Representations of Interpolating QUE Algebras

Representation Theory of Quantized Enveloping
Algebras with Interpolating Real Structure\star\star$\star$This paper is a
contribution to the Special Issue on Noncommutative Geometry and Quantum
Groups in honor of Marc A. Rieffel. The full collection is available at
http://www.emis.de/journals/SIGMA/Rieffel.html

Kenny DE COMMER

K. De Commer

Department of Mathematics, University of Cergy-Pontoise,
UMR CNRS 8088, F-95000 Cergy-Pontoise, France<EMAIL_ADDRESS>http://kdecommer.u-cergy.fr

Received August 18, 2013, in final form December 18, 2013; Published online
December 24, 2013

Let $\mathfrak{g}$ be a compact simple Lie algebra. We modify the quantized
enveloping ∗-algebra associated to $\mathfrak{g}$ by a real-valued character
on the positive part of the root lattice. We study the ensuing Verma module
theory, and the associated quotients of these modified quantized enveloping
∗-algebras. Restricting to the locally finite part by means of a natural
adjoint action, we obtain in particular examples of quantum homogeneous spaces
in the operator algebraic setting.

compact quantum homogeneous spaces; quantized universal enveloping algebras;
Hopf–Galois theory; Verma modules

17B37; 20G42; 46L65

## Introduction

This paper reports on preliminary work related to the quantization of non-
compact semi-simple Lie groups. The main idea behind such a quantization is
based on the reflection technique developed in [5] and [11] (see also [7] and
[6] for concrete, small-dimensional examples relevant to the topic of this
paper). Briefly, this technique works as follows. Let $\mathbb{G}$ be a
compact quantum group acting on a compact quantum homogeneous space
$\mathbb{X}$. Assume that the von Neumann algebra
$\mathscr{L}^{\infty}(\mathbb{X})$ associated to $\mathbb{X}$ is a type $I$
factor. Then the action of $\mathbb{G}$ on $\mathscr{L}^{\infty}(\mathbb{X})$
can be interpreted as a projective representation of $\mathbb{G}$, and one can
deform $\mathbb{G}$ with the ‘obstruction’ associated to this projective
representation to form a new _locally_ compact quantum group $\mathbb{H}$.
More generally, if $\mathscr{L}^{\infty}(\mathbb{X})$ is only a finite direct
sum of type $I$-factors, one can construct $\mathbb{H}$ as a locally compact
quantum group _oid_ (of a particularly simple type). Our idea is to fit the
quantizations of non-compact semi-simple Lie groups into this framework,
obtaining them as a reflection of the quantization of their compact companion.
For this, one needs the proper quantum homogeneous spaces to feed the
machinery with.

It is natural to expect the needed quantum homogeneous space to be a
quantization of a compact symmetric space associated to the non-compact semi-
simple Lie group. By now, there is much known on the quantization of symmetric
spaces (see [24, 25] and references therein, and [31] for the non-compact
situation), but these results are mostly of an algebraic nature, and not much
seems known about corresponding operator algebraic constructions except for
special cases. In fact, in light of the motivational material presented in
Appendix B, we will instead of symmetric spaces use certain quantizations of
(co)adjoint orbits, following the approach of [10, 19, 28]. Here, one rather
constructs quantum homogeneous spaces as subquotients of (quantized) universal
enveloping algebras in certain highest weight representations. We will build
on this approach by combining it with real structures and the contraction
technique.

Our main result, Theorem 3.20, will consist in showing that the compact
quantum homogeneous spaces that we build do indeed consist of finite direct
sums of type I factors. This will give a theoretical underpinning and
motivation for the claim that the above mentioned quantizations of non-compact
semi-simple groups can indeed be constructed using the reflection technique.
Our results are however quite incomplete as of yet, as

  * $\bullet$

in the non-contracted case, we can only treat concretely the case of Hermitian
symmetric spaces,

  * $\bullet$

a more detailed analysis of the resulting quantum homogeneous spaces is
missing,

  * $\bullet$

the relation to known quantum homogeneous spaces is not elucidated,

  * $\bullet$

no precise connection with deformation quantization is provided,

  * $\bullet$

the relation with the approach of Korogodsky [23] towards the quantization of
non-compact Lie groups remains to be clarified.

We hope to come back to the above points in future work.

The structure of this paper is as follows. In Section 1, we introduce the
‘modified’ quantized universal enveloping algebras we will be studying, and
state their main properties in analogy with the ordinary quantized universal
enveloping algebras. In Section 2, we introduce a theory of Verma modules, and
study the associated unitarization problem. In Section 3, we study
subquotients of our generalized quantized universal enveloping algebras, and
show how they give rise to $C^{*}$-algebraic quantum homogeneous spaces whose
associated von Neumann algebras are direct sums of type $I$ factors. In
Section 4, we briefly discuss a case where the associated von Neumann algebra
is simply a type $I$-factor itself.

In the appendices, we give some further comments on the structures appearing
in this paper. In Appendix A, we recall the notion of cogroupoids [2] which is
very convenient for our purposes. In Appendix B, we discuss the Lie algebras
which are implicitly behind the constructions in the main part of the paper.

##  1 Two-parameter deformations of quantized
enveloping algebras

Let $\mathfrak{g}$ be a complex simple Lie algebra of rank $l$, with fixed
Cartan subalgebra $\mathfrak{h}$ and Cartan decomposition
$\mathfrak{g}=\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+}$. Let
$\Delta\subseteq\mathfrak{h}^{*}$ be the associated finite root system,
$\Delta^{+}$ the set of positive roots, and $\Phi^{+}=\\{\alpha_{r}\,|\,r\in
I\\}$ the set of simple positive roots. We identify $I$ and $\Phi^{+}$ with
the set $\\{1,\ldots,l\\}$ whenever convenient. Let
$\mathfrak{h}_{\mathbb{R}}^{*}\subseteq\mathfrak{h}^{*}$ be the real linear
span of the roots, and let $(\;,\,)$ be an inner product on
$\mathfrak{h}_{\mathbb{R}}^{*}$ for which $A=(a_{rs})_{r,s\in
I}=\left((\alpha^{\vee}_{r},\alpha_{s})\right)_{r,s\in I}$ is the Cartan
matrix of $\mathfrak{g}$, where
$\alpha^{\vee}=\frac{2}{(\alpha,\alpha)}\alpha$ for $\alpha\in\Delta$. Let
$\mathfrak{h}_{\mathbb{R}}\subseteq\mathfrak{h}$ be the real linear span of
the coroots $h_{\alpha}$, where $\beta(h_{\alpha})=(\beta,\alpha^{\vee})$ for
$\alpha,\beta\in\Delta$.

We further use the following notation. We write $\\{\omega_{r}\,|\,r\in I\\}$
for the fundamental weights in $\mathfrak{h}^{*}_{\mathbb{R}}$, so
$(\omega_{r},\alpha^{\vee}_{s})=\delta_{rs}$. The $\mathbb{Z}-$lattice spanned
by $\\{\omega_{r}\\}$ is denoted $P\subseteq\mathfrak{h}^{*}_{\mathbb{R}}$,
and $P^{+}$ denotes elements expressed as positive linear combinations of this
basis. Similarly, the root lattice spanned by the $\alpha_{r}$ is denoted $Q$,
and its positive span by $Q^{+}$. We write
$\operatorname{\mathrm{Char}}_{K}(F)$ for the monoid of monoid homomorphisms
from a commutative (additive) monoid $(F,+)$ to a commutative (multiplicative)
monoid $(K,\cdot)$. For
$\varepsilon\in\operatorname{\mathrm{Char}}_{K}(Q^{+})$ we will abbreviate
$\varepsilon_{\alpha_{r}}=\varepsilon_{r}$. The unit element of
$\operatorname{\mathrm{Char}}_{K}(Q^{+})$ will be denoted $+$, while the
element $\varepsilon$ such that $\varepsilon_{r}=0$ for all $r$ will be
denoted 0.

We use the following notation for $q$-numbers, where $0<q<1$ is fixed for the
rest of the paper:

  * $\bullet$

for $r\in\Phi^{+}$, $q_{r}=q^{\frac{(\alpha_{r},\alpha_{r})}{2}}$,

  * $\bullet$

for $n\geq 0$, $[n]_{r}=\frac{q_{r}^{n}-q_{r}^{-n}}{q_{r}-q_{r}^{-1}}$,

  * $\bullet$

for $n\geq 0$, $[n]_{r}!=\prod\limits_{k=1}^{n}[k]_{r}$,

  * $\bullet$

for $m\geq n\geq 0$, $\left[\begin{matrix}m\\\
n\end{matrix}\right]_{r}=\frac{[m]_{r}!}{[n]_{r}![m-n]_{r}!}$,

  * $\bullet$

for $\alpha\in\mathfrak{h}^{*}$, we define
$q^{\alpha}\in\operatorname{\mathrm{Char}}_{\mathbb{C}}(P)$ by
$q^{\alpha}(\omega)=q^{(\alpha,\omega)}$ (where $(\;,\,)$ has been
$\mathbb{C}$-linearly extended to $\mathfrak{h}$).

We will only work with unital algebras defined over $\mathbb{C}$, and
correspondingly all tensor products are algebraic tensor products over
$\mathbb{C}$. By a  _∗ -algebra_ $A$ will be meant an algebra $A$ endowed with
an anti-linear, anti-multiplicative involution ∗: $A\rightarrow A$. We further
assume that the reader is familiar with the theory of Hopf algebras. A _Hopf
∗-algebra_ $(H,\Delta)$ is a Hopf algebra whose underlying algebra $H$ is a
∗-algebra, and whose comultiplication $\Delta$ is a ∗-homomorphism. This
implies that the counit is a ∗-homomorphism, and that the antipode $S$ is
invertible with $S^{-1}(h)=S(h^{*})^{*}$ for all $h\in H$.

######  Definition 1.1.

For $\varepsilon,\eta\in\operatorname{\mathrm{Char}}_{\mathbb{R}}(Q^{+})$, we
define $U_{q}(\mathfrak{g};\varepsilon,\eta)$ as the universal unital
∗-algebra generated by couples of elements $X_{r}^{\pm}$, $r\in I$, as well as
elements $K_{\omega}$, $\omega\in P$, such that for all $r,s\in I$ and
$\omega,\chi\in P$, we have $K_{\omega}$ self-adjoint,
$(X_{r}^{+})^{*}=X_{r}^{-}$ and

  1. (K)q

$K_{\omega}$ is invertible and $K_{\chi}K_{\omega}^{-1}=K_{\chi-\omega}$,

  2. (T)q

$K_{\omega}X_{r}^{\pm}K_{\omega}^{-1}=q^{\frac{\pm(\omega,\alpha_{r})}{2}}X_{r}^{\pm}$,

  3. (S)q

$\sum\limits_{k=0}^{1-a_{rs}}(-1)^{k}\left[\begin{matrix}1-a_{rs}\\\
k\end{matrix}\right]_{r}(X_{r}^{\pm})^{k}X_{s}^{\pm}(X_{r}^{\pm})^{1-a_{rs}-k}=0$
for $r\neq s$,

  4. (C)${}^{q}_{\varepsilon,\eta}$

$[X_{r}^{+},X_{s}^{-}]=\delta_{rs}\frac{\varepsilon_{r}K_{\alpha_{r}}^{2}-\eta_{r}K_{\alpha_{r}}^{-2}}{q_{r}-q_{r}^{-1}}$.

When $\varepsilon=\eta=+$ (i.e. $\varepsilon_{r}=\eta_{r}=1$ for all $r$), we
will denote the underlying algebra as $U_{q}(\mathfrak{g})$. This is a slight
variation, obtained by considering $\frac{1}{2}P$ instead of $P$, of the
simply-connected version of the ordinary quantized universal enveloping
algebra of $\mathfrak{g}$ [4, Remark 9.1.3].

The $U_{q}(\mathfrak{g};\varepsilon,\varepsilon)$ are quantizations of the
∗-algebras $U(\mathfrak{g}_{\varepsilon})$, introduced in Appendix B. Note
also that the algebras $U_{q}(\mathfrak{g};0,+)$ are well-known to
algebraists, see [20, Section 3] and [17].

Remark that the algebras $U_{q}(\mathfrak{g};\varepsilon,\eta)$ with
$\prod_{r}\varepsilon_{r}\eta_{r}\neq 0$ are mutually isomorphic as unital
algebras. Indeed, as $A$ is invertible, we can choose
$b\in\operatorname{\mathrm{Char}}_{\mathbb{C}}(P)$ such that
$b_{\alpha_{r}}^{4}=\eta_{r}/\varepsilon_{r}$ for all $r\in\Phi^{+}$. We can
also choose $a\in\operatorname{\mathrm{Char}}_{\mathbb{C}}(Q^{+})$ such that
$a_{r}$ is a square root of
$b_{\alpha_{r}}^{2}\varepsilon_{r}=\frac{\eta_{r}}{b_{\alpha_{r}}^{2}}$. Then

$\displaystyle\phi:\ U_{q}(\mathfrak{g};\varepsilon,\eta)\rightarrow
U_{q}(\mathfrak{g}),\qquad\begin{cases}X_{r}^{+}\mapsto a_{r}X_{r}^{+},\\\
X_{r}^{-}\mapsto a_{r}X_{r}^{-},\\\ K_{\omega}\mapsto
b_{\omega}K_{\omega}\end{cases}$ (1.1)

is a unital isomorphism. However, unless
$\operatorname{\mathrm{sgn}}(\varepsilon_{r}\eta_{r})=+$ for all $r$, in which
case we can choose $b\in\operatorname{\mathrm{Char}}_{\mathbb{R}^{+}}(P)$ and
$a\in\operatorname{\mathrm{Char}}_{\mathbb{R}^{+}}(Q^{+})$, this rescaling
will not respect the ∗-structure.

We list some properties of the $U_{q}(\mathfrak{g};\varepsilon,\eta)$. The
proofs do not differ from those for the well-known case $U_{q}(\mathfrak{g})$,
cf. [30] and [16, Section 4].

######  Proposition 1.2.

  1. $1.$

Let $U_{q}(\mathfrak{n}^{\pm})$ be the unital subalgebra generated by the
$X_{r}^{\pm}$ inside $U_{q}(\mathfrak{g};\varepsilon,\eta)$. Then
$U_{q}(\mathfrak{n}^{\pm})$ is the universal algebra generated by elements
$X_{r}^{\pm}$ satisfying the relations $({\rm S})^{q}$, and in particular does
not depend on $\varepsilon$ or $\eta$.

  2. $2.$

Similarly, let $U_{q}(\mathfrak{b}^{\pm})$ be the subalgebra generated by
$U_{q}(\mathfrak{n}^{\pm})$ and $U(\mathfrak{h})$, where $U(\mathfrak{h})$ is
the algebra generated by all $K_{\omega}$. Then $U_{q}(\mathfrak{b}^{\pm})$ is
universal with respect to the relations $({\rm K})^{q}$, $({\rm T})^{q}$ and
$({\rm S})^{q}$.

  3. $3.$

$($Triangular decomposition$)$ The multiplication map gives an isomorphism

$\displaystyle U_{q}\big{(}\mathfrak{n}^{+}\big{)}\otimes
U(\mathfrak{h})\otimes U_{q}(\mathfrak{n}^{-})\rightarrow
U_{q}(\mathfrak{g};\varepsilon,\eta).$ (1.2)

In particular, the $U_{q}(\mathfrak{g};\varepsilon,\eta)$ are non-trivial.
Note that the above proposition allows one to identify
$U_{q}(\mathfrak{g};\varepsilon,\eta)$ and $U_{q}(\mathfrak{g})$ as _vector
spaces_ ,

$\displaystyle i_{\varepsilon,\eta}:\
U_{q}(\mathfrak{g};\varepsilon,\eta)\rightarrow U_{q}(\mathfrak{g}),\qquad
x_{+}x_{0}x_{-}\mapsto x_{+}x_{0}x_{-},\qquad x_{\pm}\in
U_{q}\big{(}\mathfrak{n}^{\pm}\big{)},\qquad x_{0}\in U(\mathfrak{h}).$

Hence $U_{q}(\mathfrak{g};\varepsilon,\eta)$ may also be viewed as
$U_{q}(\mathfrak{g})$ with a deformed product $m_{\varepsilon,\eta}$. Indeed,
arguing as in [21, Section 4], one can show that

$\displaystyle m_{\varepsilon,\eta}(x\otimes
y)=\omega_{\varepsilon}(x_{(1)},y_{(1)})x_{(2)}y_{(2)}\omega_{\eta}(x_{(3)},y_{(3)})$

for certain cocycles $\omega_{\varepsilon}$ on $U_{q}(\mathfrak{g})$.

Also the following proposition is immediate.

######  Proposition 1.3.

For each
$\varepsilon,\mu,\eta\in\operatorname{\mathrm{Char}}_{\mathbb{R}}(Q^{+})$,
there exists a unique unital ∗-homomorphism

$\displaystyle\Delta_{\varepsilon,\eta}^{\mu}:\
U_{q}(\mathfrak{g};\varepsilon,\eta)\rightarrow
U_{q}(\mathfrak{g};\varepsilon,\mu)\otimes U_{q}(\mathfrak{g};\mu,\eta)$

such that $\Delta_{\varepsilon,\eta}^{\mu}(K_{\omega})=K_{\omega}\otimes
K_{\omega}$ for all $\omega\in P$ and

$\displaystyle\Delta_{\varepsilon,\eta}^{\mu}\big{(}X_{r}^{\pm}\big{)}=X_{r}^{\pm}\otimes
K_{\alpha_{r}}+K_{\alpha_{r}}^{-1}\otimes
X_{r}^{\pm},\qquad\forall\,r\in\Phi^{+}.$

###### Proof.

From Proposition 1.2 and the case $\varepsilon=\eta=+$, we know that
$\Delta_{\varepsilon,\eta}^{\mu}$ respects the relations $({\rm K})^{q}$,
$({\rm T})^{q}$ and $({\rm S})^{q}$. The compatibility with the relations
$({\rm C})^{q}_{\varepsilon,\eta}$ is verified directly. ∎

######  Lemma 1.4.

The collection
$\big{(}\\{U_{q}(\mathfrak{g};\varepsilon,\eta)\\},\\{\Delta_{\varepsilon,\eta}^{\mu}\\}\big{)}$
forms a connected cogroupoid over the index set
$\operatorname{\mathrm{Char}}_{\mathbb{R}}(Q^{+})\cong\mathbb{R}^{l}$ $($cf.
Appendix A).

###### Proof.

One immediately checks the coassociativity condition on the generators. One
further can define uniquely a unital ∗-homomorphism

$\displaystyle\epsilon_{\varepsilon}:\
U_{q}(\mathfrak{g};\varepsilon,\varepsilon)\rightarrow\mathbb{C},\qquad\begin{cases}X_{r}^{\pm}\mapsto
0,\\\ K_{\omega}\mapsto 1\end{cases}$

and unital anti-homomorphism

$\displaystyle S_{\varepsilon,\eta}:\
U_{q}(\mathfrak{g};\varepsilon,\eta)\rightarrow
U_{q}(\mathfrak{g};\eta,\varepsilon),\qquad\begin{cases}X_{r}^{\pm}\mapsto-q^{\pm
1}X_{r}^{\pm},\\\ K_{\omega}\mapsto K_{\omega}^{-1}.\end{cases}$

Again, one verifies on generators that these maps satisfy the counit and
antipode condition on generators, hence on all elements. ∎

We will denote by $\lhd$ the associated adjoint action of
$U_{q}(\mathfrak{g};\eta,\eta)$ on $U_{q}(\mathfrak{g};\varepsilon,\eta)$, cf.
Appendix A, Definition A.2.

##  2 Verma module theory

We keep the notation of the previous section.

######  Definition 2.1.

For $\lambda\in\operatorname{\mathrm{Char}}_{\mathbb{C}}(P)$, we denote by
$\mathbb{C}_{\lambda}$ the one-dimensional left
$U_{q}(\mathfrak{b}^{+})$-module associated to the character

$\displaystyle\chi_{\lambda}:\
U_{q}\big{(}\mathfrak{b}^{+}\big{)}\rightarrow\mathbb{C},\qquad\begin{cases}X_{r}^{+}\mapsto
0,\\\ K_{\omega}\mapsto\lambda_{\omega}.\end{cases}$

We denote

$\displaystyle
M_{\lambda}^{\varepsilon,\eta}=U_{q}(\mathfrak{g};\varepsilon,\eta)\underset{U_{q}(\mathfrak{b}^{+})}{\otimes}\mathbb{C}_{\lambda}.$

We denote by $V_{\lambda}^{\varepsilon,\eta}$ the simple quotient of
$M_{\lambda}^{\varepsilon,\eta}$ by its maximal non-trivial submodule. Note
that such a maximal submodule exists by the triangular decomposition (1.2), as
then any non-trivial submodule is a sum of weight spaces with weights distinct
from $\lambda$. We denote by $v_{\lambda}^{\varepsilon,\eta}$ the highest
weight vector $1\otimes 1$ in either $M_{\lambda}^{\varepsilon,\eta}$ or
$V_{\lambda}^{\varepsilon,\eta}$.

For $\lambda\in\operatorname{\mathrm{Char}}_{\mathbb{R}}(P)$, we can introduce
on $V_{\lambda}^{\varepsilon,\eta}$ a non-degenerate Hermitian form
$\langle\;,\,\rangle$ such that $\langle xv,w\rangle=\langle v,x^{*}w\rangle$
for all $x\in U_{q}(\mathfrak{g};\varepsilon,\eta)$ and $v,w\in
V_{\lambda}^{\varepsilon,\eta}$. We will call a form satisfying this property
_invariant_. Such a form is then unique up to a scalar. The construction of
this form is the same as in the case of $U_{q}(\mathfrak{g})$ [16, Section 5],
[31, Section 2.1.5]. Namely, let $U_{q}(\mathfrak{n}^{\pm})_{+}\subseteq
U_{q}(\mathfrak{n}^{\pm})$ be the kernel of the restriction of the counit on
$U_{q}(\mathfrak{b}^{\pm})$, and consider the orthogonal decomposition

$\displaystyle
U_{q}(\mathfrak{g};\varepsilon,\eta)=U(\mathfrak{h})\oplus\big{(}U_{q}\big{(}\mathfrak{n}^{-}\big{)}_{+}U_{q}(\mathfrak{g};\varepsilon,\eta)+U_{q}(\mathfrak{g};\varepsilon,\eta)U_{q}(\mathfrak{n}^{+})_{+}\big{)}.$
(2.1)

Let $\tau$ denote the projection onto the first summand. Then one first
observes that one has a Hermitian form on $M_{\lambda}^{\varepsilon,\eta}$ by
defining

$\displaystyle\langle
xv_{\lambda}^{\varepsilon,\eta},yv_{\lambda}^{\varepsilon,\eta}\rangle=\chi_{\lambda}(\tau(x^{*}y)),\qquad
x,y\in U_{q}(\mathfrak{g}).$

This is clearly a well-defined invariant form. It necessarily descends to a
non-degenerate Hermitian form on $V_{\lambda}^{\varepsilon,\eta}$.

The goal is to find necessary and sufficient conditions for this Hermitian
form to be positive-definite, in which case the module is called
_unitarizable_. This is in general a hard problem. In the following, we
present some partial results, restricting to the case $\eta=+$. We always
assume that $\lambda$ is real-valued, unless otherwise mentioned, and that
$M_{\lambda}^{\varepsilon,\eta}$ and $V_{\lambda}^{\varepsilon,\eta}$ have
been equipped with the above canonical Hermitian form.

We first consider the case $\varepsilon=\eta=+$, for which the following
result is well-known.

######  Lemma 2.2.

Let $\lambda\in\operatorname{\mathrm{Char}}_{\mathbb{R}^{+}}(P)$. Then the
following are equivalent.

  1. $1.$

The module $V_{\lambda}^{+,+}$ is unitarizable.

  2. $2.$

For all $r\in I$, $\lambda_{\alpha_{r}}^{4}\in q_{r}^{2\mathbb{N}}$.

  3. $3.$

$V_{\lambda}^{+,+}$ is finite-dimensional.

###### Proof.

For the implications $(2)\Rightarrow(3)\Rightarrow(1)$, see e.g. [4, Corollary
10.1.15, Proposition 10.1.21]. Assume now that $V_{\lambda}^{+,+}$ is
unitarizable. Then $V_{\lambda_{r}}^{+,+}$ is a unitarizable representation of
$U_{q_{r}}(\mathfrak{su}(2))$. By a simple computation using the commutation
between the $X_{r}^{\pm}$ (cf. [9]), we have that

$\displaystyle\langle(X_{r}^{+})^{k}(X_{r}^{-})^{k}v_{\lambda_{r}}^{+,+},v_{\lambda_{r}}^{+,+}\rangle=\prod_{l=1}^{k}\frac{\big{(}q_{r}^{l}-q_{r}^{-l}\big{)}\big{(}q_{r}^{-l+1}\lambda_{\alpha_{r}}^{2}-q_{r}^{l-1}\lambda_{\alpha_{r}}^{-2}\big{)}}{\big{(}q_{r}-q_{r}^{-1}\big{)}^{2}}.$

Hence, by unitarity and the fact that $0<q_{r}<1$, we find that
$\prod\limits_{l=1}^{k}\big{(}q_{r}^{l-1}\lambda_{\alpha_{r}}^{-2}-q_{r}^{-l+1}\lambda_{\alpha_{r}}^{2}\big{)}\geq
0$ for all $k\geq 0$. This is only possible if eventually one of the factors
becomes zero, i.e. when $\lambda_{\alpha_{r}}^{4}\in q_{r}^{2\mathbb{N}}$. ∎

By a limit argument, we now extend this result to the case
$\varepsilon\in\operatorname{\mathrm{Char}}_{\\{0,1\\}}(Q^{+})$.

######  Proposition 2.3.

Suppose $\varepsilon\in\operatorname{\mathrm{Char}}_{\\{0,1\\}}(Q^{+})$, and
let $\lambda\in\operatorname{\mathrm{Char}}_{\mathbb{R}^{+}}(P)$. Then
$V_{\lambda}^{\varepsilon,+}$ is unitarizable if and only if
$\lambda_{\alpha_{r}}^{4}\in q_{r}^{2\mathbb{N}}$ for all $r\in I$ with
$\varepsilon_{r}=1$.

###### Proof.

The ‘only if’ part of the proposition is obvious, since the
$K_{\alpha_{r}},X_{r}^{\pm}$ with $\varepsilon_{r}=1$ generate a quantized
enveloping algebra
$\widetilde{U}_{q}(\mathfrak{k}_{\operatorname{\mathrm{ss}}})$ in its compact
(non-simply connected) form.

To show the opposite direction, consider first general
$\varepsilon,\eta\in\operatorname{\mathrm{Char}}_{\mathbb{R}}(Q^{+})$. For
$\alpha\in Q^{+}$, denote by $U_{q}(\mathfrak{n}^{-})(\alpha)$ the finite-
dimensional space of elements $X\in U_{q}(\mathfrak{n}^{-})$ with
$K_{\omega}XK_{\omega}^{-1}=q^{-\frac{1}{2}(\omega,\alpha)}X$. Identifying
$U_{q}(\mathfrak{n}^{-})$ with $M_{\lambda}^{\varepsilon,\eta}$ in the
canonical way by means of $x\mapsto xv_{\lambda}^{\varepsilon,\eta}$, we may
interpret the Hermitian forms on the $M_{\lambda}^{\varepsilon,\eta}$ as a
family of Hermitian forms
$\langle\,\cdot,\cdot\,\rangle^{\varepsilon,\eta}_{\lambda}$ on
$U_{q}(\mathfrak{n}^{-})$. It is easily seen that these vary continuously with
$\varepsilon,\eta$ and $\lambda$ on each $U_{q}(\mathfrak{n}^{-})(\alpha)$.
Assume now that $\eta=+$ and $\varepsilon_{r}>0$ for all $r$. In this case
$U_{q}(\mathfrak{g};\varepsilon,+)$ is isomorphic to $U_{q}(\mathfrak{g})$ as
a ∗-algebra by a rescaling of the generators by positive numbers, cf. (1.1).
By Lemma 2.2, $\langle\,\cdot,\cdot\,\rangle^{+,+}_{\lambda}$ is positive
semi-definite if and only if $V_{\lambda}^{+,+}$ is finite-dimensional, and
the the latter happens if and only if $\lambda_{\alpha_{r}}^{4}\in
q_{r}^{2\mathbb{N}}$ for all $r\in I$. Hence we get by the above rescaling
that $\langle\,\cdot,\cdot\,\rangle^{\varepsilon,+}_{\lambda}$ is positive
semi-definite if and only if $\lambda_{\alpha_{r}}^{4}\in
q_{r}^{2\mathbb{N}}\varepsilon_{r}^{-1}$ for all $r$.

Fix now a subset $S$ of the simple positive roots, and put $\varepsilon_{r}=1$
for $r\in S$. Assume that $\lambda_{\alpha_{r}}^{4}\in q_{r}^{2\mathbb{N}}$
when $r\in S$. For $r\notin S$ and $m_{r}\in\mathbb{N}$, define
$\varepsilon_{r}=q_{r}^{2m_{r}}\lambda_{\alpha_{r}}^{-4}$. From the above, we
obtain that $\langle\,\cdot,\cdot\,\rangle^{\varepsilon,+}_{\lambda}$ is
positive semi-definite. Taking the limit $m_{r}\rightarrow\infty$ for $r\notin
S$, we deduce that $\langle\,\cdot,\cdot\,\rangle^{\varepsilon,+}_{\lambda}$
is positive semi-definite for $\varepsilon_{r}=\delta_{r\in S}$. ∎

The above $V_{\lambda}^{\varepsilon,+}$ with pre-Hilbert space structure can
also be presented more concretely as generalized Verma modules (from which it
will be clear that they are not finite dimensional when $\varepsilon_{r}=0$
for some $r$). We will need some preparations, obtained from modifying
arguments in [16]. Note that to pass from the conventions in [16] to ours, the
$q$ in [16] has to be replaced by $q^{1/2}$, and the coproduct in [16] is also
opposite to ours. However, as [16] gives preference to the left adjoint
action, while we work with the right adjoint action, most of our formulas will
in fact match.

We start with recalling a basic fact.

######  Lemma 2.4.

Let $\omega\in P^{+}$,
$\varepsilon\in\operatorname{\mathrm{Char}}_{\mathbb{R}}(Q^{+})$, and consider
$K_{\omega}^{-4}\in U_{q}(\mathfrak{g};\varepsilon,+)$. Then
$K_{\omega}^{-4}\lhd U_{q}(\mathfrak{g})$ is finite-dimensional, where $\lhd$
is the adjoint action $($cf. Definition A.2).

###### Proof.

One checks that the arguments of [16, Lemma 6.1, Lemma 6.2, Proposition 6.3,
Proposition 6.5] are still valid in our setting. ∎

Let $\varepsilon\in\operatorname{\mathrm{Char}}_{\mathbb{R}}(Q^{+})$. Recall
that the map $\tau$ denoted the projection onto the first summand in (2.1).
Let $\tau_{Z,\varepsilon}$ be the restriction of $\tau$ to the center
$\mathscr{Z}(U_{q}(\mathfrak{g};\varepsilon,+))$ of
$U_{q}(\mathfrak{g};\varepsilon,+)$. This is an $\varepsilon$-modified Harish-
Chandra map. The usual reasoning shows that this is a homomorphism into
$U(\mathfrak{h})$.

######  Lemma 2.5.

The map $\tau_{Z,\varepsilon}$ is a bijection between
$\mathscr{Z}(U_{q}(\mathfrak{g};\varepsilon,+))$ and the linear span of the
set $\left\\{\sum\limits_{w\in
W}\varepsilon_{\omega-w\omega}q^{(-2w\omega,\rho)}K_{-4w\omega}\,\Big{|}\,\omega\in
P^{+}\right\\}$, where $\rho=\sum_{r}\omega_{r}$ is the sum of the fundamental
weights and $W$ denotes the Weyl group of $\Delta$ with its natural action on
$\mathfrak{h}^{*}$.

###### Proof.

Note first that $\omega-w\omega$ is inside $Q^{+}$ for all $\omega\in P^{+}$
and $w\in W$, so that $\varepsilon_{\omega-w\omega}$ is well-defined.

Arguing as in [16, Section 8], we can assign to any $\omega\in P^{+}$ an
element $z_{\omega}$ in $\mathscr{Z}(U_{q}(\mathfrak{g};\varepsilon,+))$,
uniquely determined up to a non-zero scalar, such that

$\displaystyle K_{\omega}^{-4}\lhd
U_{q}(\mathfrak{g})=\mathbb{C}z_{\omega}+\left(K_{\omega}^{-4}\lhd
U_{q}(\mathfrak{g})_{+}\right),$

where $U_{q}(\mathfrak{g})_{+}$ is the kernel of the counit and where $\lhd$
denotes the adjoint action. More concretely, as $K_{\omega}^{-4}\lhd
U_{q}(\mathfrak{g})$ is a finite-dimensional right
$U_{q}(\mathfrak{g})$-module, it is semi-simple, and we have a projection $E$
of $K_{\omega}^{-4}\lhd U_{q}(\mathfrak{g})$ onto the space of its invariant
elements. We can then take $z_{\omega}=E\big{(}K_{\omega}^{-4}\big{)}$, and
$z_{\omega}\in\mathscr{Z}(U_{q}(\mathfrak{g};\varepsilon,+))$ by Lemma A.3.

Suppose now first that $\varepsilon_{r}\neq 0$ for all $r$, and choose
$b\in\operatorname{\mathrm{Char}}_{\mathbb{C}}(P)$ such that
$b_{\alpha_{r}}^{4}=\varepsilon_{r}^{-1}$. Consider

$\displaystyle\Psi_{\varepsilon}:\
U_{q}(\mathfrak{g};\varepsilon,+)\rightarrow
U_{q}(\mathfrak{g}),\qquad\begin{cases}X_{r}^{\pm}\mapsto
b_{\alpha_{r}}^{-1}X_{r}^{\pm},\\\ K_{\beta}\mapsto
b_{\beta}K_{\beta},\qquad\beta\in P.\end{cases}$

This is a unital $\lhd$-equivariant algebra isomorphism, and
$\tau_{Z,\varepsilon}=\Psi_{\varepsilon}^{-1}\circ\tau_{Z,+}\circ\Psi_{\varepsilon}$.
Hence by $\lhd$-equivariance, we find from the computations in [16, Section 8]
that, for a non-zero $\varepsilon$-independent scalar $c_{\omega}$,

$\displaystyle\tau_{Z,\varepsilon}(z_{\omega})=c_{\omega}\sum_{\nu\in
P^{+}}\dim((V_{\omega})_{\nu})\left(\sum_{w\in
W}b_{\omega-w\nu}^{-4}q^{-2(w\nu,\rho)}K_{-4w\nu}\right),$

where $(V_{\omega})_{\nu}$ denotes the weight space at $q^{\frac{1}{2}\nu}$
(i.e. the space of vectors on which the $K_{\omega}$ act as
$q^{\frac{1}{2}(\nu,\omega)}$) of the finite-dimensional
$U_{q}(\mathfrak{g})$-module $V_{\omega}$ with highest weight
$q^{\frac{1}{2}\omega}$. But clearly we then only have to sum over those $\nu$
with $\omega-\nu\in Q^{+}$, so that
$b_{\omega-w\nu}^{-4}=\varepsilon_{\omega-w\nu}$, and we can write

$\displaystyle\tau_{Z,\varepsilon}(z_{\omega})=c_{\omega}\underset{\omega-\nu\in
Q^{+}}{\sum_{\nu\in P^{+}}}\dim((V_{\omega})_{\nu})\left(\sum_{w\in
W}\varepsilon_{\omega-w\nu}q^{-2(w\nu,\rho)}K_{-4w\nu}\right).$ (2.2)

Recall now that $U_{q}(\mathfrak{g};\varepsilon,\eta)$ can be identified with
$U_{q}(\mathfrak{g})$ as a vector space by a map $i_{\varepsilon,\eta}$. Let
us denote by $\lhd_{\varepsilon,\eta}$ the image of $\lhd$ under this map.
Then for $x,y\in U_{q}(\mathfrak{g})$ fixed, it is easily seen from the
triangular decomposition that the $x\lhd_{\varepsilon,\eta}y$ live in a fixed
finite-dimensional subspace of $U_{q}(\mathfrak{g})$ as the $\varepsilon,\eta$
vary, and the resulting map $(\varepsilon,\eta)\mapsto
x\lhd_{\varepsilon,\eta}y$ is then continuous. Furthermore, if $V$ is a
finite-dimensional right $U_{q}(\mathfrak{g})$-module with space of fixed
elements $V_{\operatorname{\mathrm{triv}}}$, we can find $p\in
U_{q}(\mathfrak{g})$ such that for any $v\in V$, the element $vp$ is the
projection of $v$ onto $V_{\operatorname{\mathrm{triv}}}$. It follows from the
previous paragraph and the above remarks that when $\varepsilon_{r}\neq 0$ for
any $r$, we have
$i_{\varepsilon,+}(z_{\omega})=K_{\omega}^{-4}\lhd_{\varepsilon,+}p_{\omega}$
for some _fixed_ $p_{\omega}\in U_{q}(\mathfrak{g})$. By continuity, it then
follows that (2.2) in fact holds for arbitrary
$\varepsilon\in\operatorname{\mathrm{Char}}_{\mathbb{R}}(Q^{+})$.

The conclusion of the argument now follows as in [16, Theorem 8.6]. ∎

Let now $S\subseteq\Phi^{+}$, and let $\varepsilon$ extend the characteristic
function of $S$. Let $U_{q}(\mathfrak{t}_{S})$ be the Hopf ∗-subalgebra of
$U_{q}(\mathfrak{g};\varepsilon,+)$ generated by the $K_{\omega_{r}}^{\pm}$
with $1\leq r\leq l$ and $X_{r}^{\pm}$ with $r\in S$. Let
$U_{q}(\mathfrak{q}^{+}_{S})$ be the Hopf subalgebra of
$U_{q}(\mathfrak{g};\varepsilon,+)$ generated by $U_{q}(\mathfrak{t}_{S})$ and
all $X_{r}^{+}$ with $r\in I$. It is easy to see that
$U_{q}(\mathfrak{q}^{+}_{S})$ can be isomorphically imbedded into
$U_{q}(\mathfrak{g})$. Let $V$ be a finite-dimensional highest weight
representation of $U_{q}(\mathfrak{t}_{S})$ associated to a character in
$\operatorname{\mathrm{Char}}_{\mathbb{R}^{+}}(P)$. Then we can extend this to
a representation of $U_{q}(\mathfrak{q}^{+}_{S})$ on $V$ [31, Section 2.3.1],
and hence we can form

$\displaystyle\operatorname{\mathrm{Ind}}_{\varepsilon}(V)=U_{q}(\mathfrak{g};\varepsilon,+)\underset{U_{q}(\mathfrak{q}^{+}_{S})}{\otimes}V.$

The following proposition complements Proposition 2.3.

######  Proposition 2.6.

Let $S\subseteq\Phi^{+}$, and let $\varepsilon$ restrict to the characteristic
function of $S$. Let $V$ be an irreducible highest weight representation of
$U_{q}(\mathfrak{t}_{S})$ associated to a character
$\lambda\in\operatorname{\mathrm{Char}}_{\mathbb{R}^{+}}(P)$. Then the
$U_{q}(\mathfrak{g};\varepsilon,+)$-representation
$\operatorname{\mathrm{Ind}}_{\varepsilon}(V)$ is irreducible.

###### Proof.

By its universal property, $\operatorname{\mathrm{Ind}}_{\varepsilon}(V)$ can
be identified with a quotient of $M_{\lambda}^{\varepsilon,+}$. We want to
show that this quotient coincides with $V_{\lambda}^{\varepsilon,+}$.

Suppose that $v_{\lambda^{\prime}}$ is a highest weight vector inside
$M_{\lambda}^{\varepsilon,+}$ at weight $\lambda^{\prime}$ different from
$\lambda$. By Lemma 2.5, it follows that

$\displaystyle\sum_{w\in
W}\varepsilon_{\omega-w\omega}q^{(-2w\omega,\rho)}\lambda_{-4w\omega}^{\prime}=\sum_{w\in
W}\varepsilon_{\omega-w\omega}q^{(-2w\omega,\rho)}\lambda_{-4w\omega}$

for all $\omega\in P^{+}$. Now if $w=s_{\alpha_{i_{1}}}\cdots
s_{\alpha_{i_{p}}}$ in reduced form, with $s_{\alpha}$ the reflection across
the root $\alpha$, we have

$\displaystyle\omega-w\omega=\sum_{t=1}^{p}s_{\alpha_{i_{1}}}\cdots
s_{\alpha_{i_{t-1}}}(\omega-s_{\alpha_{i_{t}}}\omega),$

where each term is positive. It follows that we have

$\displaystyle\sum_{w\in
W_{S}}q^{(-2w\omega,\rho)}\lambda_{-4w\omega}^{\prime}=\sum_{w\in
W_{S}}q^{(-2w\omega,\rho)}\lambda_{-4w\omega}$

for all _strictly_ dominant $\omega$, where $W_{S}$ is the Coxeter group
generated by reflections around simple roots $\alpha_{s}$ with $s\in S$.

Taking $\omega=\rho+\omega_{r}$ with $r\notin S$, we get
$(\lambda_{-4\omega_{r}}^{\prime}-\lambda_{-4\omega_{r}})C=0$ with

$\displaystyle C=\sum_{w\in W_{S}}q^{(-2w\rho,\rho)}\lambda_{-4w\rho}>0.$

Hence $\lambda_{\omega_{r}}^{\prime}=\lambda_{\omega_{r}}$ for all $r\notin
S$. We deduce that $v_{\lambda^{\prime}}\in
U_{q}(\mathfrak{t}_{S})v_{\lambda}$, and so the image of
$v_{\lambda^{\prime}}$ in $\operatorname{\mathrm{Ind}}_{\varepsilon}(V)$ is
zero. This implies that
$\operatorname{\mathrm{Ind}}_{\varepsilon}(V)=V_{\lambda}^{\varepsilon,+}$. ∎

The case $\varepsilon\in\operatorname{\mathrm{Char}}_{\\{-1,0,1\\}}(Q^{+})$ is
not so easy to treat in general. In the following, we will restrict ourselves
to the case where we have one $\varepsilon_{t}<0$ while $\varepsilon_{r}\geq
0$ for $r\neq t$. This will in particular comprise the ‘symmetric Hermitian’
case.

######  Theorem 2.7.

Let $\varepsilon$ be such that there is a unique simple root $\alpha_{t}$ with
$\varepsilon_{t}<0$, while $\varepsilon_{r}\in\\{0,1\\}$ for $r\neq t$. Let
$\lambda\in\operatorname{\mathrm{Char}}_{\mathbb{R}^{+}}(P)$. Then
$V_{\lambda}^{\varepsilon,+}$ is unitarizable if and only if
$\lambda_{\alpha_{r}}^{4}\in q_{r}^{2\mathbb{N}}$ for all $r$ with
$\varepsilon_{r}\neq 1$.

###### Proof.

By a same kind of limiting argument as in Proposition 2.3, the general case
can be deduced from the case with $\varepsilon_{r}=1$ for $r\neq t$.

Suppose then that $\varepsilon_{t}<0$ and $\varepsilon_{r}=1$ for $r\neq t$.
Write $S=\Phi^{+}\setminus\\{t\\}$. We have the algebra automorphism
$\phi:U_{q}(\mathfrak{g};\varepsilon,+)\rightarrow U_{q}(\mathfrak{g})$
appearing in (1.1). By means of this isomorphism, we obtain a natural
isomorphism $M_{\lambda}^{\varepsilon,+}\cong M_{\gamma}^{+,+}$, where
$\gamma\in\operatorname{\mathrm{Char}}_{\mathbb{C}}(P)$ is such that
$\gamma_{\alpha_{r}}^{4}=\varepsilon_{r}\lambda_{\alpha_{r}}^{4}$ for all $r$.
In particular, $\gamma_{\alpha_{t}}^{2}\in\mathbb{C}\setminus\mathbb{R}$. It
follows from [18, Proposition 5.13] that
$\operatorname{\mathrm{Ind}}_{\varepsilon}(V_{\lambda})$ is irreducible, where
$V_{\lambda}$ denotes the irreducible representation of
$U_{q}(\mathfrak{q}^{+}_{S})$ at highest weight $\lambda$. Hence the
signatures of the Hermitian inner products on the
$\operatorname{\mathrm{Ind}}_{\varepsilon}(V_{\lambda})$ are constant as
$\varepsilon_{t}<0$ varies. Indeed, these spaces can be identified canonically
with a fixed quotient of $U_{q}(\mathfrak{n}^{-})$, see [31, Proposition
2.81], and then the Hermitian inner products clearly form a continuous family
as $\varepsilon$ varies.

From Proposition 2.6, we know that the Hermitian inner product on
$\operatorname{\mathrm{Ind}}_{\varepsilon}(V_{\lambda})$ for
$\varepsilon_{t}=0$ is positive definite. It follows that the Hermitian inner
product on a weight space of
$\operatorname{\mathrm{Ind}}_{\varepsilon}(V_{\lambda})$ is positive for
$\varepsilon_{t}<0$ small. As the signature is constant, this holds for all
$\varepsilon_{t}<0$. ∎

##  3 Quantized homogeneous spaces

######  Definition 3.1.

For $\varepsilon,\eta\in\operatorname{\mathrm{Char}}_{\mathbb{R}}(Q^{+})$, we
denote by
$U_{q}(\mathfrak{g};\varepsilon,\eta)_{\operatorname{\mathrm{f\/in}}}$ the
space of locally finite vectors in $U_{q}(\mathfrak{g};\varepsilon,\eta)$ with
respect to the right adjoint action by $U_{q}(\mathfrak{g};\eta,\eta)$ (cf.
Definition A.2),

$\displaystyle
U_{q}(\mathfrak{g};\varepsilon,\eta)_{\operatorname{\mathrm{f\/in}}}=\big{\\{}x\in
U_{q}(\mathfrak{g};\varepsilon,\eta)\,\big{|}\dim(x\lhd
U_{q}(\mathfrak{g};\eta,\eta))<\infty\big{\\}}.$

It is easily seen that the space
$U_{q}(\mathfrak{g};\varepsilon,\eta)_{\operatorname{\mathrm{f\/in}}}$ is a
∗-subalgebra of $U_{q}(g;\varepsilon,\eta)$ (cf. [16, Corollary 2.3]), and in
the following it will always be treated as a right
$U_{q}(\mathfrak{g})$-module by $\lhd$.

A similar definition of
${}_{\operatorname{\mathrm{f\/in}}}U_{q}(\mathfrak{g};\varepsilon,\eta)$ can
be made with respect to the left adjoint action of
$U_{q}(\mathfrak{g};\varepsilon,\varepsilon)$, and the two resulting algebras
$U_{q}(\mathfrak{g};\varepsilon,\eta)_{\operatorname{\mathrm{f\/in}}}$ and
${}_{\operatorname{\mathrm{f\/in}}}U_{q}(\mathfrak{g};\varepsilon,\eta)$
should in some sense be seen as dual to each other. For example, the
$U_{q}(\mathfrak{g};\varepsilon,+)_{\operatorname{\mathrm{f\/in}}}$ will lead
to compact quantum homogeneous spaces, while the
${}_{\operatorname{\mathrm{f\/in}}}U_{q}(\mathfrak{g};\varepsilon,+)$ should
lead to non-compact quantum homogeneous spaces such as quantum bounded
symmetric domains [31]. However, in this paper we will restrict ourselves to
the compact case.

The $U_{q}(\mathfrak{g};\varepsilon,\eta)_{\operatorname{\mathrm{f\/in}}}$ are
sufficiently large, as the next proposition shows, extending Lemma 2.4.

######  Proposition 3.2.

As a right $U_{q}(\mathfrak{g})$-module,
$U_{q}(\mathfrak{g};\varepsilon,+)_{\operatorname{\mathrm{f\/in}}}$ is
generated by the $K_{\omega}^{-4}$ with $\omega\in P^{+}$. The algebra
generated by
$U_{q}(\mathfrak{g};\varepsilon,+)_{\operatorname{\mathrm{f\/in}}}$ and the
$K_{\omega_{r}}^{4}$ equals the subalgebra of
$U_{q}(\mathfrak{g};\varepsilon,+)$ generated by the $K_{\omega_{r}}^{\pm 4}$
and the $K_{\alpha_{r}}X_{r}^{\pm}$.

###### Proof.

Again, the proof of [16, Theorem 6.4] can be directly modified. ∎

Note that for $\varepsilon_{r}\neq 0$ for all $r$, the above proposition
follows more straightforwardly from [16] by a rescaling argument.

The dependence of
$U_{q}(\mathfrak{g};\varepsilon,\eta)_{\operatorname{\mathrm{f\/in}}}$ on
$\varepsilon$ and $\eta$ is weaker than for
$U_{q}(\mathfrak{g};\varepsilon,\eta)$ itself. We consider a special case in
the following lemma. Recall that $A$ denotes the Cartan matrix.

######  Lemma 3.3.

Consider
$\varepsilon,\eta\in\operatorname{\mathrm{Char}}_{\mathbb{R}\setminus\\{0\\}}(Q^{+})$,
and write
$\operatorname{\mathrm{sgn}}(\varepsilon_{r}/\eta_{r})=(-1)^{\chi_{r}}$. If
$\chi$ is in the range of $A\,\mathrm{mod}\,2$, then
$U_{q}(\mathfrak{g};\varepsilon,+)_{\operatorname{\mathrm{f\/in}}}\cong
U_{q}(\mathfrak{g};\eta,+)_{\operatorname{\mathrm{f\/in}}}$ as right
$U_{q}(\mathfrak{g})$-module ∗-algebras.

###### Proof.

Choose $b\in\operatorname{\mathrm{Char}}_{\mathbb{C}}(P)$ such that
$b_{\alpha_{r}}^{4}=\eta_{r}/\varepsilon_{r}$. Then

$\displaystyle\Psi_{\varepsilon,\eta}:\
U_{q}(\mathfrak{g};\varepsilon,+)\rightarrow
U_{q}(\mathfrak{g};\eta,+),\qquad\begin{cases}X_{r}^{\pm}\mapsto
b_{\alpha_{r}}^{-1}X_{r}^{\pm},\\\ K_{\omega}\mapsto
b_{\omega}K_{\omega}\end{cases}$

is a unital $\lhd$-equivariant algebra isomorphism. Hence
$\Psi_{\varepsilon,\eta}$ induces a unital $\lhd$-equivariant algebra
isomorphism
$\psi_{\varepsilon,\eta}:U_{q}(\mathfrak{g};\varepsilon,+)_{\operatorname{\mathrm{f\/in}}}\rightarrow
U_{q}(\mathfrak{g};\eta,+)_{\operatorname{\mathrm{f\/in}}}$.

As the $K_{\omega}^{-4}$ with $\omega\in P^{+}$ generate
$U_{q}(\mathfrak{g};\varepsilon,+)_{\operatorname{\mathrm{f\/in}}}$ as a
module, $\psi_{\varepsilon,\eta}$ will be ∗-preserving if and only if
$b_{\alpha_{r}}^{4}\in\mathbb{R}$ for all $r$. This can be realized if we can
find $c_{r}\in\\{-1,1\\}$ such that
$\prod_{s}c_{s}^{a_{sr}}=\operatorname{\mathrm{sgn}}(\eta_{r}/\varepsilon_{r})$,
which is equivalent with the condition appearing in the statement of the
lemma. ∎

In particular, we find for example that
$U_{q}(\mathfrak{sl}(2m+1);\varepsilon,+)_{\operatorname{\mathrm{f\/in}}}$ for
$m\in\mathbb{N}_{0}$ is independent of the choice of
$\varepsilon\in\operatorname{\mathrm{Char}}_{\mathbb{R}\setminus\\{0\\}}(Q^{+})$.
On the other hand,
$U_{q}(\mathfrak{sl}(2);\varepsilon,+)_{\operatorname{\mathrm{f\/in}}}$ are
mutually non-isomorphic as ∗-algebras for $\varepsilon\in\\{-1,0,1\\}$, see
[8].

######  Definition 3.4.

Let $\varepsilon,\eta\in\operatorname{\mathrm{Char}}_{\mathbb{R}}(Q^{+})$,
$\lambda\in\operatorname{\mathrm{Char}}_{\mathbb{R}^{+}}(P)$, and let
$V_{\lambda}^{\varepsilon,\eta}$ be the irreducible highest weight module of
$U_{q}(\mathfrak{g};\varepsilon,\eta)$ at $\lambda$ with associated
representation $\pi_{\lambda}^{\varepsilon,\eta}$. We write

$\displaystyle
B_{\lambda}(\mathfrak{g};\varepsilon,\eta)=\pi_{\lambda}^{\varepsilon,\eta}(U_{q}(\mathfrak{g};\varepsilon,\eta))$

and

$\displaystyle
B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,\eta)=\pi_{\lambda}^{\varepsilon,\eta}(U_{q}(\mathfrak{g};\varepsilon,\eta)_{\operatorname{\mathrm{f\/in}}}).$

######  Remark 3.5.

The space
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,\eta)$
_is not_ defined as the space
$B_{\lambda}(\mathfrak{g};\varepsilon,\eta)_{\operatorname{\mathrm{f\/in}}}$
of locally finite $\lhd$-elements in
$B_{\lambda}(\mathfrak{g};\varepsilon,\eta)$, although conceivably they are
the same in many cases. In the case $q=1$, the equality of these two algebras
goes by the name of the Kostant problem, cf. [19, Remark 3].

######  Notation 3.6.

We will use the following notation for particular elements in the
$B_{\lambda}(\mathfrak{g};\varepsilon,+)$:

$\displaystyle
Z_{r}=\pi_{\lambda}^{\varepsilon,+}\big{(}K_{\omega_{r}}^{-4}\big{)},\qquad
X_{r}=q_{r}^{1/2}(q_{r}^{-1}-q_{r})\pi_{\lambda}^{\varepsilon,+}\big{(}K_{\alpha_{r}}K_{\omega_{r}}^{-4}X^{+}_{r}\big{)},$
$\displaystyle Y_{r}=X_{r}^{*},\qquad
W_{r}=\pi_{\lambda}^{\varepsilon,+}\big{(}K_{\alpha_{r}}^{4}K_{\omega_{r}}^{-8}\big{)},$
$\displaystyle
T_{r}=\big{(}q_{r}-q_{r}^{-1}\big{)}^{2}\pi_{\lambda}^{\varepsilon,+}\big{(}K_{\alpha_{r}}^{2}K_{\omega_{r}}^{-4}X_{r}^{+}X_{r}^{-}\big{)}+\varepsilon_{r}q_{r}^{-1}\pi_{\lambda}^{\varepsilon,+}\big{(}K_{\alpha_{r}}^{4}K_{\omega_{r}}^{-4}\big{)}+q_{r}\pi_{\lambda}^{\varepsilon,+}\big{(}K_{\omega_{r}}^{-4}\big{)}$
$\displaystyle\phantom{T_{r}}{}=\big{(}q_{r}-q_{r}^{-1}\big{)}^{2}\pi_{\lambda}^{\varepsilon,+}\big{(}K_{\alpha_{r}}^{2}K_{\omega_{r}}^{-4}X_{r}^{-}X_{r}^{+}\big{)}+\varepsilon_{r}q_{r}\pi_{\lambda}^{\varepsilon,+}\big{(}K_{\alpha_{r}}^{4}K_{\omega_{r}}^{-4}\big{)}+q_{r}^{-1}\pi_{\lambda}^{\varepsilon,+}\big{(}K_{\omega_{r}}^{-4}\big{)}.$

The following commutation relations will be needed later on.

######  Lemma 3.7.

The elements $W_{r}$ and $T_{r}$ commute with $X_{r}$, $Y_{r}$, $Z_{r}$,
$T_{r}$ and $W_{r}$. Moreover,

$\displaystyle X_{r}Z_{r}=q_{r}^{2}Z_{r}X_{r},\qquad
Y_{r}Z_{r}=q_{r}^{-2}Z_{r}Y_{r}$

and

$\displaystyle
X_{r}Y_{r}=-\varepsilon_{r}W_{r}+q_{r}T_{r}Z_{r}-q_{r}^{2}Z_{r}^{2},\qquad
Y_{r}X_{r}=-\varepsilon_{r}W_{r}+q_{r}^{-1}T_{r}Z_{r}-q_{r}^{-2}Z_{r}^{2}.$

We further have that $T_{r}$ and $W_{r}$ are invariant under $\lhd
X_{r}^{\pm}$ and $\lhd K_{\omega}$, while

$\displaystyle X_{r}\lhd X_{r}^{+}=0,\qquad Y_{r}\lhd
X_{r}^{+}=-q_{r}^{1/2}\big{(}q_{r}^{-1}+q_{r}\big{)}Z_{r}+q_{r}^{1/2}T_{r},$
$\displaystyle X_{r}\lhd
K_{\omega}=q^{-\frac{(\omega,\alpha_{r})}{2}}X_{r},\qquad Y_{r}\lhd
K_{\omega}=q^{\frac{(\omega,\alpha_{r})}{2}}X_{r},$ $\displaystyle X_{r}\lhd
X_{r}^{-}=q_{r}^{-1/2}\big{(}q_{r}^{-1}+q_{r}\big{)}Z_{r}-q_{r}^{-1/2}T_{r},\qquad
Y_{r}\lhd X_{r}^{-}=0$

and

$\displaystyle Z_{r}\lhd X_{r}^{+}=q_{r}^{1/2}X_{r},\qquad Z_{r}\lhd
K_{\omega}=0,\qquad Z_{r}\lhd X_{r}^{-}=-q_{r}^{-1/2}Y_{r}.$

Finally, all elements $X_{r}$, $Y_{r}$, $Z_{r}$, $T_{r}$, $W_{r}$ are inside
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$.

###### Proof.

All these assertions follow from straightforward computations. As the
$Z_{r}=\pi_{\lambda}^{\varepsilon,+}(K_{\omega_{r}}^{-4})$ and $W_{r}$ are in
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ by
Proposition 3.2, and the latter is $\lhd$-stable, it follows from the above
computations that also $X_{r}$, $Y_{r}$ and $T_{r}$ are in
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$. ∎

######  Proposition 3.8.

The only $\lhd$-invariant elements in
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ are
scalar multiples of the unit element.

###### Proof.

Assume that $x\in
U_{q}(\mathfrak{g};\varepsilon,+)_{\operatorname{\mathrm{f\/in}}}$ with
$\pi_{\lambda}^{\varepsilon,+}(x)$ invariant. As
$U_{q}(\mathfrak{g};\varepsilon,+)_{\operatorname{\mathrm{f\/in}}}$ is a semi-
simple right $U_{q}(\mathfrak{g})$-module, we have an equivariant projection
$E$ of $U_{q}(\mathfrak{g};\varepsilon,+)_{\operatorname{\mathrm{f\/in}}}$
onto the ∗-algebra of its invariant elements. The latter is simply the center
$\mathscr{Z}(U_{q}(\mathfrak{g};\varepsilon,+))$ of
$U_{q}(\mathfrak{g};\varepsilon,+)$, by Lemma A.3. As
$\pi_{\lambda}^{\varepsilon,+}$ is $\lhd$-equivariant by construction, we
deduce that
$\pi_{\lambda}^{\varepsilon,+}(x)=\pi_{\lambda}^{\varepsilon,+}(E(x))$. But
the latter is a scalar. ∎

######  Remark 3.9.

An alternative proof consists in applying Schur’s lemma to the simple module
$V_{\lambda}^{\varepsilon,+}$. Indeed, $x\in
B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ is
$\lhd$-invariant if and only if it commutes with all
$\pi_{\lambda}^{\varepsilon,+}(y)$ for $y\in
U_{q}(\mathfrak{g};\varepsilon,+)$. As $V_{\lambda}^{\varepsilon,+}$ is
simple, Schur’s lemma implies that the algebra of $\lhd$-invariant elements in
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$
forms a field of countable dimension over $\mathbb{C}$, hence coincides with
$\mathbb{C}$. (I would like the referee for pointing out this approach).

######  Proposition 3.10.

Let $(V,\pi)$ be a ∗-representation of
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ on a
pre-Hilbert space. Then $\pi$ is bounded.

The proof is based on an argument which is well-known in the setting of
compact quantum groups.

###### Proof.

As $B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$
consists of locally finite elements, any $b\in
B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ can
be written as a finite linear combination of elements $b_{i}\in
B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ for
which there exists a finite-dimensional ∗-representation $\pi$ of
$U_{q}(\mathfrak{g})$ on a Hilbert space such that $b_{i}\lhd
h=\sum_{j}\pi_{ij}(h)b_{j}$ for all $h\in U_{q}(\mathfrak{g})$, the $\pi_{ij}$
being the matrix components with respect to some orthogonal basis. An easy
computation shows that $\sum_{i}b_{i}^{*}b_{i}$ is an invariant element, hence
a scalar by Proposition 3.8. Hence there exists $C\in\mathbb{R}^{+}$ such that
for any $\xi\in V$ and any $i$, we have $\|\pi(b_{i})\xi\|\leq C\|\xi\|$. We
deduce that the element $\pi(b)$ is bounded. ∎

######  Definition 3.11.

A
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$-module
$V$ is called a  _highest weight module_ if there exists a cyclic vector $v\in
V$ which is annihilated by all $X_{r}$ and which is an eigenvector for all
$Z_{r}$ with non-zero eigenvalue. A pre-Hilbert space structure on $V$ is
called _invariant_ if $\langle x\xi,\eta\rangle=\langle\xi,x^{*}\eta\rangle$
for all $\xi,\eta\in V$ and $x\in
B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$.

We aim to show that the
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ have
only a finite number of non-equivalent irreducible highest weight modules. Of
course, each
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$
admits at least the highest weight module $V_{\lambda}^{\varepsilon,+}$. Also
note that, by an easy argument, each highest weight module decomposes into a
direct sum of joint weight spaces for the $Z_{r}$.

######  Proposition 3.12.

Each $B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$
admits only a finite number of non-equivalent irreducible highest weight
modules.

###### Proof.

As the statement does not depend on the ∗-structure, we may by rescaling
restrict to the case that $\varepsilon_{r}\in\\{0,1\\}$ for all $r$ upon
allowing $\lambda\in\operatorname{\mathrm{Char}}_{\mathbb{C}}(P)$.

By Proposition 3.2 and the fact that any highest weight module is semi-simple
for the torus part, it is easily argued that any irreducible highest weight
module of
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ is
obtained by restriction of a $U_{q}(\mathfrak{g};\varepsilon,+)$-module
$V_{\lambda^{\prime}}^{\varepsilon,+}$ for some
$\lambda^{\prime}\in\operatorname{\mathrm{Char}}_{\mathbb{C}}(P)$. As the
center of $U_{q}(\mathfrak{g};\varepsilon,+)$ acts by the same character on
$V_{\lambda}^{\varepsilon,+}$ and $V_{\lambda^{\prime}}^{\varepsilon,+}$, we
find by Lemma 2.5 that the expression $\sum\limits_{w\in
W}\varepsilon_{\omega-w\omega}q^{-2(w\omega,\rho)}\lambda_{-4w\omega}$ remains
the same upon replacing $\lambda$ by $\lambda^{\prime}$, for each $\omega\in
P^{+}$. Writing $S$ for the set of $r$ with $\varepsilon_{r}=0$, it follows as
in the proof of Lemma 2.5 that

$\displaystyle\sum_{w\in
W_{S}}q^{-2(w\omega,\rho)}\lambda_{-4w\omega}=\sum_{w\in
W_{S}}q^{-2(w\omega,\rho)}\lambda_{-4w\omega}^{\prime}$

for all $\omega\in P^{++}$, the strictly dominant weights. As (invertible)
characters on a commutative semi-group are linearly independent, and as
$P^{++}-P^{++}=P$, it follows that the functions $\omega\rightarrow
q^{-2(\omega,\rho)}\lambda_{-4\omega}$ and $\omega\rightarrow
q^{-2(\omega,\rho)}\lambda_{-4\omega}^{\prime}$ on $P$ lie in the same
$W_{S}$-orbit. As the highest weight vector in an irreducible highest weight
module is uniquely determined up to a scalar, and as the equivalence classes
of such highest weight modules are then determined by the associated
eigenvalue of the $Z_{r}=\pi_{\lambda}^{\varepsilon,+}(K_{\omega_{r}}^{-4})$,
this is sufficient to prove the proposition. ∎

Remark that the above proof also gives the upper bound $|W_{S}|$ for the
number of inequivalent highest weight representations, but of course this
estimate is not sharp if one only considers unitarizable representations.

######  Proposition 3.13.

Let $\pi$ be a ∗-representation of
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ on a
Hilbert space $\mathscr{H}$. If $0$ is not in the point-spectrum of any of the
$Z_{r}$, then $\mathscr{H}$ is a $($possibly infinite$)$ direct sum of
completions $(\mathscr{H}_{k},\pi_{k})$ of unitarizable highest weight modules
of $B^{\operatorname{\mathrm{f\/in}}}_{\lambda}(\mathfrak{g};\varepsilon,+)$.

###### Proof.

By a direct integral decomposition, and using Proposition 3.12, it is
sufficient to show that any such _irreducible_ ∗-representation $\pi$ of
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ on a
Hilbert space $\mathscr{H}$ is the completion of a highest weight module with
invariant pre-Hilbert space structure. We then argue as in [27, Section 3].
Write $\chi_{X}$ for the characteristic function of a set. By assumption,
there exists $t\in\mathbb{R}^{l}$ with $t_{r}\neq 0$ for all $r$ and
$P_{t}=\chi_{\prod_{r}[q_{r}t_{r},t_{r}]}(\pi(Z_{1}),\ldots,\pi(Z_{l}))$ non-
zero. Suppose now that $r$ is such that $\pi(X_{r})P_{t}\neq 0$. From the
commutation relations between the $X_{r}$ and the $Z_{s}$, we deduce that
$P_{(t_{1},\ldots,q_{r}^{-2}t_{r},\ldots,t_{l})}\neq 0$. As the $\pi(Z_{r})$
are bounded, this process must necessarily stop. Hence we may choose $t$ such
that $P_{t}\neq 0$ but $\pi(X_{r})P_{t}=0$ for all $r$.

Let $V$ be the union of the images of the spectral projections of
$(Z_{1},\ldots,Z_{l})$ corresponding to the
$\prod_{r}\left(\mathbb{R}\setminus(-\frac{1}{n},\frac{1}{n})\right)$ with
$n\in\mathbb{N}$. As
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ is
spanned by elements which skew-commute with the $Z_{r}$, it follows that $V$
is a
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$-module
on which the $\pi(Z_{r})$ are invertible linear maps. This entails that the
restriction of $\pi$ to $V$ can be extended to a representation
$\widetilde{\pi}$ of
$B^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)_{\operatorname{\mathrm{ext}}}$,
the sub∗-algebra of $B(\mathfrak{g};\varepsilon,+)$ generated by $X_{r}$,
$Y_{r}$ and the $Z_{r}^{\pm 1}$ (which contains
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ by
Proposition 3.2). Note that this ∗-algebra admits a triangular decomposition
(in the obvious way with respect to the above generators).

Pick now a non-zero $\xi\in P_{t}\mathscr{H}$. Suppose that $\xi$ were not in
the pure point spectrum of some $\pi(Z_{r})$. Then we can find
$q_{r}t_{r}<a<t_{r}$ such that $\chi_{[q_{r}t_{r},a]}(\pi(Z_{r}))\xi\neq
0\neq\chi_{(a,t_{r}]}(\pi(Z_{r}))\xi$. However,
$[q_{1}t_{1},t_{1}]\times\cdots\times[q_{r}t_{r},a]\times\cdots\times[q_{l}t_{l},t_{l}]\cap\prod_{s}[q_{s}^{2k_{s}+1}t_{s},q_{s}^{2k_{s}}t_{s}]=\varnothing$
for all $k_{s}\in\mathbb{N}$ with at least one $k_{s}>0$. From the commutation
relations between the $Y_{s}$ and $Z_{s^{\prime}}$, and the fact that
$\pi(X_{s})\xi=0$ for all $s$, we deduce that
$\chi_{[q_{r}t_{r},a]}(\pi(Z_{r}))\xi$ is orthogonal to the
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)_{\operatorname{\mathrm{ext}}}$-module
spanned by $\chi_{(a,t_{r}]}(\pi(Z_{r}))\xi$. As $\pi$ is irreducible, this
would entail $\chi_{[q_{r}t_{r},a]}(\pi(Z_{r}))\xi=0$. Having arrived at a
contradiction, we conclude that $\xi$ is a joint eigenvector of all
$\pi(Z_{r})$.

As $\xi$ is annihilated by all $\pi(X_{r})$ and is a joint eigenvector of all
$\pi(Z_{r})$, the module generated by it is a highest weight module. As $\pi$
was irreducible, this module must necessarily be dense in $\mathscr{H}$, and
the proposition is proven. ∎

We now want to consider analytic versions of the
$B^{\operatorname{\mathrm{f\/in}}}_{\lambda}(\mathfrak{g};\varepsilon,+)$.

######  Definition 3.14.

Let $B$ be a unital ∗-algebra. We say that $B$ admits a  _universal
$C^{*}$-envelope_ if there exists a non-trivial unital $C^{*}$-algebra $C$
together with a unital ∗-homomorphism $\pi_{u}:B\rightarrow C$ of unital
∗-algebras such that any ∗-homomorphism $B\rightarrow D$ with $D$ a unital
$C^{*}$-algebra factors through $C$.

Of course, the above $C^{*}$-algebra $C$ is then uniquely determined up to
isomorphism.

######  Definition 3.15.

We define $\operatorname{\mathrm{Pol}}(\mathbb{G}_{+}^{q})$ to be the Hopf
∗-algebra inside the dual of $U_{q}(\mathfrak{g})$ which is spanned by the
matrix coefficients of finite-dimensional highest weight representations of
$U_{q}(\mathfrak{g})$ associated to positive characters. We define

$\displaystyle\alpha_{\lambda}^{\varepsilon,+}:\
B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)\rightarrow\operatorname{\mathrm{Pol}}(\mathbb{G}_{+}^{q})\otimes
B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$

as the comodule ∗-algebra structure dual to the module ∗-algebra structure
$\lhd$ by $U_{q}(\mathfrak{g})$.

Note that the latter definition makes sense, since
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ is
integrable as a right $U_{q}(\mathfrak{g})$-module.

It is known [26] that $\operatorname{\mathrm{Pol}}(\mathbb{G}_{+}^{q})$ admits
a universal $C^{*}$-algebraic envelope $C(\mathbb{G}_{+}^{q})$, which becomes
a compact quantum group in the sense of [32]. We will denote by
$\varphi_{\mathbb{G}_{+}^{q}}$ the invariant state on $C(\mathbb{G}_{+}^{q})$,
which is faithful by co-amenability of $\mathbb{G}_{+}^{q}$.

######  Lemma 3.16.

Assume that
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$
admits at least one ∗-representation on a Hilbert space. Then
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$
admits a $C^{*}$-algebraic envelope $C_{\lambda}(\mathfrak{g};\varepsilon,+)$.

###### Proof.

The universal $C^{*}$-algebraic envelope of
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$
exists for precisely the same reason as in Proposition 3.10, since for any
element $b\in
B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ there
exists a universal constant $C_{b}$ such that $\|\pi(b)\|\leq C$ for all
∗-representations $\pi$ of
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ by
bounded operators on a Hilbert space. As
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$
admits at least one ∗-representation, we have
$C_{\lambda}(\mathfrak{g};\varepsilon,+)\neq 0$. ∎

######  Remark 3.17.

If $V_{\lambda}^{\varepsilon,+}$ is unitarizable, it is of course clear that
we get a  _faithful_ map from
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ into
$C_{\lambda}(\mathfrak{g};\varepsilon,+)$.

######  Lemma 3.18.

Let $C_{\lambda}(\mathfrak{g};\varepsilon,+)$ be the universal
$C^{*}$-envelope of
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$,
whenever it exists. Then $\alpha_{\lambda}^{\varepsilon,+}$ induces a
$C^{*}$-algebraic coaction by $C(\mathbb{G}_{+}^{q})$ on
$C_{\lambda}(\mathfrak{g};\varepsilon,+)$.

###### Proof.

The map $\alpha_{\lambda}^{\varepsilon,+}$ gives a $C^{*}$-representation of
$B^{\operatorname{\mathrm{f\/in}}}_{\lambda}(\mathfrak{g};\varepsilon,+)$ into
$C(\mathbb{G}_{+}^{q})\otimes C_{\lambda}(\mathfrak{g};\varepsilon,+)$, which
hence factors over $C_{\lambda}(\mathfrak{g};\varepsilon,+)$. It is
straightforward to argue that this is a $C^{*}$-algebraic coaction.∎

As $B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$
only had scalar multiples of the unit as invariants, it follows that the
coaction $\alpha_{\lambda}^{\varepsilon,+}$ on
$C_{\lambda}(\mathfrak{g};\varepsilon,+)$ is _ergodic_ [3], i.e. if
$\alpha_{\lambda}^{\varepsilon,+}(x)=1\otimes x$, then $x\in\mathbb{C}1$. We
will write $\varphi_{\lambda}^{\varepsilon,+}$ for the unique invariant state
on $C_{\lambda}(\mathfrak{g};\varepsilon,+)$, so

$\displaystyle\big{(}\varphi_{\mathbb{G}_{+}^{q}}\otimes\iota\big{)}\alpha_{\lambda}^{\varepsilon,+}(x)=\varphi_{\lambda}^{\varepsilon,+}(x)1,\qquad\forall\,x\in
C_{\lambda}(\mathfrak{g};\varepsilon,+).$

As $\varphi_{\mathbb{G}_{+}^{q}}$ is faithful, also
$\varphi_{\lambda}^{\varepsilon,+}$ is faithful.

######  Notation 3.19.

We write $\theta_{\lambda,\operatorname{\mathrm{reg}}}^{\varepsilon,{+}}$ for
the GNS-representation of
$(C_{\lambda}(\mathfrak{g};\varepsilon,\\!{+}),\varphi_{\lambda}^{\varepsilon,{+}})$,
and $W_{\lambda}(\mathfrak{g};\varepsilon,\\!{+})$ for the von Neumann
algebraic completion of $C_{\lambda}(\mathfrak{g};\varepsilon,\\!{+})$ in this
GNS-representation.

From Lemma 3.16, it follows that a $C^{*}$-algebraic envelope of
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$
exists if $V_{\lambda}^{\varepsilon,+}$ is unitarizable. In this case, we can
say something explicit about $W_{\lambda}(\mathfrak{g};\varepsilon,+)$.

######  Theorem 3.20.

Assume that $V_{\lambda}^{\varepsilon,+}$ is unitarizable. Then
$W_{\lambda}(\mathfrak{g};\varepsilon,+)$ is a finite direct sum of type $I$
factors.

The proof will make use of the following standard lemma.

######  Lemma 3.21.

Let $A$ be a unital $C^{*}$-algebra with faithful state $\varphi$. Let $M$ be
the von Neumann algebra closure of $A$ in its GNS-representation with respect
to $\varphi$. Let $\pi$ be a representation of $A$ on a Hilbert space
$\mathscr{H}$ such that there exists a faithful state $\omega\in
B(\mathscr{H})_{*}$ with $\omega\circ\pi=\varphi$. Then $\pi$ extends to a
normal faithful ∗-representation of $M$.

###### Proof.

As $\omega$ is faithful, the bicommutant $\pi(A)^{\prime\prime}$ is faithfully
represented on the GNS-space $\mathscr{L}^{2}(\pi(A),\omega)$. The unitary
$U:\mathscr{L}^{2}(A,\varphi)\rightarrow\mathscr{L}^{2}(\pi(A),\omega)$
induced by $\pi$ then provides an isomorphism
$M\rightarrow\pi(A)^{\prime\prime}$ extending $\pi$. ∎

###### Proof of Theorem 3.20.

As mentioned, the $C^{*}$-algebraic envelope
$C_{\lambda}(\mathfrak{g};\varepsilon,+)$ certainly exists.

To prove the remaining part of the theorem, we first make some preparations.
Recall that the dual of $U_{q}(\mathfrak{sl}(2,\mathbb{C}))$ can be identified
with $\operatorname{\mathrm{Pol}}(SU_{q}(2))$, Woronowicz’s twisted quantum
$SU(2)$-group [33]. It is well-known that its von Neumann algebraic envelope
$\mathscr{L}^{\infty}(SU_{q}(2))$ is isomorphic to
$B(l^{2}(\mathbb{N}))\otimes\mathscr{L}(\mathbb{Z})$, and as such admits a
faithful representation on the Hilbert space
$\mathscr{H}_{+}=l^{2}(\mathbb{N})\otimes l^{2}(\mathbb{Z})$ (cf. [26, 29]).

More generally, write $U_{q_{r}}(\mathfrak{su}(2))$ for the sub-Hopf-∗-algebra
generated by the $X_{r}^{\pm}$ and $K_{\alpha_{r}}^{\pm 1}$ inside
$U_{q}(\mathfrak{g})$. By duality, one obtains a surjective ∗-homomorphism
$\gamma_{r}:\operatorname{\mathrm{Pol}}(\mathbb{G}_{+}^{q})\rightarrow\operatorname{\mathrm{Pol}}(SU_{q_{r}}(2))$.
This induces a ∗-representation of
$\operatorname{\mathrm{Pol}}(\mathbb{G}_{+}^{q})$ on $\mathscr{H}_{+}$, which
we will denote by the same symbol $\gamma_{r}$. Suppose now that
$t=(r_{1},\ldots,r_{n})$ is an ordered $n$-tuple of elements in $I$. Then we
obtain a ∗-representation of $\operatorname{\mathrm{Pol}}(G_{+}^{q})$ on
$\mathscr{H}_{+}^{\otimes n}$ by means of the ∗-homomorphism

$\displaystyle\gamma_{t}=(\gamma_{r_{1}}\otimes\cdots\otimes\gamma_{r_{n}})\circ\Delta_{\mathbb{G}_{+}^{q}}^{(n)},$

where $\Delta_{\mathbb{G}_{+}^{q}}^{(n)}$ denotes the $n$-fold coproduct. Let
now $t_{0}=(r_{1},\ldots,r_{N})$ be such that $w_{0}=s_{r_{1}}\cdots
s_{r_{N}}$ is a reduced expression for the longest element in the Weyl group
of $\mathfrak{g}$. By [29], we know that $\varphi_{\mathbb{G}_{+}^{q}}$ can be
realized as $\omega\circ\gamma_{t_{0}}$ for some faithful normal state
$\omega\in B\big{(}\mathscr{H}_{+}^{\otimes N}\big{)}_{*}$. By Lemma 3.21,
this implies that $\gamma_{t_{0}}$ can be extended to a faithful normal
∗-representation of $\mathscr{L}^{\infty}(G_{+}^{q})$.

Let us turn now to the proof of the theorem. The main step is to prove that
the point-spectrum of
$\theta_{\lambda,\operatorname{\mathrm{reg}}}^{\varepsilon,+}\left(\prod\limits_{r=1}^{l}Z_{r}\right)$
does not contain zero. Indeed, if this is the case, then we can use
Proposition 3.13 to conclude that
$\theta_{\lambda,\operatorname{\mathrm{reg}}}^{\varepsilon,+}$ decomposes into
a direct integral of highest weight modules for
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$. It
then follows that $W_{\lambda}(\mathfrak{g};\varepsilon,+)$ is simply
$\oplus_{k=1}^{m}B(\mathscr{H}_{k})$ with $\mathscr{H}_{k}$ the Hilbert space
completion of the highest weight modules which appear in
$\theta_{\lambda,\operatorname{\mathrm{reg}}}^{\varepsilon,+}$.

To show that
$\theta_{\lambda,\operatorname{\mathrm{reg}}}^{\varepsilon,+}\left(\prod\limits_{r=1}^{l}Z_{r}\right)$
does not contain zero in its pointspectrum, it is sufficient to show that the
operator
$(\gamma_{t_{0}}\otimes\theta_{\lambda}^{\varepsilon,+})\left(\alpha_{\lambda}^{\varepsilon,+}\left(\prod\limits_{r=1}^{l}Z_{r}\right)\right)$
does not contain zero in its pointspectrum, where
$\theta_{\lambda}^{\varepsilon,+}$ is the ∗-representation of
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ on
the Hilbert space completion $\mathscr{H}_{\lambda}^{\varepsilon,+}$ of
$V_{\lambda}^{\varepsilon,+}$. Indeed, the invariant state
$\varphi_{\lambda}^{\varepsilon,+}=(\varphi_{\mathbb{G}_{+}^{q}}\otimes\operatorname{id})\alpha_{\lambda}^{\varepsilon,+}$
can be extended to a faithful normal functional on
$B(\mathscr{H}_{\lambda}^{\varepsilon,+}\otimes\mathscr{H_{+}}^{\otimes N})$,
which implies, again by Lemma 3.21, that
$(\gamma_{t_{0}}\otimes\theta_{\lambda}^{\varepsilon,+})\alpha_{\lambda}^{\varepsilon,+}$
extends to a faithful normal ∗-representation of
$W_{\lambda}(\mathfrak{g};\varepsilon,+)$.

Finally, to show that
$(\theta_{\lambda}^{\varepsilon,+}\otimes\gamma_{t_{0}})\left(\alpha_{\lambda}^{\varepsilon,+}\left(\prod\limits_{r=1}^{l}Z_{r}\right)\right)$
does not contain zero in its pointspectrum, we can reason by induction, using
the following lemma. ∎

######  Lemma 3.22.

Let $(V,\pi)$ be an irreducible highest weight module for
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$ with
an invariant pre-Hilbert space structure, and let $\mathscr{H}$ be the
completion of $V$. Fix $r\in I$, and put
$\pi_{r}=(\pi\otimes\gamma_{r})\alpha_{\lambda}^{\varepsilon,+}$. Then
$\pi_{r}\left(\prod\limits_{s=1}^{l}Z_{s}\right)$ does not contain $0$ in its
point spectrum.

###### Proof.

It is easy to see that $\pi_{r}(Z_{s})=1\otimes\pi(Z_{s})$ for $r\neq s$, so
that none of these operators have zero in their point-spectrum. Let now
$A_{r}$ be the sub-∗-algebra of
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$
generated by $Z_{r}$, $T_{r}$, $X_{r}$, $Y_{r}$, $W_{r}$, see Lemma 3.7. By
that lemma, $A_{r}$ is stable under the right action by
$U_{q_{r}}(\mathfrak{su}(2))$, and $W_{r}$ and $T_{r}$ are invariants in the
center of $A_{r}$. By invariance, $\pi_{r}(W_{r})=1\otimes\pi(W_{r})$ and
$\pi_{r}(T_{r})=1\otimes\pi(T_{r})$ are bounded self-adjoint operators. Hence,
to investigate the spectrum of $\pi_{r}(Z_{r})$, we may by disintegration
treat the above operators as scalars, say $w_{r}$ and $t_{r}$. Denote the
resulting quotient of $A_{r}$ by $A_{r}(w_{r},t_{r})$.

From Lemma 3.7, we find commutation relations between the generators $X_{r}$,
$Y_{r}$ and $Z_{r}$ of $A_{r}(w_{r},t_{r})$, as well as the resulting action
of $U_{q_{r}}(\mathfrak{su}(2))$. It follows that $A_{r}(w_{r},t_{r})$ is an
equivariant quotient of a generalized Podleś sphere $S_{q_{r},\tau}^{2}$ for
$SU_{q_{r}}(2)$ [27], for some $\tau$ depending on $t_{r}$ and $w_{r}$.
Moreover, as $\varphi_{SU_{q_{r}}(2)}$ can be realized on the Hilbert space
$\mathscr{H}_{+}$, the von Neumann algebraic envelope of $A_{r}(w_{r},t_{r})$
will be isomorphic to the von Neumann algebraic envelope of
$S_{q_{r},\tau}^{2}$, which is equal to $M_{n}(\mathbb{C})$,
$B(l^{2}(\mathbb{N}))$ or $B(l^{2}(\mathbb{N}))\oplus B(l^{2}(\mathbb{N}))$,
depending on whether $\varepsilon_{r}w_{r}$ is positive, zero or negative (cf.
[27]). In any case, the corresponding image of $Z_{r}$ will not contain 0 in
its point-spectrum. ∎

##  4 More on the
$\boldsymbol{\varepsilon\in\operatorname{\mathrm{Char}}_{\\{0,1\\}}(Q^{+})}$-case

The case of the non-standard Podleś spheres already shows that
$W_{\lambda}(\mathfrak{g};\varepsilon,+)$ is in general not a factor. However,
for $\varepsilon\in\operatorname{\mathrm{Char}}_{\\{0,1\\}}(Q^{+})$ and
$\lambda\in\operatorname{\mathrm{Char}}_{\mathbb{R}^{+}}(P)$ such that
$W_{\lambda}(\mathfrak{g};\varepsilon,+)$ is well-defined, we show that
$W_{\lambda}(\mathfrak{g};\varepsilon,+)$ _does_ become a type $I$-factor, and
we can then also say something more about the invariant integral
$\varphi_{\lambda}^{\varepsilon,+}$ on
$W_{\lambda}(\mathfrak{g};\varepsilon,+)$.

######  Proposition 4.1.

Let $\varepsilon\in\operatorname{\mathrm{Char}}_{\\{0,1\\}}(Q^{+})$,
$\lambda\in\operatorname{\mathrm{Char}}_{\mathbb{R}^{+}}(P)$, and suppose
$V_{\lambda}^{\varepsilon,+}$ is unitarizable with completion
$\mathscr{H}_{\lambda}^{\varepsilon,+}$. Identify
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)\subseteq
B(\mathscr{H}_{\lambda}^{\varepsilon,+})$. Then this inclusion completes to a
natural identification $W_{\lambda}(\mathfrak{g};\varepsilon,+)\cong
B(\mathscr{H}_{\lambda}^{\varepsilon,+})$.

###### Proof.

From the proof of Theorem 3.20 and Lemma 3.22, and from the commutation
relations in Lemma 3.7, we get that the $A_{r}(w_{r},t_{r})$ appearing in the
proof of Lemma 3.22 can only be matrix algebras or standard Podleś spheres. As
the $Z_{r}$ in the von Neumann algebraic completion of these algebras are
always positive operators, it follows by induction that the components
appearing in $W_{\lambda}(\mathfrak{g};\varepsilon,+)$ arise from restrictions
of highest weight modules $V_{\lambda^{\prime}}^{\varepsilon,+}$ of
$U_{q}(\mathfrak{g};\varepsilon,+)$ with
$\lambda^{\prime}\in\operatorname{\mathrm{Char}}_{\mathbb{R}^{+}}(P)$. Our aim
is to show that necessarily $\lambda^{\prime}=\lambda$.

Let $S$ be the set of $r$ with $\varepsilon_{r}=1$. Suppose that
$\lambda^{\prime}\in\operatorname{\mathrm{Char}}_{\mathbb{R}^{+}}(P)$ is such
that the representation of $U_{q}(\mathfrak{g};\varepsilon,+)$ on
$V_{\lambda^{\prime}}^{\varepsilon,+}$ factors over
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$.
Suppose that $V_{\lambda^{\prime}}^{\varepsilon,+}$ then admits an invariant
pre-Hilbert space structure as a
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$-module,
hence as a $U_{q}(\mathfrak{g};\varepsilon,+)$-module by Proposition 3.2. From
the proof of Proposition 3.12, we deduce that there exists $w\in W_{S}$ such
that
$q^{(-2w\omega,\rho)}\lambda_{-4w\omega}=q^{(-2\omega,\rho)}\lambda_{-4\omega}^{\prime}$
for all $\omega\in P$. In particular,
$\lambda_{\omega_{r}}=\lambda_{\omega_{r}}^{\prime}$ for $r\notin S$.

On the other hand, let
$\widetilde{U}_{q}(\mathfrak{k}_{\operatorname{\mathrm{ss}}})$ be the
subalgebra of $U_{q}(\mathfrak{g};\varepsilon,+)$ generated by the
$X_{r}^{\pm}$ and $K_{\alpha_{r}}^{\pm 1}$ with $r\in S$. Let
$\widetilde{V}_{\lambda}^{\varepsilon,+}$ be the
$\widetilde{U}_{q}(\mathfrak{k}_{\operatorname{\mathrm{ss}}})$-module spanned
by $v_{\lambda}^{\varepsilon,+}$, and similarly for
$V_{\lambda^{\prime}}^{\varepsilon,+}$. Then by Proposition 2.6, these are
irreducible highest weight modules for
$U_{q}(\mathfrak{k}_{\operatorname{\mathrm{ss}}})$ associated to the
restrictions of $\lambda$ and $\lambda^{\prime}$ to the root lattice $Q_{S}$
of $\mathfrak{k}_{\operatorname{\mathrm{ss}}}$. But as
$V_{\lambda^{\prime}}^{\varepsilon,+}$ admits an invariant pre-Hilbert space
structure (and $\mathfrak{k}_{\operatorname{\mathrm{ss}}}$ is compact), it is
necessarily finite-dimensional. However, as the restriction of
$\lambda^{\prime}$ lies in the $W_{S}$-orbit of $\lambda$ (for the so-called
‘dot’-action), it is well-known that this can happen only if the restrictions
of $\lambda$ and $\lambda^{\prime}$ to $Q_{S}$ coincide. Combined with the
observation at the end of the previous paragraph, this forces
$\lambda=\lambda^{\prime}$ inside
$\operatorname{\mathrm{Char}}_{\mathbb{R}^{+}}(P)$. ∎

######  Proposition 4.2.

Under the assumptions of Proposition 4.1, the invariant state on
$W_{\lambda}(\mathfrak{g};\varepsilon,+)$ is given by

$\displaystyle\varphi_{\lambda}^{\varepsilon,+}(x)=\frac{\mathrm{Tr}(xZ_{\rho})}{\mathrm{Tr}(Z_{\rho})},$

where $Z_{\rho}=\prod\limits_{r=1}^{l}Z_{r}$.

###### Proof.

Consider the projection $E^{\varepsilon,+}$ of
$U_{q}(\mathfrak{g};\varepsilon,+)_{\operatorname{\mathrm{f\/in}}}$ onto its
direct summand $\mathscr{Z}(U_{q}(\mathfrak{g};\varepsilon,+))$, the space of
$\lhd$-invariants. For $\varepsilon_{r}>0$ for all $r$, it follows from [16,
Chapter 7] that $E^{\varepsilon,+}(xy)=E^{\varepsilon,+}(y\sigma(x))$ for all
$x,y\in U_{q}(\mathfrak{g};\varepsilon,+)_{\operatorname{\mathrm{f\/in}}}$,
where $\sigma(x)=K_{-4\rho}xK_{-4\rho}^{-1}$. By continuity, this then holds
for all $\varepsilon\in\operatorname{\mathrm{Char}}_{\mathbb{R}}(Q^{+})$.
Clearly $\sigma$ induces an automorphism of
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$,
which we will denote by the same symbol. As
$\varphi_{\lambda}^{\varepsilon,+}$ factors over $E^{\varepsilon,+}$, we have
as well that
$\varphi_{\lambda}^{\varepsilon,+}(xy)=\varphi_{\lambda}^{\varepsilon}(y\sigma(x))$.

However, by general theory [3] we know that the modular automorphism group
$\sigma_{t}$ of $\varphi_{\lambda}^{\varepsilon,+}$ leaves
$B_{\lambda}^{\operatorname{\mathrm{f\/in}}}(\mathfrak{g};\varepsilon,+)$
invariant and is diagonalizable (and hence analytic) on it. From the above, we
can hence conclude that $\sigma_{t}=\mathrm{Ad}(Z_{\rho}^{it})$ (where we use
that $Z_{\rho}$ is a positive operator by Proposition 4.1). As
$W_{\lambda}(\mathfrak{g};\varepsilon,+)$ is a type $I$-factor, again by
Proposition 4.1, we know that $\varphi_{\lambda}^{\varepsilon,+}$ is
completely determined by its modular automorphism group up to a scalar. Hence
we obtain our expression for $\varphi_{\lambda}^{\varepsilon,+}$ as in the
statement of the proposition. ∎

##  Appendix A Cogroupoids

In this section, we recall the notion of cogroupoid due to J. Bichon [1, 2]
(cf. also the notion of _face algebra_ [13]).

######  Definition A.1.

Let $I$ be an index set, and let $\\{H_{ij}\,|\,i,j\in I\\}$ be a collection
of ∗-algebras. Suppose that for each triple of indices $i,j,k\in I$, we are
given a unital ∗-homomorphism

$\displaystyle\Delta_{ij}^{k}:\ H_{ij}\rightarrow H_{ik}\otimes H_{kj},\qquad
h\mapsto h_{(1)ik}\otimes h_{(2)kj}.$

We call $\big{(}\\{H_{ij}\\},\\{\Delta_{ij}^{k}\\}\big{)}$ a connected
_cogroupoid_ over the index set $I$ if the following conditions are satisfied:

  * $\bullet$

(Connectedness) None of the $H_{ij}$ are the zero algebra.

  * $\bullet$

(Coassociativity) For each quadruple $i$, $j$, $k$, $l$ of indices, we have

$\displaystyle\big{(}\operatorname{id}\otimes\Delta_{jl}^{k}\big{)}\Delta_{il}^{j}=\big{(}\Delta_{ik}^{j}\otimes\operatorname{id}\big{)}\Delta_{il}^{k},$

  * $\bullet$

(Counits) There exist unital ∗-homomorphisms
$\epsilon_{i}:H_{ii}\rightarrow\mathbb{C}$ such that, for all indices $i$,
$j$,

$\displaystyle(\epsilon_{i}\otimes\operatorname{id})\Delta_{ij}^{i}=\operatorname{id}_{H_{ij}}=(\operatorname{id}\otimes\epsilon_{j})\Delta_{ij}^{j},$

  * $\bullet$

(Antipodes) There exist anti-homomorphisms $S_{ij}:H_{ij}\rightarrow H_{ji}$
such that, for all indices $i$, $j$ and all $h\in H_{ii}$, we have

$\displaystyle
S_{ij}(h_{(1)ij})h_{(2)ji}=\epsilon_{i}(h)=h_{(1)ij}S_{ji}(h_{(2)ji}).$

As for Hopf ∗-algebras, it is easy to show that the $S_{ij}$ are unique, and
that $S_{ji}(S_{ij}(h)^{*})^{*}=h$ for each $h\in H_{ij}$. Note that each
$\big{(}H_{ii},\Delta_{ii}^{i}\big{)}$ defines a Hopf ∗-algebra.

######  Definition A.2.

Let $\big{(}H_{ij},\Delta_{ij}^{k}\big{)}$ be a cogroupoid. The _right adjoint
action_ (or Miyashita–Ulbrich action) $\lhd$ of $H_{jj}$ on $H_{ij}$ is given
by

$\displaystyle x\lhd h=S_{ji}(h_{(1)ji})xh_{(2)ij}.$

One easily proves that $\lhd$ defines a right $H_{jj}$-module ∗-algebra
structure on $H_{ij}$. The compatibility with the ∗-structure means that
$(x\lhd h)^{*}=x^{*}\lhd S_{jj}(h)^{*}$.

######  Lemma A.3.

The space of $\lhd$-invariant elements in $H_{ij}$ coincides with the center
of $H_{ij}$.

###### Proof.

(Cf. [16, Lemma 2.4].) If $x\in H_{ij}$ is in the center, clearly $x\lhd
h=\varepsilon_{j}(h)x$ for all $h\in H_{jj}$, by definition of the antipode.
Conversely, if $S_{ji}(h_{(1)ji})xh_{(2)ij}=\varepsilon_{j}(h)x$ for all $h\in
H_{jj}$, we have for $y\in H_{ij}$ that
$xy=y_{(1)ij}S_{ji}(y_{(2)ji})xy_{(3)ij}=yx$. ∎

Suppose now that $B$ is a unital ∗-algebra, and that for some $i$, $j$ we have
a unital ∗-homomorphism $\pi:H_{ij}\rightarrow B$. As the right action of
$H_{jj}$ on $H_{ij}$ is inner, it descends to a right action on $B$,

$\displaystyle b\lhd h=\pi(S_{ji}(h_{(1)ji}))b\pi(h_{(2)ij}).$

Note that the central elements in $B$ are invariant for the action.

A particular case of cogroupoid can be constructed from a Hopf algebra
$(H,\Delta)$ together with a collection of real 2-cocycle functionals
$\\{\omega_{i}\,|\,i\in I\\}$ on it. Here we mean by real 2-cocycle functional
an element $\omega\in(H\otimes H)^{*}$ which is convolution invertible and
such that $\omega(1,h)=\varepsilon(h)=\omega(h,1)$ for all $h\in H$, while
$\omega(h^{*},k^{*})=\overline{\omega(k,h)}$ and

$\displaystyle\omega(h_{(1)},k_{(1)})\omega(h_{(2)}k_{(2)},l)=\omega(k_{(1)},l_{(1)})\omega(h,k_{(2)}l_{(2)}),\qquad\forall\,h,k,l\in
H.$

Let us write $H_{ij}$ for the vector space $H$ with the new multiplication

$\displaystyle m_{ij}:\ H\otimes H\rightarrow H,\qquad h\otimes
k\mapsto\omega_{i}(h_{(1)},k_{(1)})h_{(2)}k_{(2)}\omega_{j}^{-1}(h_{(3)},k_{(3)}),$

and write $\Delta_{ij}^{k}$ for the given coproduct $\Delta$ seen as a map
$H_{ij}\rightarrow H_{ik}\otimes H_{kj}$. Then the $(H_{ij},\Delta_{ij}^{k})$
form a connected cogroupoid.

##  Appendix B Continuous one-parameter families of Lie algebras

We introduce the real Lie algebras whose quantizations we studied in Section
1. We refer to standard works as [14, 15, 22] for the basic background on Lie
algebras and Lie groups.

We keep the notation as in Section 1. We further write
$\\{h_{r}\\}\subseteq\mathfrak{h}$ for the basis dual to $\\{\omega_{r}\\}$,
and write $\mathfrak{h}_{\mathbb{R}}$ for its real span, which we may identify
with the real dual of $\mathfrak{h}_{\mathbb{R}}^{*}$. We write
$\mathfrak{g}_{c}\subseteq\mathfrak{g}$ for the compact real form of
$\mathfrak{g}$, and † for the corresponding anti-linear anti-automorphism such
that $x^{{\dagger}}=-x$ for $x\in\mathfrak{g}_{c}$.

Finally, for each $\alpha\in\Delta^{+}$, we choose root vectors
$X^{\pm}_{\alpha}\in\mathfrak{n}^{\pm}$ such that
$(X_{\alpha}^{+})^{\dagger}=X_{\alpha}^{-}$ for all $\alpha\in\Delta^{+}$ and
$[X_{r}^{+},X_{r}^{-}]=h_{r}$ for all $\alpha_{r}\in\Phi^{+}$.

Fix $\varepsilon\in\operatorname{\mathrm{Char}}_{\mathbb{R}}(Q^{+})$. We can
define on $\mathfrak{g}$ the linear maps

$\displaystyle S_{\varepsilon}^{\pm}:\
\mathfrak{g}\rightarrow\mathfrak{g}\colon\begin{cases}X_{\alpha}^{\pm}\mapsto\varepsilon_{\alpha}X_{\alpha}^{\pm},&\alpha\in\Delta^{+},\\\
z\mapsto z,&z\in\mathfrak{h}+\mathfrak{n}^{\mp}.\end{cases}$

######  Definition B.1.

We define $\mathfrak{g}_{\varepsilon}$ to be the complex vector space

$\displaystyle\\{(S_{\varepsilon}^{+}(z),S_{\varepsilon}^{-}(z))\,|\,z\in\mathfrak{g}\\}\subseteq\mathfrak{g}\oplus\mathfrak{g}.$

######  Proposition B.2.

The vector space $\mathfrak{g}_{\varepsilon}$ is a Lie ∗-subalgebra of the
direct sum Lie algebra $\mathfrak{g}\oplus\mathfrak{g}$ equipped with the
involution $(w,z)^{*}=(z^{{\dagger}},w^{{\dagger}})$. Moreover,
$\dim_{\mathbb{C}}(\mathfrak{g}_{\varepsilon})=\dim_{\mathbb{C}}(\mathfrak{g})$.

###### Proof.

From the definition, we see that $\mathfrak{g}_{\varepsilon}$ is the linear
space generated by elements of the form
$(\varepsilon_{\alpha}X_{\alpha}^{+},X_{\alpha}^{+})$,
$(X_{\alpha}^{-},\varepsilon_{\alpha}X_{\alpha}^{-})$ and $(h_{r},h_{r})$.
Accordingly, $\mathfrak{g}_{\varepsilon}$ has dimension
$\dim_{\mathbb{C}}(\mathfrak{g})$ and inherits the ∗-operation from
$\mathfrak{g}\oplus\mathfrak{g}$. Since
$\varepsilon_{\alpha+\beta}=\varepsilon_{\alpha}\varepsilon_{\beta}$ whenever
$\alpha$, $\beta$ and $\alpha+\beta$ are positive roots, we find that
$\mathfrak{g}_{\varepsilon}$ is closed under the bracket operation. ∎

In the following, we consider $\mathfrak{g}_{\varepsilon}$ with its
∗-operation inherited from $\mathfrak{g}\oplus\mathfrak{g}$.

######  Remark B.3.

By rescaling the $X_{\alpha}^{\pm}$, we see that
$\mathfrak{g}_{\varepsilon}\cong\mathfrak{g}_{\eta}$ as Lie ∗-algebras
whenever $\varepsilon_{r}=\lambda_{r}\eta_{r}$ for certain $\lambda_{r}>0$.
Hence we may in principle always assume that $\varepsilon\in\\{-1,0,1\\}^{l}$.
It is however sometimes convenient to keep the continuous deformation aspect
into the game. In the physics literature, this type of deformation goes by the
name of ‘the contraction method’. (In low dimensions, it can easily be
visualized, cf. [12, Chapter 13].)

######  Definition B.4.

We define
$\mathfrak{g}_{\varepsilon}^{\mathbb{R}}=\\{z\in\mathfrak{g}_{\varepsilon}\,|\,z^{*}=-z\\}$.

Hence $\mathfrak{g}_{\varepsilon}^{\mathbb{R}}$ is a real Lie algebra with
$\mathfrak{g}_{\varepsilon}$ as its complexification. We can also realize
$\mathfrak{g}^{\mathbb{R}}_{\varepsilon}$ more conveniently inside
$\mathfrak{g}$ as follows.

######  Proposition B.5.

Write

$\displaystyle
X_{\alpha}^{(\varepsilon)}=X_{\alpha}^{+}-\varepsilon_{\alpha}X_{\alpha}^{-},\qquad
Y_{\alpha}^{(\varepsilon)}=i\big{(}X_{\alpha}^{+}+\varepsilon_{\alpha}X_{\alpha}^{-}\big{)}$

as elements in $\mathfrak{g}$. Consider the $\mathbb{R}$-linear span of the
$X_{\alpha}^{(\varepsilon)}$, $Y_{\alpha}^{(\varepsilon)}$ and $ih_{r}$. Then
this space is closed under the Lie bracket of $\mathfrak{g}$, and forms a real
Lie algebra isomorphic to $\mathfrak{g}^{\mathbb{R}}_{\varepsilon}$.

###### Proof.

As a real Lie algebra, one can embed $\mathfrak{g}$ inside the direct sum
$\mathfrak{g}\oplus\mathfrak{g}$ by means of the map

$\displaystyle\mathfrak{g}\rightarrow\mathfrak{g}\oplus\mathfrak{g},\qquad
z\mapsto\big{(}z,-z^{{\dagger}}\big{)}.$

Under this identification, it is immediately verified that the elements in the
statement of the proposition get sent to a basis of
$\mathfrak{g}_{\varepsilon}^{\mathbb{R}}$. ∎

######  Proposition B.6.

Suppose $\varepsilon_{r}\neq 0$ for all $r$. Then
$\mathfrak{g}_{\varepsilon}^{\mathbb{R}}$ is a semi-simple real Lie algebra.

###### Proof.

In this case, the projection onto the first coordinate of
$\mathfrak{g}\oplus\mathfrak{g}$ provides an isomorphism between
$\mathfrak{g}_{\varepsilon}$ and $\mathfrak{g}$. ∎

Hence if $\varepsilon\in\operatorname{\mathrm{Char}}_{\\{-1,1\\}}(Q^{+})$, the
Lie algebra $\mathfrak{g}_{\varepsilon}^{\mathbb{R}}$ is the real form of
$\mathfrak{g}$ corresponding to the involution

$\displaystyle(X_{r}^{+})^{*}=\varepsilon_{r}X_{r}^{-},\qquad
h_{r}^{*}=h_{r}.$

Accordingly, $\mathfrak{g}_{\varepsilon}^{\mathbb{R}}$ is a real Lie algebra
of equal rank, meaning that we can take a Cartan decomposition
$\mathfrak{g}^{\mathbb{R}}=\mathfrak{t}\oplus\mathfrak{p}$ with $\mathfrak{t}$
compact and $i\mathfrak{h}_{\mathbb{R}}\subseteq\mathfrak{t}$. Conversely, it
is easy to see that any equal rank semi-simple real Lie algebra can be
realized as some $\mathfrak{g}_{\varepsilon}^{\mathbb{R}}$. We recall that the
equal rank semi-simple real Lie algebras are precisely those which admit an
irreducible unitary representation of discrete type.

As an example, let us present the simple equal rank real Lie algebras of type
$(A)$ (see [14, Chapter X]). For $r\in\\{0,\ldots,l\\}$, choose
$\sigma_{r}\in\\{\pm 1\\}$ such that $\varepsilon_{r}=\sigma_{r-1}\sigma_{r}$
for $r\in\\{1,\ldots,l\\}$. Then $\mathfrak{g}_{\varepsilon}^{\mathbb{R}}$ is
always of type $(AIII)$, and $\varepsilon$ corresponds to
$\mathfrak{su}(p,l+1-p)$ with $p$ the number of negative $\sigma_{r}$. This
precise correspondence between the $\varepsilon$ and the various real Lie
algebras is easy to determine explicitly from the standard descriptions of the
Cartan decompositions.

More generally, we obtain from Proposition B.5 the following characterization
of which Lie algebras appear as some $\mathfrak{g}_{\varepsilon}$.

######  Corollary B.7.

Let $\mathfrak{p}=\mathfrak{l}\oplus\mathfrak{u}$ be a parabolic Lie
subalgebra of $\mathfrak{g}$ with Levi factor $\mathfrak{l}$ and largest
nilpotent ideal $\mathfrak{u}$. Let $\mathfrak{l}^{\mathbb{R}}$ be a real form
of $\mathfrak{l}$ which is of equal rank on each simple summand and compact on
the center. Then
$\mathfrak{l}^{\mathbb{R}}\oplus\mathfrak{u}\cong\mathfrak{g}_{\varepsilon}^{\mathbb{R}}$
as real Lie algebras for some
$\varepsilon\in\operatorname{\mathrm{Char}}_{\mathbb{R}}(Q^{+})$, and all
$\mathfrak{g}_{\varepsilon}^{\mathbb{R}}$ arise in this way.

Our next step is to present $\mathfrak{g}_{\varepsilon}$ by means of
generators and relations.

######  Definition B.8.

Fix $\varepsilon\in\operatorname{\mathrm{Char}}_{\mathbb{R}}(Q^{+})$. We
define $\widetilde{\mathfrak{g}}_{\varepsilon}$ as a universal Lie algebra in
the following way. A set of generators is given by a triple of elements
$X_{r}^{\pm}$ and $H_{r}$ for each simple root $\alpha_{r}$. The relations can
be grouped into four parts, which we label as ‘(H)-condition’, ‘(T)orus
action’, ‘(S)erre relations’ and ‘(C)oupling conditions’: for all $r,s\in I$,

  1. (H)

$[H_{r},H_{s}]=0$,

  2. (T)

$[H_{r},X_{s}^{\pm}]=\pm a_{rs}X_{s}^{\pm}$,

  3. (S)

$\mathrm{ad}(X_{r}^{\pm})^{1-a_{rs}}(X_{s}^{\pm})=0$ when $r\neq s$,

  4. (C)ε

$[X_{r}^{+},X_{s}^{-}]=\delta_{rs}\varepsilon_{r}H_{r}$.

It is immediate that $\widetilde{\mathfrak{g}}_{\varepsilon}$ can be endowed
with a ∗-operation such that $(X_{r}^{+})^{*}=X_{r}^{-}$ and
$H_{r}^{*}=H_{r}$.

######  Proposition B.9.

The Lie ∗-algebras $\widetilde{\mathfrak{g}}_{\varepsilon}$ and
$\mathfrak{g}_{\varepsilon}$ are isomorphic.

###### Proof.

It is straightforward to verify that there is a unique Lie ∗-algebra
homomorphism
$\phi:\widetilde{\mathfrak{g}}_{\varepsilon}\rightarrow\mathfrak{g}_{\varepsilon}\subseteq\mathfrak{g}\oplus\mathfrak{g}$
such that $\phi(X_{r}^{+})=(\varepsilon_{r}X_{r}^{+},X_{r}^{+})$,
$\phi(X_{r}^{-})=(X_{r}^{-},\varepsilon_{r}X_{r}^{-})$ and
$\phi(H_{r})=(h_{r},h_{r})$. It is obviously surjective. On the other hand, it
is easy to see by induction that each element of
$\widetilde{\mathfrak{g}}_{\varepsilon}$ can be written in the form $x+y+z$
with $x$ in the Lie algebra generated by the $X_{r}^{+}$’s, $z$ in the Lie
algebra generated by the $X_{r}^{-}$’s, and $y$ in the linear span of the
$H_{r}$. Since $\mathfrak{n}^{\pm}$ are universal with respect to the
relations (S), we find that
$\dim(\widetilde{\mathfrak{g}}_{\varepsilon})\leq\dim(\mathfrak{g}_{\varepsilon})$,
hence $\phi$ is bijective. ∎

### Acknowledgments

It is a pleasure to thank the following people for discussions on topics
related to the subject of this paper: J. Bichon, P. Bieliavsky, H.P. Jakobsen,
E. Koelink, S. Kolb, U. Krähmer and S. Neshveyev.

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