directly that the ranges of our multiplication maps are already closed under multiplication.

Fix $\epsilon \in \mathbb {R}^N$ , and recall the notations (0.18), (1.22) and (4.1). Define $\mathscr {E}_{\epsilon } \in \mathscr {U}_q^{{\mathrm {pos}}}(\mathfrak {gl}(N,\mathbb {C}))$ by

$$\begin{align*}\mathscr{E}_{\epsilon}\xi = (\eta_{\epsilon})_{\mathrm{wt}(\xi)}\xi, \end{align*}$$

which is meaningful since $\xi \in V_{\lambda }$ has $\mathrm {wt}(\xi ) \in P_{{\mathrm {pos.int}}}$ for $\lambda \in P_{{\mathrm {pos.int}}}^{{\mathrm {dom}}}$ . In the above, we put $0^0=1$ by convention. For example, for the vector module we get

Lemma 4.5. There is a unique $*$ -homomorphism

(4.9) $$ \begin{align} i_{B}^{\epsilon}: \mathcal{O}_q(H(N)) \rightarrow \mathcal{O}_{q,\epsilon}^{\mathbb{R}}(B(N)),\quad Z \mapsto T^{*}(E_{\epsilon}\otimes 1) T. \end{align} $$

Proof. This follows directly from the defining relations, using the easily computed identity

$$\begin{align*}(1\otimes E_{\epsilon})\hat{R}_{\epsilon}(1\otimes E_{\epsilon})\hat{R} = \hat{R}(1\otimes E_{\epsilon})\hat{R}_{\epsilon}(1\otimes E_{\epsilon}). \end{align*}$$

We denote then also

(4.10) $$ \begin{align} i_{T}^{\epsilon}: \mathcal{O}_q(H(N)) \rightarrow \mathcal{O}_{q}^{\epsilon}(T(N)),\quad Z \mapsto T^{*}(E_{\epsilon}\otimes 1) T. \end{align} $$

We can also give again a more global form of these maps as follows, using notation as in (2.32) and viewing $T_V \in {\mathrm {End}}(V)\otimes \mathcal {O}_q(B(N))$ :

Lemma 4.6. The map $i_{B}^{\epsilon }$ from (4.9) is given by

(4.11) $$ \begin{align} i_{B}^{\epsilon}: \mathcal{O}_q(H(N)) \rightarrow \mathcal{O}_{q,\epsilon}^{\mathbb{R}}(B(N)),\quad Z_V(\xi,\eta) \mapsto \sum_{i,j} T_V(e_i,\xi)^* \langle e_i,\mathscr{E}_{\epsilon}e_j\rangle T_V(e_j,\eta), \end{align} $$

where $\{e_i\}$ is an arbitrary orthonormal basis of V.

Proof. Viewing $U_q(\mathfrak {gl}(N,\mathbb {C})) \subseteq \mathscr {U}_q^{{\mathrm {pos}}}(\mathfrak {gl}(N,\mathbb {C}))$ , it is easily seen that $\epsilon _{i+1} \mathscr {E}_{\epsilon } E_i = E_i \mathscr {E}_{\epsilon }$ . Hence, with $\nu _{\epsilon }$ as in (4.8), we have $\mathscr {E}_{\epsilon } \nu _{\epsilon }(X) = X\mathscr {E}_{\epsilon }$ for $X \in U_q(\mathfrak {b}(N))$ , and so

$$\begin{align*}(\mathscr{E}_{\epsilon}\otimes 1)\mathscr{R}_{\epsilon} = \mathscr{R}(\mathscr{E}_{\epsilon}\otimes 1),\qquad \textrm{with } \mathscr{R}_{\epsilon} = (\nu_{\epsilon}\otimes {\mathrm{id}})\mathscr{R}. \end{align*}$$

Since also $\mathscr {E}_{\epsilon }$ is grouplike, that is, $\Delta ^{{\mathrm {pos}}}(\mathscr {E}_{\epsilon }) = \mathscr {E}_{\epsilon }\otimes \mathscr {E}_{\epsilon }$ , we find that

$$\begin{align*}\mathscr{R}\Delta^{{\mathrm{pos}}}(\mathscr{E}_{\epsilon}) = (\mathscr{E}_{\epsilon} \otimes 1)\mathscr{R}_{\epsilon}(1\otimes \mathscr{E}_{\epsilon}). \end{align*}$$

Noting that $({\mathrm {id}}\otimes S)\mathscr {R}_{\epsilon }$ is the inverse of $\mathscr {R}_{\epsilon }$ in $\mathscr {U}_q(\mathfrak {gl}(N,\mathbb {C}))\widehat {\otimes } \mathscr {U}_q(\mathfrak {gl}(N,\mathbb {C}))^{\mathrm {op}}$ , it is now easily verified through the global formula (2.7) for the product of $\mathcal {O}_q(H(N))$ , that the map defined by (4.11) is a $*$ -homomorphism. To show that it coincides with $i_{B}^{\epsilon }$ it is sufficient to verify equality on the generating matrix Z, which is immediate.

Recall now the localisation $\mathcal {O}_q^{{\mathrm {loc}}}(H(N))$ from (2.22). More generally, recalling Definition 2.13, we can consider the localisation

$$\begin{align*}\mathcal{O}_{q,\leq M}^{{\mathrm{loc}}}(H(N)) = \mathcal{O}_{q}^{\leq M}(H(N))[z_1^{-1},\ldots,z_M^{-1}]. \end{align*}$$

Our aim is to give a different description of $\mathcal {O}_{q,\leq M}^{{\mathrm {loc}}}(H(N))$ . We start with the following lemma.

Lemma 4.7. We have

Proposition 4.8. Let $\epsilon \in (\mathbb {R}\setminus \{0\})^M$ . Then the map $i_T^{\epsilon }$ from (4.10) factors through an injective $*$ -homomorphism

$$\begin{align*}j_T^{\epsilon}: \mathcal{O}_{q,\leq M}^{{\mathrm{loc}}}(H(N)) \rightarrow \mathcal{O}_q^{\epsilon}(T(N)). \end{align*}$$

Moreover, the range of $j_T^{\epsilon }$ is the unital $*$ -algebra generated by all $t_i^{\pm 2}$ for $1\leq i \leq M$ and $t_it_{ij}$ with $1\leq i\leq M$ and $i<j\leq N$ .

Note that the proof will actually show that the restriction of $j_T^{\epsilon }$ to $\mathcal {O}_{q}^{\leq M}(H(N))$ is injective, so that it indirectly also shows that $\mathcal {O}_{q}^{\leq M}(H(N))$ embeds in its localisation.

Proof. Let $\widetilde {\epsilon }$ be as in (4.4). It is clear by the formula for (4.11), the definition of $I_{M+1}$ in Definition 2.13 and Lemma 4.7 that $i_B^{\epsilon }$ factors through a $*$ -homomorphism

(4.12) $$ \begin{align} j_B^{\epsilon}: \mathcal{O}_{q,\leq M}^{{\mathrm{loc}}}(H(N)) \rightarrow \mathcal{O}_{q,\epsilon}^{\mathbb{R}}(B(N))\subseteq \mathcal{O}_{q,\widetilde{\epsilon}}^{\mathbb{R}}(B(N)) \end{align} $$

Choose now for each $\lambda \in P_{{\mathrm {pos.int}}}^{{\mathrm {dom}}}$ an orthonormal basis of weight vectors $\{\xi _{\lambda ,i}\mid i \in I_{\lambda }\}$ for $V_{\lambda }$ , and put $\omega _{\lambda ,i} = \mathrm {wt}(\xi _{\lambda ,i})$ and $\alpha _{\lambda ,i} = \lambda - \omega _{\lambda ,i}$ . Recalling the notation (1.34), we can then choose $a_{\lambda ,i} \in U_q(\mathfrak {n})_{\alpha _{\lambda ,i}}$ with $a_{\lambda ,i}\xi _{\lambda ,j} = \delta _{ij}\xi _{\lambda }$ whenever $\alpha _{\lambda ,i} = \alpha _{\lambda ,j}$ , where $\xi _{\lambda }$ is a chosen highest weight vector of $V_{\lambda }$ (of unit length).

Assume then that we have a nontrivial finite linear combination

$$\begin{align*}C = \sum_{\lambda\in P_{{\mathrm{pos.int}}}^{{\mathrm{dom}}}}\sum_{k,l\in I_{\lambda}} a_{\lambda,k,l} z_{\lambda}(\xi_{\lambda,k},\xi_{\lambda,l}) \in {\mathrm{Ker}}(i_B^{\epsilon}), \end{align*}$$

with $a_{\lambda ,k,l} =0$ for all but finitely many indices. We will show that $C\in I_{M+1}$ .

Note first that $i_B^{\epsilon }(C)=0$ gives

$$\begin{align*}\sum_{\lambda}\sum_{k,l,r\in I_{\lambda}} a_{\lambda,k,l} T_{\lambda}(\xi_{\lambda,r},\xi_{\lambda,k})^* \langle \xi_{\lambda,r},\mathscr{E}_{\widetilde{\epsilon}}\xi_{\lambda,r}\rangle T_{\lambda}(\xi_{\lambda,r},\xi_{\lambda,l}) =0. \end{align*}$$

From (4.5) we deduce that

(4.13) $$ \begin{align} \sum_{\lambda}\sum_{k,l,r\in I_{\lambda}} \langle \xi_{\lambda,r},\mathscr{E}_{\widetilde{\epsilon}}\xi_{\lambda,r}\rangle a_{\lambda,k,l} T_{\lambda}(\xi_{\lambda,r},\xi_{\lambda,k})^* \otimes T_{\lambda}(\xi_{\lambda,r},\xi_{\lambda,l}) =0. \end{align} $$

Let now $K_{\alpha ,\beta }$ , for $\alpha ,\beta \in Q^+$ , be the set of $(\lambda ,k,l)$ with

$$\begin{align*}a_{\lambda,k,l}\neq0,\qquad \mathrm{wt}(\xi_{\lambda}) - \mathrm{wt}(\xi_{\lambda,k})= \alpha,\qquad \mathrm{wt}(\xi_{\lambda}) - \mathrm{wt}(\xi_{\lambda_i,l})= \beta. \end{align*}$$

Let K be the set of $(\alpha ,\beta )$ with $K_{\alpha ,\beta }\neq \emptyset $ , and pick $(\alpha _0,\beta _0)\in K$ maximal with respect to the coordinatewise partial order $(\alpha ,\beta )\preccurlyeq (\alpha ',\beta ') \Leftrightarrow \alpha \preccurlyeq \alpha '$ and $\beta \preccurlyeq \beta '$ . Pick $(\lambda _0,k_0,l_0) \in K_{\alpha _0,\beta _0}$ . Then $a_{\lambda _0,k_0}\xi _{\lambda ,k}\otimes a_{\lambda _0,l_0}\xi _{\lambda ,l}$ will be a multiple of $\xi _{\lambda }\otimes \xi _{\lambda }$ for each $(\lambda ,k,l) \in K_{\alpha _0,\beta _0}$ , and zero for all other $(\lambda ,k,l)\in K_{\alpha ,\beta }$ with $(\alpha ,\beta )\neq (\alpha _0,\beta _0)$ . Hence upon applying $({\mathrm {id}}\otimes \tau (-,a_{\lambda _{i_0},l_0}))\Delta $ to the second tensorand and $({\mathrm {id}}\otimes \overline {\tau (a_{\lambda _{i_0},k_0},-*)})\Delta $ to the first tensorand of (4.13), we see that there exist $c_{\lambda ,k,l} \in \mathbb {C}$ with $c_{\lambda _0,k_0,l_0} = 1$ , with all other $c_{\lambda _0,k,l}= 0$ , and

$$\begin{align*}\sum_{(\lambda,k,l) \in K_{\alpha_0,\beta_0}}\sum_{r\in I_{\lambda}} \langle \xi_{\lambda,r},\mathscr{E}_{\widetilde{\epsilon}}\xi_{\lambda,r}\rangle a_{\lambda,k,l}c_{\lambda ,k,l} T_{\lambda}(\xi_{\lambda,r},\xi_{\lambda})^* \otimes T_{\lambda}(\xi_{\lambda,r},\xi_{\lambda}) =0, \end{align*}$$

which simplifies to

$$\begin{align*}\sum_{(\lambda,k,l) \in K_{\alpha_0,\beta_0}} \langle \xi_{\lambda},\mathscr{E}_{\widetilde{\epsilon}}\xi_{\lambda}\rangle a_{\lambda,k,l}c_{\lambda,k,l} T_{\lambda}(\xi_{\lambda},\xi_{\lambda})^* \otimes T_{\lambda}(\xi_{\lambda},\xi_{\lambda}) =0. \end{align*}$$

Since the $T_{\lambda }(\xi _{\lambda },\xi _{\lambda })$ are linearly independent, we deduce that

$$\begin{align*}\langle \xi_{\lambda_{0}},\mathscr{E}_{\widetilde{\epsilon}}\xi_{\lambda_{0}} \rangle = (\eta_{\widetilde{\epsilon}})_{\lambda_0} = 0. \end{align*}$$

By our assumption on $\epsilon $ , this can only happen if $(\lambda _0)_k>0$ for some $k\geq M+1$ .

Now the linear span of all the $X_{\lambda }(\xi ,\eta )$ with $\lambda _k>0$ for some $k\geq M+1$ equals the $2$ -sided ideal generated by all for $k\geq M+1$ in $\mathcal {O}_q(M_N(\mathbb {C}))$ , since any $V_{\lambda }$ can be identified with the representation spanned by the tensor product of the highest weight vectors in . It follows that $I_{M+1}$ is the linear span of all the $Z_{\lambda }(\xi ,\eta )$ with $\lambda _k>0$ for some $k\geq M+1$ . In particular, $Z_{\lambda _0}(\xi _{\lambda _0,k_0},\xi _{\lambda _0,l_0}) \in I_{M+1}$ . By induction on the number of nonzero terms in C, we now find that $C \in I_{M+1}$ . It follows that (4.12) is injective.

Let us now determine the range of (4.12). Let $\mathscr {A}$ be the $*$ -subalgebra of $\mathcal {O}_{q,\widetilde {\epsilon }}^{\mathbb {R}}(B(N))$ generated by the $(t_{ii}^{*}t_{ii})^{-1}$ and $t_{ii}^*t_{ij}$ for $1\leq i \leq M$ and $1\leq j \leq N$ . Since

(4.14)

it follows by induction that $j^{\epsilon }_B(\mathcal {O}_{q,\leq M}^{{\mathrm {loc}}}(H(N)))$ contains all , and hence $\mathscr {A} \subseteq j^{\epsilon }_B(\mathcal {O}_{q,\leq M}^{{\mathrm {loc}}}(H(N)))$ . But (4.14) also shows that this inclusion holds in the other direction, so $\mathscr {A} = j^{\epsilon }_B(\mathcal {O}_{q,\leq M}^{{\mathrm {loc}}}(H(N)))$ .

The proposition now follows since the map $\pi _T: \mathscr {A} \rightarrow \mathcal {O}_q^{\epsilon }(T(N))$ is injective.