# Banach space representations of Drinfeld-Jimbo algebras and their complex-
analytic forms

O. Yu. Aristov<EMAIL_ADDRESS>
###### Abstract.

We prove that every non-degenerate Banach space representation of the
Drinfeld-Jimbo algebra $U_{q}(\mathfrak{g})$ of a semisimple complex Lie
algebra $\mathfrak{g}$ is finite dimensional when $|q|\neq 1$. As a corollary,
we find an explicit form of the Arens-Michael envelope of
$U_{q}(\mathfrak{g})$, which is similar to that of $U(\mathfrak{g})$ obtained
by Joseph Taylor in 70s. In the case when
$\mathfrak{g}=\mathfrak{s}\mathfrak{l}_{2}$, we also consider the
representation theory of the corresponding analytic form
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ (with $e^{\hbar}=q$) and
show that it is simpler than for $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$. For
example, all irreducible continuous representations of
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ are finite dimensional
for every admissible value of the complex parameter $\hbar$, while
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ has a topologically irreducible infinite
dimensional representation when $|q|=1$ and $q$ is not a root of unity.

This work was supported by the RFBR grant no. 19-01-00447.

## Introduction

Besides well-known general representation theory of semisimple complex Lie
algebras, a specific theory of their Banach space representations was also
developed (see a detailed treatment of the latter in [BS01]). On the other
hand, representations of Drinfeld-Jimbo algebras (quantum deformations of
universal enveloping algebras) were being studied only in the algebraic
context. See, e.g., the monograph [KS97] for finite-dimensional
representations; a rich infinite-dimensional theory is also elaborated.

Here we are interested in Banach space representations of the Drinfeld-Jimbo
algebras $U_{q}(\mathfrak{g})$, $q\in\mathbb{C}\setminus\\{0,-1,1\\}$,
associated with a semisimple complex Lie algebra $\mathfrak{g}$ as well as
that of their complex-analytic forms $\widetilde{U}(\mathfrak{g})_{\hbar}$,
$\hbar\in\mathbb{C}$. (The latter series of topological algebras is defined in
my article [Ar20b].) Our main results asserts that every non-degenerate Banach
space representation of $U_{q}(\mathfrak{g})$ is finite dimensional when
$|q|\neq 1$ and the same is true for
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ when $e^{\hbar}$ is a
not root of unity (Theorems 1.1 and 2.3, respectively).

We also prove some results for other values of parameters. Note that for
$U_{q}(\mathfrak{g})$ three options arise naturally: $|q|\neq 1$, $q$ is a
root of unity, and $|q|=1$ but $q$ is not a root of unity. The alternatives
for $\widetilde{U}(\mathfrak{g})_{\hbar}$ are more traditional: $e^{\hbar}$ is
a root of unity or not.

We show that when $|q|=1$ and $q$ is not a root of unity,
$U_{q}(\mathfrak{g})$ admits infinite-dimensional topologically irreducible
Banach space representations (Proposition 1.9). We show that when $e^{\hbar}$
is not a root of unity, every continuous finite-dimensional representation of
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ is completely reducible
(Theorem 2.10) and when $e^{\hbar}$ is a root of unity the dimensions of
irreducible continuous representations of
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ are bounded (Theorem
2.6). Moreover, in the first case we classify continuous irreducible
representations of $\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$
(Theorem 2.8).

Combining results in this paper with some standard representation theory of
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ (see [Ka95] or [KS97]), we obtain Tables
1 and 2 (f.d. stands for ‘finite dimensional’).

Table 1. Representations of $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ $\mathbf{q}$ | representations | irr. representations
---|---|---
$|q|\neq 1$ | f.d. compl. reducible | f.d.
not root, $|q|=1$ | no restriction known |  $\exists$ top. irr. inf. d.
root |  $\exists$ inf. d. | f.d. bounded degree
Table 2. Representations of $\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ $\mathbf{e^{\hbar}}$ | representations | irr. representations
---|---|---
not root | f.d. compl. reducible | f.d.
root |  $\exists$ inf .d | f.d. bounded degree

As an application of our results on Banach space representations we describe
the structure of the Arens-Michael envelope of $U_{q}(\mathfrak{g})$ for
$|q|\neq 1$ and the structure of
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ in the case when
$e^{\hbar}$ is a not root of unity (Theorems 1.7 and 2.11, respectively).
Recall that the Arens-Michael envelope of an associative algebra over
$\mathbb{C}$ is a universal object connected with the problem of finding
homomorphisms with range in a Banach algebra; see the definition at the end of
Section 1. Considering finitely-generated associative algebras over
$\mathbb{C}$ as the main subject of study in Noncommutative complex affine
algebraic geometry, one can treat their Arens-Michael envelopes as “algebras
of noncommutative holomorphic functions” and thus as a subject of study in
Noncommutative complex-analytic geometry. The same can be said for
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$, which is a
holomorphically finitely generated algebra in the sense of Pirkovskii [Pi14,
Pi15].

Finding an explicit description of the Arens-Michael envelope of a finitely-
generated algebra seems easy only at first glance. On the contrary, in some
cases it is necessary to overcome some technical difficulties. One of the
first results was obtained by Joseph Taylor, who considered the classical
(undeformed) case and proved in [Ta72] that the Arens-Michael envelope of
$U(\mathfrak{g})$ is topologically isomorphic to the direct product of a
countable family of full matrix algebras, where each of the multiples
corresponds to a finite-dimensional irreducible representation of
$\mathfrak{g}$. We refer the reader for contemporary results on Arens-Michael
envelopes to the papers of Pirkovskii [Pi06, Pi11, Pi08] and also the papers
of the author [Ar19+, Ar20a, Ar20b]. Analytic forms of quantum algebras over
non-archimedean fields (including Arens-Michael envelopes) were considered in
[Sm18+] and [Du19]. Note that in the non-archimedean case such completions can
be described on a more direct way than in the classical; see, e.g., [Ly13].

In his Master thesis [Pe15], Pedchenko found the following description of the
Arens-Michael envelope of $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ in the case
when $|q|=1$. Let $K$, $F$ and $E$ denote the standard generators of
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ (see Section 1). Then it follows from a
PBW-type theorem that

$\mathbb{C}[u,z,z^{-1},v]\to
U_{q}(\mathfrak{s}\mathfrak{l}_{2})\\!:u^{n}z^{j}v^{m}\mapsto
F^{n}K^{j}E^{m}\qquad(j\in\mathbb{Z},\,n,m\in\mathbb{Z}_{+})$

determines a well-defined linear map (so-called ordered calculus) and,
moreover, it can be extended to a continuous linear map

$\mathcal{O}(\mathbb{C}\times\mathbb{C}^{\times}\times\mathbb{C})\to\widehat{U}_{q}(\mathfrak{s}\mathfrak{l}_{2})$

from the space of holomorphic functions to the Arens-Michael envelope. (The
standard notation for the Arens-Michael envelope of an algebra $A$ is
$\widehat{A}$.) Pedchenko proved that the latter map is a topological
isomorphism when $|q|=1$. In the argument he used a method proposed by
Pirkovskii in [Pi08], which is based on iterated analytic Ore extensions. This
approach can be applied only under some additional analytic conditions, which
do not hold when $|q|\neq 1$, and Pedchenko left the question open in this
case.

In his proof of the theorem on the Arens-Michael envelope of $U(\mathfrak{g})$
Taylor used an analytic approach based on the representation theory of compact
Lie groups. On the contrary, here we use basically an algebraic technique.
Note that Taylor’s result can be derived from the following assertion: If
$\mathfrak{g}$ is a semisimple complex Lie algebra, then the range of any
homomorphism from $U(\mathfrak{g})$ to a Banach algebra is finite dimensional
[BS01, §30, Theorem 2, p. 196]. The proof of this assertion is essentially
algebraic; it is based on the fact that for any
$\mathfrak{s}\mathfrak{l}_{2}$-triple, i.e., elements satisfying

$[H,E]=2E,\quad[H,F]=-2F,\quad[E,F]=H,$

the relation $[[E,F],E]=2E$ holds. In a Banach algebra this relation has some
algebraic consequences, which imply that every element of the completion is
algebraic. To prove our main result, Theorem 1.1, we use a modification of
this approach and show that for a quantum
$\mathfrak{s}\mathfrak{l}_{2}$-triple (see (1.1)) and every $m$ there is a
non-trivial Laurent polynomial in $K$ that belongs to the ideal generated by
$E^{m}$ (Lemma 1.5).

The reader can find some open questions and discussion in Section 3.

##  1\. Banach space representations and the Arens-Michael envelope of
$U_{q}(\mathfrak{g})$

Let $q\in\mathbb{C}$ such that $q\neq 0$ and $q^{2}\neq 1$. Recall that the
quantum algebra $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ is defined as the
universal complex associative algebra with generators $E$, $F$, $K$, $K^{-1}$
subject to

$KEK^{-1}=q^{2}E,\quad KFK^{-1}=q^{-2}F,\quad[E,F]=\frac{K-K^{-1}}{q-q^{-1}},$
(1.1)

and $KK^{-1}=K^{-1}K=1$; see [KS97, § 3.1.1, p. 53]. The first two relations
can be rewritten as $KE=E\sigma(K)$ and $KF=F\sigma^{-1}(K)$, where $\sigma$
is the automorphism of $\mathbb{C}[K,K^{-1}]$ determined by
$\sigma(K)=q^{2}K$. It is easy to see that

$KE^{n}=E^{n}\sigma^{n}(K)\quad\text{and}\quad KF^{n}=F\sigma^{-n}(K)$ (1.2)

for every $n\in\mathbb{Z}$. These equalities will be useful in what follows.

Let $\mathfrak{g}$ be a semisimple complex Lie algebra of rank $l$ and
$q\in\mathbb{C}\setminus\\{0\\}$. The condition $q^{2d_{j}}\neq 1$ is also
assumed for some non-zero $d_{1},\ldots,d_{l}$ given by the weight theory.
Recall that $U_{q}(\mathfrak{g})$ is the algebra with generators $E_{j}$,
$F_{j}$ and $K_{j}$ ($j=1,\ldots l$) such that every $K_{j}$ is invertible and
a number of relations is satisfied (see details in [KS97, § 6.1.2, p. 161]).
We need only the facts that $K_{j}$ pairwise commute and for each $j$ the
elements $E_{j}$, $F_{j}$, $K_{j}$ form a quantum
$\mathfrak{s}\mathfrak{l}_{2}$-triple (in the exponentiated form), that is,
the relations in (1.1) are satisfied with $E$, $F$, $K$ and $q$ replaced by
$E_{j}$, $F_{j}$, $K_{j}$ and $q^{d_{j}}$, respectively.

### The case when $|q|\neq 1$

We begin with a result that is similar to the assertion that the range of any
homomorphism from $U(\mathfrak{g})$ to a Banach algebra is finite dimensional
(see [BS01, §30, Theorem 2, p. 196]). In the end of the section, we use this
result to describe the structure of the Arens-Michael envelope of
$U_{q}(\mathfrak{g})$ in the case when $|q|\neq 1$.

######  Theorem 1.1.

Let $\mathfrak{g}$ be a semisimple complex Lie algebra and $|q|\neq 1$.

_(A)_ The range of any homomorphism from $U_{q}(\mathfrak{g})$ to a Banach
algebra is finite dimensional.

_(B)_ Every non-degenerate representation of $U_{q}(\mathfrak{g})$ on a Banach
space is finite dimensional.

The argument is in two parts, analytic and algebraic. The analytic part (Lemma
1.2) is simple but the algebraic (Proposition 1.3) is a bit more involved.

######  Lemma 1.2.

Let $a$ and $c$ be elements of a Banach algebra. Suppose that $c$ is
invertible and $aca^{-1}=\gamma c$ for some $\gamma\in\mathbb{C}$ such that
$|\gamma|\neq 1$ and $\gamma\neq 0$. Then $c$ is nilpotent.

###### Proof.

Assume the opposite, i.e., that $c^{n}\neq 0$ for every $n\in\mathbb{N}$. Let
$\|\cdot\|$ denote the norm on the Banach algebra. Since
$ac^{n}a^{-1}=\gamma^{n}c^{n}$ and $\|\cdot\|$ is submultiplicative, we have
$\|\gamma^{n}c^{n}\|\leqslant\|a\|\,\|c^{n}\|\,\|a^{-1}\|$ and so
$|\gamma|^{n}\leqslant\|a\|\,\|a^{-1}\|$ for all $n\in\mathbb{N}$. Therefore
$|\gamma|\leqslant 1$. Letting $d\\!:=aca^{-1}$ and using the equality
$a^{-1}da=\gamma^{-1}d$, we obtain similarly that $|\gamma|^{-1}\leqslant 1$.
This contradicts the hypothesis. ∎

Fix a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$. Let
$\beta_{1},\ldots,\beta_{n}$ be a tuple of positive roots of $\mathfrak{g}$
and let $E_{\beta_{r}}$ and $F_{\beta_{r}}$ ($r=1,\ldots,n$) be the root
elements of $U_{q}(\mathfrak{g})$ [KS97, § 6.2.3, p. 175, (65)].

We use the following proposition twice, right now in the proof of Theorem 1.1
and below for $\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$.

######  Proposition 1.3.

Suppose that $q$ is not a root of unity and $\pi$ is a homomorphism
$U_{q}(\mathfrak{g})$ to some algebra. If $\pi(E_{\beta_{r}})$ and
$\pi(F_{\beta_{r}})$ are nilpotent for every $r$, then the range of $\pi$ is
finite dimensional.

Assuming for the moment that the proposition is true, we can easily prove the
theorem.

###### Proof of Theorem 1.1.

(A) Fix $r\in\\{1,\ldots,n\\}$. For
$\lambda=n_{1}\alpha_{1}+\cdots+n_{l}\alpha_{l}$, where $n_{j}\in\mathbb{Z}$
and $\alpha_{1},\ldots,\alpha_{l}$ are the simple roots corresponding to
$K_{1},\ldots,K_{l}$, put $K_{\lambda}=K_{1}^{n_{1}}\cdots K_{l}^{n_{l}}$.
Then by [KS97, § 6.2.3, p. 176, Proposition 23(ii)],

$K_{\lambda}E_{\beta_{r}}K_{\lambda}^{-1}=q^{(\lambda,\beta_{r})}E_{\beta_{r}}\quad\text{and}\quad
K_{\lambda}F_{\beta_{r}}K_{\lambda}^{-1}=q^{-(\lambda,\beta_{r})}F_{\beta_{r}},$

where $(\cdot,\cdot)$ is the symmetric bilinear form induced by the Killing
form. Putting $\lambda=\beta_{r}$ we have $|q^{(\lambda,\lambda)}|\neq 1$.

Let $\pi$ be a homomorphism from $U_{q}(\mathfrak{g})$ to a Banach algebra. It
follows from Lemma 1.2 with $\gamma=q^{(\lambda,\lambda)}$ that
$\pi(E_{\beta_{r}})$ and $\pi(F_{\beta_{r}})$ are nilpotent for every $r$.
Since $q$ is not a root of unity, we can apply Proposition 1.3.

Part (B) follows immediately from Part (A). ∎

For the proof of Proposition 1.3 we need auxiliary lemmas. Suppose that $E$,
$F$ and $K$ are elements of some algebra that satisfy the relations in (1.1)
for given $q$. The first lemma is a simple observation. Consider the
automorphism $\sigma$ of $\mathbb{C}[K,K^{-1}]$ determined by
$\sigma(K)\\!:=q^{2}K$.

######  Lemma 1.4.

If $q$ is not a root of unity and

$R=(\alpha K^{k}-\beta K^{-k})(\gamma K^{k}-\delta K^{-k})$

for some $k\in\mathbb{N}$ and
$\alpha,\beta,\gamma,\delta\in\mathbb{C}\setminus\\{0\\}$, then

$\sigma^{-1}(R)-R=\alpha^{\prime}K^{2k}-\beta^{\prime}K^{-2k},$

where $\alpha^{\prime},\beta^{\prime}\in\mathbb{C}\setminus\\{0\\}$.

###### Proof.

Note that

$\sigma^{-1}(R)-R=(\alpha q^{-2k}K^{k}-\beta q^{2k}K^{-k})(\gamma
q^{-2k}K^{k}-\delta q^{2k}K^{-k})\\\ -(\alpha K^{k}-\beta K^{-k})(\gamma
K^{k}-\delta K^{-k})\\\
=\alpha\gamma(q^{-2k}-1)K^{2k}-\beta\delta(1-q^{2k})K^{-2k}.$

Since $q$ is not a root of unity, the assertion is proved. ∎

######  Lemma 1.5.

Let $m\in\mathbb{N}$. Suppose that $q$ is not a root of unity and for each
$n\in\\{0,\ldots,m\\}$ there are non-zero $t_{n},s_{n}\in\mathbb{C}$ such that

$[E^{n},F]=E^{n-1}P_{n},$ (1.3)

where $P_{n}\\!:=t_{n}K-s_{n}K^{-1}$. Then there is a non-trivial Laurent
polynomial in $K$ that belongs to the ideal generated by $E^{m}$.

###### Proof.

Denote the ideal generated by $E^{m}$ by $I$. We claim that for each
$n\in\\{0,\ldots,m-1\\}$ there is a Laurent polynomial $R_{n}$ in $K$ such
that

(1) $E^{n}R_{n}\in I$;

(2) $R_{m-1}=P_{m}$ and for $n<m-1$

$R_{n}=(\alpha_{n}K^{k}-\beta_{n}K^{-k})(a^{\prime}_{n}K^{k}-b^{\prime}_{n}K^{-k}),$

where $k=2^{m-n}$ and
$\alpha_{n},\beta_{n},\alpha_{n}^{\prime},\beta_{n}^{\prime}\in\mathbb{C}\setminus\\{0\\}$.

Then the claim with $n=0$ implies the assertion of the lemma.

We proceed by reverse induction. Put $R_{m-1}=P_{m}$. By (1.3), Part (1) of
the claim holds when $n=m-1$. So the claim holds for $n=m-1$.

Suppose that the claim has been proved for some $n\leqslant m-1$. Put

$R_{n-1}\\!:=(\alpha_{n}K^{2k}-\beta_{n}K^{-2k})(\sigma^{-1}(R_{n})-R_{n}),$
(1.4)

where $k=2^{m-n}$, $\alpha_{n}\\!:=t_{n}^{2k}$ and $\beta_{n}\\!:=s_{n}^{2k}$.
It follows from Lemma 1.4 that the second factor in (1.4) is of the form
$\alpha_{n}^{\prime}K^{2k}-\beta_{n}^{\prime}K^{-2k}$, where
$\alpha_{n}^{\prime},\beta_{n}^{\prime}\neq 0$. So Part (2) holds for $n-1$.

Since $k>0$, the first factor in (1.4) is of the form $P_{n}Q_{n}$ for some
$Q_{n}$. The second equality in (1.2) implies that

$[E^{n}R,F]=[E^{n},F](\sigma^{-1}(R)-R)$

for every Laurent polynomial $R$ in $K$. Hence, by (1.4) and (1.3),

$E^{n-1}R_{n-1}=E^{n-1}P_{n}Q_{n}(\sigma^{-1}(R_{n})-R_{n})\\\
=[E^{n},F](\sigma^{-1}(R_{n})-R_{n})Q_{n}=[E^{n}R_{n},F]Q_{n}.$

The last element belongs to $I$ because $E^{n}R_{n}\in I$ by the induction
hypothesis. Thus Part (1) holds for $n-1$ and this completes induction. ∎

######  Lemma 1.6.

Let $A$ be an algebra generated by an element $a$. Then $a$ is algebraic
(i.e., there is a non-trivial polynomial $p$ such that $p(a)=0$) if and only
if $A$ is finite dimensional.

The proof is straightforward.

###### Proof of Proposition 1.3.

Recall that by definition

$E_{\beta_{r}}\\!:=T_{j_{1}}T_{j_{2}}\cdots
T_{j_{r-1}}(E_{j_{r}})\quad\text{and}\quad
F_{\beta_{r}}\\!:=T_{j_{1}}T_{j_{2}}\cdots T_{j_{r-1}}(F_{j_{r}}),$ (1.5)

where $j_{1},\ldots,j_{n}\in\\{1,\ldots,l\\}$ and $T_{j_{1}},\ldots,T_{j_{n}}$
are the generators of the braid group, which acts on $U_{q}(\mathfrak{g})$
[KS97, § 6.2.3, p. 175, (65)].

Fix $m$ such that $\pi(E_{\beta_{r}})^{m}=\pi(F_{\beta_{r}})^{m}=0$ for every
$r$ and denote by $I$ the ideal of $U_{q}(\mathfrak{g})$ generated by
$\\{E_{\beta_{r}},\,F_{\beta_{r}};\,r=1,\ldots,n\\}$. Since $\pi$ factors on
the quotient homomorphism $U_{q}(\mathfrak{g})\to U_{q}(\mathfrak{g})/I$, it
suffices to show that the image of $U_{q}(\mathfrak{g})/I$ is finite
dimensional. The set

$\\{F_{\beta_{1}}^{r_{1}}\cdots F_{\beta_{n}}^{r_{n}}K_{1}^{k_{1}}\cdots
K_{l}^{k_{l}}E_{\beta_{n}}^{s_{n}}\cdots E_{\beta_{1}}^{s_{1}}\\},$

where $r_{1},\ldots,r_{n}$, $s_{1},\ldots,s_{n}$ run through $\mathbb{Z}_{+}$
and $k_{1},\ldots,k_{l}$ run through $\mathbb{Z}$, is a linear basis of
$U_{q}(\mathfrak{g})$ (see [KS97, § 6.2.3, p. 176, Theorem 24] for the
statement and [Lu93] for the proof). So it suffices to show that the
subalgebra $A$ of $U_{q}(\mathfrak{g})/I$ generated by

$\\{K_{j}+I,\,K_{j}^{-1}+I:\,j=1,\ldots,l\\}$

is finite dimensional.

First, we claim that the subalgebra generated by $K_{j}+I$ and $K_{j}^{-1}+I$
is finite dimensional for every $j$. Indeed, since $E_{j}$, $F_{j}$, $K_{j}$
is a quantum $\mathfrak{s}\mathfrak{l}_{2}$-triple with parameter
$q_{j}\\!:=q^{d_{j}}$, we have for any $m\geqslant 1$ that

$[E_{j}^{m},F_{j}]=E_{j}^{m-1}(t_{mj}K_{j}-s_{mj}K_{j}^{-1}),$ (1.6)

where

$t_{mj}\\!:=\frac{q_{j}(q_{j}^{2m}-1)}{(q_{j}^{2}-1)^{2}},\qquad
s_{mj}\\!:=\frac{q_{j}^{-1}(q_{j}^{-2m}-1)}{(q_{j}^{-2}-1)^{2}};$

see, e.g., [Ka95, Lemma VI.1.3]. Since $q$ is not a root of unity, all the
coefficients $t_{mj}$ and $s_{mj}$ are non-trivial. Therefore by Lemma 1.5,
for every $j$ there is a non-trivial Laurent polynomial in $K_{j}$ that
belongs to $I$. This implies that $K_{j}+I$ is algebraic in
$U_{q}(\mathfrak{g})/I$ and so by Lemma 1.6, the subalgebra of
$U_{q}(\mathfrak{g})/I$ generated by $K_{j}+I$ and $K_{j}^{-1}+I$ is finite
dimensional.

Recall that $\beta_{1},\ldots,\beta_{n}$ is the sequence of positive roots as
defined in [KS97, § 6.2.3, p. 175]. Let $j_{i},\ldots,j_{n}$ be the
corresponding sequence of indices. For $r\in\\{1,\ldots,n\\}$ denote by
$A_{r}$ the subalgebra of $U_{q}(\mathfrak{g})/I$ generated by

$\\{K_{j_{i}}+I,\,K_{j_{i}}^{-1}+I:\,i=1,\ldots,r\\}.$

To complete the proof we show by induction that $A_{r}$ is finite dimensional
for every $r$.

Assume that $r=1$. Then we have from (1.5) that $E_{\beta_{1}}=E_{j_{1}}$. It
follows from the claim proved above that $A_{1}$ is finite dimensional.

Now assume that $r>1$ and $A_{r-1}$ is finite dimensional. Denote by $J_{r}$
the ideal of $U_{q}(\mathfrak{g})$ generated by $E_{j_{r}}^{m}$. Then by
(1.5), $T_{j_{1}}T_{j_{2}}\cdots T_{j_{r-1}}(J_{r})\subset I$. Lemma 1.5
implies that there is a non-trivial Laurent polynomial $p$ such that
$p(K_{j_{r}})\in J_{r}$. In particular, $p(T_{j_{1}}T_{j_{2}}\cdots
T_{j_{r-1}}(K_{j_{r}}))\in I$. Note that $T_{i}(K_{j})=K_{j}K_{i}^{-a_{ij}}$
(where $a_{ij}$ are the entries of the Cartan matrix) for every $i$ and $j$
[KS97, § 6.2.3, p. 174, (61)]. Therefore we have a non-trivial Laurent
polynomial $\tilde{p}$ in $r$ variables such that
$\tilde{p}(K_{j_{1}}+I,\ldots,K_{j_{r}}+I)=0$. Since $A_{r}$ is commutative,
we can rewrite $\tilde{p}$ as a Laurent polynomial in $K_{j_{r}}+I$ with
coefficients in $A_{r-1}$, which is a finite-dimensional algebra by the
induction hyposthesis. It easy to see that this implies that $K_{j_{r}}+I$ is
algebraic and so $A_{r}$ is also finite dimensional as required. ∎

Now we can prove an analogue of Taylor’s theorem on the Arens-Michael envelope
of $U(\mathfrak{g})$. Recall that an _Arens-Michael algebra_ is a complete
topological algebra such that its topology can be determined by a system of
submultiplicative prenorms $\|\cdot\|$, i.e., $\|ab\|\leqslant\|a\|\,\|b\|$
holds for every $a$ and $b$. An _Arens-Michael envelope_ of an associative
algebra $A$ over $\mathbb{C}$ is a pair $(\widehat{A},\iota_{A})$, where
$\widehat{A}$ is an Arens-Michael algebra and $\iota_{A}$ is a homomorphism
$A\to\widehat{A}$, such that for any Arens-Michael algebra $B$ and for each
homomorphism $\varphi\\!:A\to B$ there exists a unique continuous homomorphism
$\widehat{\varphi}\\!:\widehat{A}\to B$ making the diagram

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{A}}$$\scriptstyle{\varphi}$$\textstyle{\widehat{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\varphi}}$$\textstyle{B}$

commutative [He93, Chapter 5]. In fact, it suffices to check this property for
Banach algebras. Note that $\widehat{A}$ is topologically isomorphic to the
completion of $A$ with respect to the topology determined by all
submultiplicative prenorms.

Let $\mathfrak{g}$ be a semisimple complex Lie algebra and $\Sigma_{q}$ be the
set of equivalence classes of irreducible finite-dimensional representations
of $U_{q}(\mathfrak{g})$ for given $q$. Then for $\sigma\in\Sigma_{q}$ we have
a homomorphism $U_{q}(\mathfrak{g})\to\mathrm{M}_{d_{\sigma}}(\mathbb{C})$,
where $d_{\sigma}$ is the dimension of $\sigma$ and
$\mathrm{M}_{d_{\sigma}}(\mathbb{C})$ is the algebra of matrices of order
$d_{\sigma}$. Denote by $\varphi$ the corresponding homomorphism

$U_{q}(\mathfrak{g})\to\prod_{\sigma\in\Sigma_{q}}\mathrm{M}_{d_{\sigma}}(\mathbb{C}).$

######  Theorem 1.7.

Let $\mathfrak{g}$ be a semisimple complex Lie algebra. If $|q|\neq 1$, then
the algebra $\prod_{\sigma\in\Sigma_{q}}\mathrm{M}_{d_{\sigma}}(\mathbb{C})$
endowed with the direct product topology together with $\varphi$ is the Arens-
Michael envelope of $U_{q}(\mathfrak{g})$.

###### Proof.

It suffices to show that every homomorphism $\pi$ from $U_{q}(\mathfrak{g})$
to a Banach algebra factors through
$\prod_{\sigma\in\Sigma_{q}}\mathrm{M}_{d_{\sigma}}(\mathbb{C})$. Denote the
range of $\pi$ by $B$. By Theorem 1.1, $B$ is finite dimensional and so it is
a finite-dimensional Banach algebra. Then $B$ becomes a
$U_{q}(\mathfrak{g})$-module with respect to $a\cdot b\\!:=\pi(a)b$. Since $q$
is not a root of unity, any finite-dimensional $U_{q}(\mathfrak{g})$-module is
completely reducible [Ro88, Theorem 2]. Therefore $\pi$ factors through some
finite product of algebras of the form $\mathrm{M}_{d_{\sigma}}(\mathbb{C})$
and hence through
$\prod_{\sigma\in\Sigma_{q}}\mathrm{M}_{d_{\sigma}}(\mathbb{C})$. ∎

In the rest of the article we suppose that
$\mathfrak{g}=\mathfrak{s}\mathfrak{l}_{2}$.

######  Remark 1.8.

Comparing the lists of irreducible finite-dimensional representations of
$U(\mathfrak{s}\mathfrak{l}_{2})$ and $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$
(see (1.7) below), we have that for any such representation of
$U(\mathfrak{s}\mathfrak{l}_{2})$ there are exactly two such representations
of $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$. So, in view of Theorem 1.7, when
$|q|\neq 1$, we can identify $\widehat{U}_{q}(\mathfrak{s}\mathfrak{l}_{2})$
with $\widehat{U}(\mathfrak{s}\mathfrak{l}_{2})\otimes\mathbb{C}^{2}$. Of
course, this isomorphism is not canonical.

### The case when $|q|=1$ but $q$ is not a root of unity

Suppose that $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_{2}$. A power series
description of the Arens-Michael envelope of
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ in the case when $|q|=1$ is given by
Pedchenko in [Pe15]. But it says nothing about Banach space representations.

We recall some representation theory of $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$.
If $q$ is not a root of unity, then as has been mentioned above, any finite-
dimensional representation of $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ is
completely reducible. Moreover, any irreducible finite-dimensional
representation of $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ is associated with a
homomorphism of the form

$U_{q}(\mathfrak{s}\mathfrak{l}_{2})\to\mathrm{M}_{n+1}(\mathbb{C})\\!:\,E\mapsto
E_{n,\varepsilon},\quad F\mapsto F_{n,\varepsilon},\quad K\mapsto
K_{n,\varepsilon},$ (1.7)

where $\varepsilon=\pm 1$, $n\in\mathbb{Z}_{+}$,

$E_{n,\varepsilon}=\varepsilon\begin{pmatrix}0&[n]_{q}&0&\ldots&0\\\
0&0&[n-1]_{q}&\ldots&0\\\ \vdots&\ddots&\ddots&\ddots&\vdots\\\
0&0&\ddots&\ddots&1\\\ 0&0&\ldots&0&0\\\ \end{pmatrix},\quad
F_{n,\varepsilon}=\begin{pmatrix}0&0&\ldots&0&0\\\ 1&0&\ldots&0&0\\\
0&[2]_{q}&\ddots&0&0\\\ \vdots&\ddots&\ddots&\ddots&\vdots\\\
0&0&\ldots&[n]_{q}&0\\\ \end{pmatrix},$
$K_{n,\varepsilon}=\varepsilon\begin{pmatrix}q^{n}&0&\ldots&0&0\\\
0&q^{n-2}&\ldots&0&0\\\ \vdots&\ddots&\ddots&\ddots&\vdots\\\
0&0&\ldots&q^{-n+2}&0\\\ 0&0&\ldots&0&q^{-n}\\\ \end{pmatrix};$

here

$[n]_{q}\\!:=\frac{q^{n}-q^{-n}}{q-q^{-1}};$

see [KS97, § 3.2, p. 62, Proposition 9]. So we have an infinite series of
irreducible (finite-dimensional) Banach space representations. We now suppose
that $|q|=1$ but $q$ is not a root of unity and consider completions of Verma
modules.

Let $\lambda\in\mathbb{C}\setminus\\{0\\}$ and $V(\lambda)$ denote the Verma
module of $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ as defined, e.g., in [Ka95, p.
129, Lemma VI.3.6]. Namely, $V(\lambda)$ is a linear space with basis
$\\{e_{n}\\!:\,n\in\mathbb{N}\\}$ and the generators of
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ are represented by matrices:

$E\mapsto\begin{pmatrix}0&-[1]_{q,\lambda}&0&\ldots&0&\ldots\\\
0&0&-[2]_{q,\lambda}&\ldots&0&\ldots\\\
\vdots&\ddots&\ddots&\ddots&\vdots&\vdots\\\
0&0&\ddots&\ddots&-[n]_{q,\lambda}&\ddots\\\
\vdots&\vdots&\ldots&\vdots&\vdots&\ddots\end{pmatrix}\,,$
$F\mapsto\begin{pmatrix}0&0&\ldots&0&\ldots\\\ 1&0&\ldots&0&\ldots\\\
0&[2]_{q}&\ddots&0&\ldots\\\ \vdots&\ddots&\ddots&\ddots&\vdots\\\
0&0&\ldots&[n]_{q}&\ddots\\\
\vdots&\vdots&\ldots&\vdots&\ddots\end{pmatrix}\,,\quad
K\mapsto\begin{pmatrix}\lambda&0&\ldots&0&\ldots\\\ 0&\lambda
q^{-2}&\ldots&0&\ldots\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\ldots&\lambda q^{-2n}&\vdots\\\
\vdots&\vdots&\ldots&\vdots&\ddots\end{pmatrix}\,,$

where

$[n]_{q,\lambda}\\!:=\frac{q^{n}\lambda^{-1}-q^{-n}\lambda}{q-q^{-1}}.$ (1.8)

(Note that $[n]_{q}\\!:=[n]_{q,1}$.)

Let $p\in[1,+\infty)$. It is easy to see that this representation can be
extended to a representation on the Banach space $\ell^{p}$ if and only if
$|q|=1$. We denote this extension by $S_{\lambda,p}$. So we have a series of
infinite-dimensional Banach space representations of
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ in contrast to the case when $|q|\neq
1$.

Recall that a representation of an algebra on a Banach space is said to be
_topologically irreducible_ if there is no proper closed invariant subspace.
In the following proposition we assume that $p=2$ because the argument uses
Wermer’s theorem on operators on a Hilbert space.

######  Proposition 1.9.

Suppose that $q\in\mathbb{C}$ is not a root of unity, $|q|=1$ and $\lambda\neq
0$. Then $S_{\lambda,2}$ is topologically irreducible if and only if $q^{2}$
is not a root of $\lambda^{2}$.

###### Proof.

Note that $S_{\lambda,2}(K)$ is a normal operator and the Hilbert space
$\ell^{2}$ has an orthonormal basis of eigenvectors of $S_{\lambda,2}(K)$.
Since the modulus of every eigenvalue equals $|\lambda|$, spectral synthesis
holds for $S_{\lambda,2}(K)$ [We52, Theorem 4], i.e., every non-trivial closed
invariant subspace coincides with the closure of all eigenvectors contained in
it.

Since $q\in\mathbb{C}$ is not a root of unity, all the numbers $[n]_{q}$ are
non-zero. It follows from the form of $S_{\lambda,2}(F)$ that if a non-trivial
closed invariant subspace contains $e_{n}$ for some $n\in\mathbb{N}$, then it
also contains $e_{n+1}$. On the other hand, it follows from the form of
$S_{\lambda,2}(E)$ that the closure of the linear span of
$\\{e_{k}\\!:\,k\geqslant n+1\\}$ is invariant if and only if
$[n]_{q,\lambda}=0$. Thus, a non-zero proper closed invariant subspace exists
if and only if there is $n$ such that $q^{2n}=\lambda^{2}$. ∎

### The case when $q$ is a root of unity

Suppose that $q$ is a root of unity. It is well known that then
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ has big centre and this implies that all
irreducible representations of $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ are
finite dimensional [KS97, § 3.3, Corollary 15, p. 67]. The following
proposition holds in contrast to the case when $|q|\neq 1$.

######  Proposition 1.10.

Let $q$ be a root of unity. Then there are infinite-dimensional Banach space
representations of $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$.

###### Proof.

It suffices to show that there is a homomorphism to the Banach algebra with
infinite-dimensional range.

Let $T_{ab\lambda}$ be the three-parameter family of $p^{\prime}$-dimensional
representations ($a,b,\lambda\in\mathbb{C}$ and $\lambda\neq 0$) of
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$, where $p^{\prime}\in\mathbb{N}$, as
defined in [KS97, § 3.3, p. 68, (34)]. We consider each representation space
as a finite-dimensional Hilbert space and assume that the standard basis is
normed to one.

Fix $a$ and $b$ and take an infinite compact subset $K$ of
$\mathbb{C}\setminus\\{0\\}$. Then there is $C>0$ such that the norms of
$T_{ab\lambda}(E)$, $T_{ab\lambda}(F)$ and $T_{ab\lambda}(K)$ are at most $C$
when $\lambda$ runs $K$. So we have a homomorphism from
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ to the Banach algebra
$C(K,\mathrm{M}_{p^{\prime}}(\mathbb{C}))$ of matrix-valued continuous
functions. It is evident from the explicit form of $T_{ab\lambda}$ that the
range is infinite dimensional. ∎

##  2\. Banach space representations and the structure of
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$

In this section we study the representations of Arens-Michael algebra
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ defined in [Ar20b, § 5].

Let $\hbar\in\mathbb{C}$ and $\sinh\hbar\neq 0$. Then
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ denotes the universal
Arens-Michael algebra generated by $E$, $F$, $H$ subject to relations

$[H,E]=2E,\quad[H,F]=-2F,\quad[E,F]=\frac{\sinh\hbar H}{\sinh\hbar}\,.$ (2.1)

(The term ‘universal’ means that for any Arens-Michael algebra $B$ containing
elements that satisfy (2.1) there is a unique continuous homomorphism
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}\to B$ sending the
generators to that elements. The universal Arens-Michael algebra is isomorphic
to the quotient of the algebra of free entire functions over the closed two-
sided ideal generated by the corresponding identities. See, e.g., [Pi15] for
the algebras of free entire functions and their quotients.)

The algebra $\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ can be
endowed with a structure of a topological Hopf algebra but we do not need it
here; see details in [Ar20b]. Note also that one can define similarly the
algebra $\widetilde{U}(\mathfrak{g})_{\hbar}$ for every semisimple complex Lie
algebra $\mathfrak{g}$ [ibid., Remark 5.5]. But here we consider here only the
case when $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_{2}$.

First, we discuss a connection of
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ and
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$.

######  Lemma 2.1.

Let $E$, $F$ and $H$ be elements of Arens-Michael algebra satisfying (2.1).
Put

$K\\!:=e^{\hbar H}\quad\text{and}\quad q\\!:=e^{\hbar}.$ (2.2)

Then $E$, $F$ and $K$ satisfy (1.1).

###### Proof.

We recall that well-known formula

$\mathop{\mathrm{ad}}\nolimits
h(Q)(T)=\sum_{n=1}^{\infty}\frac{1}{n!}(\mathop{\mathrm{ad}}\nolimits
Q)^{n}(T)h^{(n)}(Q)$ (2.3)

holds for elements $Q$ and $T$ of an Arens-Michael algebra and an entire
function $h$; cf. [BS01, § 15, p. 82, Corollary 1]. (Here
$\mathop{\mathrm{ad}}\nolimits Q(T)\\!:=[Q,T]$.)

It follows from $[H,E]=2E$ that $(\mathop{\mathrm{ad}}\nolimits
H)^{n}(E)=2^{n}E$. Hence by (2.3),

$(\mathop{\mathrm{ad}}\nolimits
h(H))E=\sum_{n=1}^{\infty}\frac{1}{n!}\,2^{n}\,E\,h^{(n)}(H)=E\,(h(H+2)-h(H)),$

i.e., $h(H)E=Eh(H+2)$. Therefore,

$KE=e^{\hbar H}E=E\,e^{\hbar(H+2)}=q^{2}EK.$

Similarly, we have $KF=q^{-2}FK$. The last relation in (1.1) is trivial. ∎

Thus it follows from Lemma 2.1 that we have a well-defined homomorphism
determined by

$\theta\\!:U_{q}(\mathfrak{s}\mathfrak{l}_{2})\to\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}\\!:E\to
E,\quad F\to F,\quad K\to e^{\hbar H}.$ (2.4)

(For simplicity of notation, we denote the generators $E$ and $F$ in
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ and
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ by the same letters.)

The following theorem is the first of two main results of this section. The
second is Theorem 2.11.

######  Theorem 2.2.

Suppose that $\sinh\hbar\neq 0$. Every irreducible continuous representation
of $\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ is finite
dimensional.

We consider separately the cases when $e^{\hbar}$ is not a root of unity and
when it is.

######  Theorem 2.3.

. Suppose that $\sinh\hbar\neq 0$ and $e^{\hbar}$ is not a root of unity.

_(A)_ The range of any continuous homomorphism from
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ to a Banach algebra is
finite dimensional.

_(B)_ Any non-degenerate continuous representation of
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ on a Banach space is
finite dimensional.

We need two lemmas. The first improves Lemma 1.6 .

######  Lemma 2.4.

Let $B$ be a Banach algebra (algebraically) generated by an element $b$. If
there is a non-trivial function $h$ holomorphic in a neighbourhood $U$ of the
spectrum of $b$ such that $h(b)=0$, then $B$ is finite dimensional.

###### Proof.

Since the spectrum $b$ is compact, we can assume that $U$ contains only a
finite number of zeros of $h$. Therefore $h=ph_{1}$, where $p$ is a polynomial
and $h_{1}$ is a function holomorphic and non-vanishing in $U$. Then
$h_{1}(b)$ is invertible and so $p(b)=0$. An application of Lemma 1.6
completes the proof. ∎

######  Lemma 2.5.

Suppose that a unital associative algebra is generated by elements $e$, $f$
and $h$ such that

$[h,e]=\alpha e,\quad[h,f]=\beta f,\quad[e,f]=p(h).$

where $\alpha,\beta\in\mathbb{C}$ and $p$ is a polynomial. Then the linear
span $L$ of $\\{e^{j}h^{k}f^{n}\\}$, where $j,k,n\in\mathbb{Z}_{+}$, coincides
with the whole algebra.

###### Proof.

It suffices to show that every product of the form
$e^{j_{1}}h^{k_{1}}f^{n_{1}}e^{j_{2}}h^{k_{2}}f^{n_{2}}$, where the exponents
are in $\mathbb{Z}_{+}$, is contained in $L$. It is obvious that $fe\in L$ and
it can be shown by induction (first in $j_{2}$ and next in $n_{1}$) that the
same is true for $f^{n_{1}}e^{j_{2}}$. It is easy to see that the subalgebra
generated by $e$ and $h$ (or $f$ and $h$) coincides with the linear span of
$\\{e^{j}h^{k}\\}$, $j,k\in\mathbb{Z}_{+}$, (the linear span of
$\\{h^{k}f^{n}\\}$, $k,n\in\mathbb{Z}_{+}$, respectively). Therefore the
product under consideration is in $L$. ∎

###### Proof of Theorem 2.3.

(A) Let $\pi$ be a continuous homomorphism from
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ to a Banach algebra $B$.
It is not hard to see that the relations $[H,E]=2E$ and $[H,F]=-2F$ imply that
$\pi(E)$ and $\pi(F)$ are nilpotent (cf. [Pi08, Example 5.1]).

Define $q$ and $K$ as in (2.2) and consider the homomorphism $\theta$ as in
(2.4). Since $q$ is not a root of unity, Proposition 1.3 implies that the
range of $\pi\theta$ is finite dimensional and so is the subalgebra generated
by $\pi(K)$. By Lemma 1.6, $\pi(K)$ is an algebraic element, i.e., there a
polynomial $p$ such that $p(\pi(K))=0$.

Since $\pi$ is continuous, $e^{\hbar\pi(H)}=\pi(K)$ and so
$p(e^{\hbar\pi(H)})=0$. It follows from Lemma 2.4 that the algebra $B_{0}$
generated by $\pi(H)$ is finite dimensional. Hence $B_{0}$ is closed and hence
it is a Banach subalgebra of $B$. Therefore $\sinh\hbar\pi(H)\in B_{0}$. Then
there a polynomial $p_{0}$ such that $\sinh\hbar\pi(H)=p_{0}(\pi(H))$. Thus
$\pi(E)$, $\pi(F)$ and $\pi(H)$ satisfy the hypotheses of Lemma 2.5 and so the
linear span of $\\{\pi(E)^{j}\pi(H)^{k}\pi(F)^{n}\\}$, where
$j,k,n\in\mathbb{Z}_{+}$, coincides with the subalgebra $B_{1}$ generated by
these three elements. Since $\pi(E)$ and $\pi(F)$ are nilpotent and $\pi(H)$
is algebraic, $B_{1}$ is finite dimensional and hence closed. Finally, note
that $E$, $F$ and $H$ are topological generators of
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ and so the range of
$\pi$ equals $B_{1}$.

Part (B) follows immediately from Part (A). ∎

Now we turn to the case when $e^{\hbar}$ is a root of unity. It has been
mentioned above that under this assumption every irreducible representation of
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ is finite dimensional [KS97, § 3.3,
Corollary 15, p. 67]. Slightly changing the argument, we obtain the same
assertion for $\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$.

######  Theorem 2.6.

Suppose that $e^{\hbar}$ is a root of unity. Then every irreducible continuous
representation of $\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ is
finite dimensional. Moreover, there is $s\in\mathbb{N}$ such that all
irreducible continuous representations have dimension at most $s$.

###### Proof.

Let $K\\!:=e^{\hbar H}$, $q\\!:=e^{\hbar}$ and
$\theta\\!:U_{q}(\mathfrak{s}\mathfrak{l}_{2})\to\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$
be the natural homomorphism. Suppose that $q$ is a root of unity and let $d$
denote the order of $q$. Put $s\\!:=d$ when $d$ is odd and $s\\!:=d/2$ when
$d$ is even. It is well known that $E^{s}$, $F^{s}$ and $K^{s}$ are in the
center of $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ [Ka95, p. 134, Lemma IV.5.3].
Since $\theta(K^{s})$ obviously commutes with $H$, it is in the center of
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$.

Let $T$ be an irreducible continuous representation of
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$. It follows from Shur’s
lemma that $e^{s\hbar T(H)}=T\theta(K^{s})=\lambda$ for some
$\lambda\in\mathbb{C}$. By Lemma 2.4, the subalgebra generated by $T(H)$ is
finite dimensional and hence closed. Therefore there is a polynomial $p$ such
that $T(\sinh\hbar H/\sinh\hbar)=p(T(H))$. Since $T(E)$ and $T(F)$ are
nilpotent (cf. the proof of Theorem 2.3), it follows from Lemma 2.5 that the
range of $T$ is finite dimensional.

To show that there is no irreducible finite-dimensional representation of
dimension greater than $s$ we use the same argument as in [Ka95, p. 134.
Proposition VI.5.2]. Indeed, assume to the contradiction that $T$ is such a
representation. Two cases can occur: there exists a non-zero eigenvector of
$T(H)$ such that $T(F)v=0$ or not. In the first case denote by $V^{\prime}$
the linear span of $\\{v,T(E)v,\ldots,T(E^{s-1})v\\}$ and in the second case
take a non-zero eigenvector of $T(H)$ such that $T(F)v\neq 0$ and denote by
$V^{\prime\prime}$ the linear span of $\\{v,T(F)v,\ldots,T(F^{s-1})v\\}$. It
follows from $[H,E]=2E$ and $[H,F]=-2F$ that $V^{\prime}$ and
$V^{\prime\prime}$, respectively, are invariant under $T(H)$. Moreover, as it
is proved in [ibid.], in the first and second cases, $V^{\prime}$ and
$V^{\prime\prime}$, respectively, are invariant under both $T(E)$ and $T(F)$.
Thus we have an invariant subspace of dimension $s$. ∎

Combining Theorem 2.6 with Part (B) of Theorem 2.3, we immediately deduce
Theorem 2.2.

### A classification of irreducible representations of
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ when $e^{\hbar}$ is not
a root of unity

Let $n\in\mathbb{Z}_{+}$, $k\in\mathbb{Z}$ and $\varepsilon\in\\{-1,1\\}$. Put

$H_{n,k,\varepsilon}\\!:=\begin{pmatrix}n+r_{k,\varepsilon}&0&\ldots&0&0\\\
0&n-2+r_{k,\varepsilon}&\ldots&0&0\\\ \vdots&\ddots&\ddots&\ddots&\vdots\\\
0&0&\ldots&-n+2+r_{k,\varepsilon}&0\\\ 0&0&\ldots&0&-n+r_{k,\varepsilon}\\\
\end{pmatrix}\,,$

where $r_{k,1}\\!:=2k\pi\hbar^{-1}i$ and $r_{k,-1}\\!:=(2k+1)\pi\hbar^{-1}i$.

It is not nard to see that $E_{n,\varepsilon}$, $F_{n,\varepsilon}$ (see
(1.7)) and $H_{n,k,\varepsilon}$ satisfy the relations in (2.1). Recall that
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ is defined as a
universal algebra; so we have a continuous $(n+1)$-dimensional representation
of it determined by

$E\mapsto E_{n,\varepsilon},\quad F\mapsto F_{n,\varepsilon},\quad H\mapsto
H_{n,k,\varepsilon}.$ (2.5)

We denote this representation by $T_{n,k,\varepsilon}$ and the representation
of $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ defined in (1.7) by
$T_{n,\varepsilon}$. It is easy to see that
$T_{n,\varepsilon}=T_{n,k,\varepsilon}\theta$.

######  Remark 2.7.

When $|e^{\hbar}|=1$ and $e^{\hbar}$ is not a root of unity, the Banach space
representation theory of $\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$
is quite different from that of $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$. For
example, the infinite-dimensional representation $S_{\lambda,2}$ in
Proposition 1.9 cannot be modified in the same way as $T_{n,\varepsilon}$ been
obtained from $T_{n,k,\varepsilon}$ because in this case we get a matrix with
unbounded sequence of eigenvalues. Moreover,
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ have no continuous
infinite-dimensional representation on a Banach space as is shown in Theorem
2.3.

######  Theorem 2.8.

Suppose that $e^{\hbar}$ is not a root of unity.

_(A)_ Every representation $T_{n,k,\varepsilon}$ defined by (2.5) is
irreducible.

_(B)_ Two representations $T_{n,k,\varepsilon}$ and
$T_{n^{\prime},k^{\prime},\varepsilon^{\prime}}$ are equivalent only when
$n=n^{\prime}$, $k=k^{\prime}$ and $\varepsilon=\varepsilon^{\prime}$.

_(C)_ Every continuous irreducible representation of
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ on a Banach space is
finite dimensional and equivalent to some $T_{n,k,\varepsilon}$, where
$n\in\mathbb{Z}_{+}$, $k\in\mathbb{Z}$ and $\varepsilon\in\\{-1,1\\}$.

We need a lemma. Let $V$ be a
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$-module. We say that an
eigenvector $v$ of $H$ in $V$ is a _weight vector_ for
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$. The corresponding
eigenvalue $\alpha$ is called a _weight_. If, in addition, $E\cdot v=0$, then
$v$ is called a _highest weight vector_ of weight $\alpha$ (cf. the versions
for $U(\mathfrak{s}\mathfrak{l}_{2})$ and
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ in [Ka95, p. 101, Definition V.4.1 and
p. 127, Definition VI.3.2])

######  Lemma 2.9.

_(cf.[Ka95, Lemmas V.4.3 and VI.3.4])_ Let $v$ be a highest weight vector for
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ of weight $\alpha$. For
any $p\in\mathbb{N}$ put $v_{p}\\!:=F^{p}\cdot v/[p]_{q}$. Then

$H\cdot v_{p}=(\alpha-2p)v_{p}\,,\qquad E\cdot
v_{p}=-[p-1]_{q,\lambda}v_{p-1}\,,\qquad F\cdot v_{p-1}=[p]_{q}v_{p}\,.$

###### Proof.

The first equality easily follows the relation $[H,F^{p}]=-2pF^{p}$. On the
other hand, $v$ is a highest weight vector for
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ of weight $e^{\hbar\alpha}$. So the
second and third equalities immediately follows from [Ka95, Lemma VI.3.4]. ∎

###### Proof of Theorem 2.8.

We use the representation theory of $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$.

(A) Since $T_{n,k,\varepsilon}\theta$ coincides with the irreducible
representation $T_{n,\varepsilon}$ of $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$,
the representation $T_{n,k,\varepsilon}$ is also irreducible.

(B) Note that $T_{n,k,\varepsilon}\theta$ and
$T_{n^{\prime},k^{\prime},\varepsilon^{\prime}}\theta$ is equivalent only when
$n=n^{\prime}$ and $\varepsilon=\varepsilon^{\prime}$ (see [Ka95, p. 128] or
[KS97, § 3.2, p. 62, Proposition 8]). On the other hand, the sets of
eigenvalues of $H_{n,k,\varepsilon}$ and $H_{n,k^{\prime},\varepsilon}$
coincide only when $k=k^{\prime}$.

(C) Let $T$ be a continuous irreducible representation of
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ on a Banach space $V$.
By Theorem 2.3, $V$ is finite dimensional. It is easy to see from the relation
$[H,E]=2E$ that any non-zero finite-dimensional
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$-module has a highest
weight vector $v$ [Ka95, p. 101, Proposition V.4.2]. Denote the corresponding
weight by $\alpha$. Then $v$ is a highest weight vector for
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ of weight $e^{\hbar\alpha}$ and we can
apply [Ka95, p. 128, Theorem VI.3.5]. In particular,
$e^{\hbar\alpha}=\varepsilon e^{\hbar n}$ for some $n\in\mathbb{Z}_{+}$ and
$\varepsilon\in\\{-1,1\\}$, $v_{p}=0$ for $p>n$ and
$\\{v_{0},v_{1},\ldots,v_{n}\\}$ is a basis of an irreducible
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$-submodule $V^{\prime}$ of $V$. Then
$\alpha=n+r_{k,\varepsilon}$ for some $k\in\mathbb{Z}$. Finally, Lemma 2.9
implies that $V^{\prime}$ is a
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$-submodule and so
$V=V^{\prime}$ being irreducible. Thus $T$ has the desired form. ∎

######  Theorem 2.10.

Suppose that $e^{\hbar}$ is not a root of unity. Then every continuous finite-
dimensional representation of
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ is completely reducible.

###### Proof.

We follow the argument for [Ka95, Theorem VII.2.2] with necessary
modifications.

Let $V$ be a continuous finite-dimensional left
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$-module and $V^{\prime}$
is a proper submodule of $V$. We need to show that $V^{\prime}$ can be
complemented

(1) Suppose that the codimension of $V^{\prime}$ is $1$. We proceed by
induction on the dimension of $V^{\prime}$.

If $\dim V^{\prime}=0$ the assertion is evident. Let $\dim V^{\prime}=1$. Then
$V^{\prime}$ and $V/V^{\prime}$ are simple one-dimensional modules of weights
$\alpha_{1}$ and $\alpha_{2}$, respectively. If $\alpha_{1}\neq\alpha_{2}$,
then it is easy to see that there is a submodule complementary to
$V^{\prime}$. If $\alpha_{1}=\alpha_{2}$, there exists a basis
$\\{v_{1},v_{2}\\}$ with $V^{\prime}=\mathbb{C}v_{1}$ such that $H\cdot
v_{1}=\alpha v_{1}$ and $H\cdot v_{2}=\alpha v_{2}+\alpha^{\prime}v_{1}$.
Since the representation is continuous, we have that $K\cdot
v_{1}=e^{\alpha}v_{1}$ and $K\cdot
v_{2}=e^{\alpha}v_{2}+e^{\alpha}\alpha^{\prime}v_{1}$. Arguing as in [Ka95,
Theorem VII.2.2], we get that $E$ and $F$ act on $V$ trivially and, moreover,
$\alpha^{\prime}=0$ and so $H$ is diagonalizable. This implies again that
there is a complementary submodule.

We now assume that $p>1$ and the assertion is proved in dimension smaller than
$p$. Let $\dim V^{\prime}=p$. If $V^{\prime}$ is not simple, then it contains
a submodule $V_{0}$ such that $\dim V_{0}<p$. So we can apply the induction
hypothesis and deduce the assertion by a standard argument (cf. the proof of
[Ka95, Theorem V.4.6]).

Suppose now that $V^{\prime}$ is simple. Recall that the quantum Casimir
element

$C_{q}\\!:=EF+\frac{q^{-1}K+qK^{-1}}{(q-q^{-1})^{2}}$

is central in $U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ [Ka95, Proposition
VI.4.1]. It is easy to see that $EF$ and $H$ commute and hence $\theta(C_{q})$
is central in $\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$. Moreover,
there is $\mu\in\mathbb{C}$ such that $\theta(C_{q})+\mu$ acts by $0$ on the
$1$-dimensional module $V/V^{\prime}$ and by a non-zero scalar on $V^{\prime}$
[Ka95, Lemma VII.2.1]. Arguing as in Part 1.b of the proof of [Ka95, Theorem
VII.2.2], we deduce that $V^{\prime}$ can be complemented.

(2) We now reduce assertion of the theorem to the case of codimension $1$.
Consider the vector space $W$ of linear maps from $V$ to $V^{\prime}$ whose
restriction to $V^{\prime}$ is multiplication by a scalar and the vector
subspace $W^{\prime}$ of linear maps such that this scalar is $0$. It is
obvious that $W^{\prime}$ has codimension $1$.

To endow $W$ and $W^{\prime}$ with module structures we need the fact that
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ is a topological Hopf
algebra [Ar20b, Proposition 5.2]. We denote the comultiplication, counit and
antipode by $\Delta$, $\varepsilon$ and $S$, respectively.

Note that
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}\mathbin{\widehat{\otimes}}V$
is a left
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}\mathbin{\widehat{\otimes}}\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$-module
with the multiplication determined by $(x\otimes y)\cdot(z\otimes
v)\\!:=xy\otimes z\cdot v$. (Here $\mathbin{\widehat{\otimes}}$ denotes the
complete projective tensor product of Fréchet spaces.) Then the vector space
of all linear maps from $V$ to $V^{\prime}$ is a left
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})$-module with respect to the
multiplication determined by

$(x\cdot f)(v)\\!:=(1\otimes f)((1\otimes S)\Delta(x)\cdot(1\otimes v)),$

where $f$ is a linear map,
$x\in\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ and $v\in V$; cf.
[Ka95, § III.5, p. 58, (5.5)]. It is not hard to see that $W$ and $W^{\prime}$
are modules with respect to this action.

Since $W^{\prime}$ has codimension $1$, it follows from Part (1) of this
proof, that there is a submodule $W^{\prime\prime}$ such that
$W=W^{\prime}\oplus W^{\prime\prime}$. Then there is $f$ such that
$\mathbb{C}f=W^{\prime\prime}$. Put $V^{\prime\prime}=\operatorname{Ker}f$. It
is clear that $V=V^{\prime}\oplus V^{\prime\prime}$ as a vector space.

To complete the proof it suffices to show that $V^{\prime\prime}$ is a
submodule. Since $W^{\prime\prime}$ has dimension $1$, it follows from Part
(C) of Theorem 2.8 that $H\cdot f=n\pi\hbar^{-1}if$ for some $n\in\mathbb{Z}$.
Since $\Delta(H)=H\otimes 1+1\otimes H$ and $S(H)=-H$ [Ar20b, Proposition
5.2], we have $(H\cdot f)(v)=H\cdot f(v)-f(H\cdot v)$ for every $v\in V$. If
$v\in V^{\prime\prime}$, then $f(v)=0$ and so $f(H\cdot v)=-(H\cdot
f)(v)=-n\pi\hbar^{-1}if(v)=0$. So $V^{\prime\prime}$ is invariant under $H$.
Arguing as in the end of the proof of [Ka95, Theorem VII.2.2], we deduce that
$V^{\prime\prime}$ is also invariant under $E$ and $F$. ∎

Denote by $\widetilde{\Sigma}_{\hbar}$ the set of equivalence classes of
irreducible finite-dimensional representations of
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ for given $\hbar$ (see
Theorem 2.8). Then for $\sigma\in\widetilde{\Sigma}_{\hbar}$ we have a
homomorphism
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}\to\mathrm{M}_{d_{\sigma}}(\mathbb{C})$,
where $d_{\sigma}$ is the dimension of $\sigma$. Denote by
$\widetilde{\varphi}$ the corresponding homomorphism

$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}\to\prod_{\sigma\in\widetilde{\Sigma}_{\hbar}}\mathrm{M}_{d_{\sigma}}(\mathbb{C}).$

######  Theorem 2.11.

_(cf. Theorem 1.7)_ Suppose that $e^{\hbar}$ is not a root of unity. Then
$\widetilde{\varphi}$ is a topological isomorphism.

###### Proof.

Let $\pi$ be a continuous homomorphism from
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ to a Banach algebra and
let $B$ be the range of $\pi$. It follows from Theorem 2.8 that $B$ is a
finite-dimensional Banach algebra. Then $B$ becomes a continuous
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$-module with respect to
$a\cdot b\\!:=\pi(a)b$. Since $e^{\hbar}$ is not a root of unity, it follows
from Theorem 2.10 that this module is completely reducible. Therefore $\pi$
factors through some finite product of algebras of the form
$\mathrm{M}_{d_{\sigma}}(\mathbb{C})$ and hence through
$\prod_{\sigma\in\widetilde{\Sigma}_{\hbar}}\mathrm{M}_{d_{\sigma}}(\mathbb{C})$.
Since $\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ is an Arens-
Michael algebra, $\widetilde{\varphi}$ is a topological isomorphism. ∎

######  Remark 2.12.

The Whitehead Lemma implies that the $\hbar$-adic formal deformation of
$U(\mathfrak{s}\mathfrak{l}_{2})$ is isomorphic to
$U(\mathfrak{s}\mathfrak{l}_{2})[[\hbar]]$ as an algebra; see [Dr89, § 4]. For
the analytic form we have a more subtle relation. Comparing the lists of
irreducible Banach space representations of
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$ and
$U(\mathfrak{s}\mathfrak{l}_{2})$ (cf. Remark 1.8), we have that when
$e^{\hbar}$ is not a root of unity, there is a (non-canonical) isomorphism

$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}\to\left(\prod_{k\in\mathbb{Z}}B_{k}\right)\otimes\mathbb{C}^{2}$

of Arens-Michael algebras, where each $B_{k}$ is isomorphic to
$\widehat{U}(\mathfrak{s}\mathfrak{l}_{2})$. Here each multiple corresponds to
a zero of the hyperbolic sine. (See a similar effect for some $2$-generated
algebras in [Ar19+].) On the other hand, if $\hbar^{\prime}$ is a root of
unity, then Theorem 2.6 implies that
$\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar^{\prime}}$ is not
isomorphic to $\widetilde{U}(\mathfrak{s}\mathfrak{l}_{2})_{\hbar}$.

##  3\. Concluding remarks and questions

It follows from Theorem 1.1 that all the irreducible Banach space
representations of $U_{q}(\mathfrak{g})$, where $\mathfrak{g}$ be a semisimple
complex Lie algebra and $|q|\neq 1$, are finite-dimensional. Moreover, it well
known that this is true (not only in the Banach-space case) when $q$ is a root
of unity. As a corollary, the irreducible Banach space representations can be
classified in both cases. On the other hand, in the third case when $|q|=1$
and $q$ is not a root of unity, there are infinite-dimensional topologically
irreducible representations; see Proposition 1.9. But the following two
questions are open.

######  Question 1.

Suppose that $|q|=1$ and $q$ is not a root of unity. Do exist infinite-
dimensional irreducible Banach space representations of $U_{q}(\mathfrak{g})$?
In particular, is it true that the representation $S_{\lambda,2}$ of
$U_{q}(\mathfrak{s}\mathfrak{l}_{2})$ considered in Proposition 1.9 is
irreducible when $q^{2}$ is not a root of $\lambda^{2}$?

######  Question 2.

Suppose that $|q|=1$ and $q$ is not a root of unity. Is it possible to give a
reasonable classification of topologically irreducible representations of
$U_{q}(\mathfrak{g})$ on Banach spaces or, at least, of topologically simple
Banach algebras that are completions of $U_{q}(\mathfrak{g})$.

Since $U_{q}(\mathfrak{g})$ has infinite-dimensional irreducible
representations, some information is lost with applying the Arens-Michael
enveloping homomorphism
$\iota\\!:U_{q}(\mathfrak{g})\to\widehat{U}_{q}(\mathfrak{g})$. The same can
be said for the natural homomorphism
$\theta\\!:U_{q}(\mathfrak{g})\to\widetilde{U}(\mathfrak{g})_{\hbar}$, where
$q=e^{\hbar}$. (See [Ar20b, Remark 5.5] for
$\widetilde{U}(\mathfrak{g})_{\hbar}$. The definition of $\theta$ is given for
$\mathfrak{s}\mathfrak{l}_{2}$ in (2.4); in the general case it is defined in
a similar way.) But there is hope at least that the kernels of $\iota$ and
$\theta$ are trivial.

######  Question 3.

Are the homomorphisms
$\iota\\!:U_{q}(\mathfrak{g})\to\widehat{U}_{q}(\mathfrak{g})$ and
$\theta\\!:U_{q}(\mathfrak{g})\to\widetilde{U}(\mathfrak{g})_{\hbar}$ are
always injective?

The first part is a partial case of Question 2 in [Ar20b].

######  Remark 3.1.

In the classical case, the assertion that the Arens-Michael enveloping
homomorphism $U(\mathfrak{g})\to\widehat{U}(\mathfrak{g})$ is injective can
easily be deduced from the well-known fact that the adjoint representation of
$U(\mathfrak{g})$ is faithful and locally finite. When $q$ is not a root of
unity, a similar argument cannot be applied to $U_{q}(\mathfrak{g})$ because
the adjoint representation is not locally finite (see [Sm92, p. 153]). On the
other hand, by the result of Pedchenko [Pe15] mentioned in the introduction,
the map
$\mathcal{O}(\mathbb{C}\times\mathbb{C}^{\times}\times\mathbb{C})\to\widehat{U}_{q}(\mathfrak{s}\mathfrak{l}_{2})$
is a topological isomorphism when $|q|=1$. Therefore
$\iota\\!:U_{q}(\mathfrak{s}\mathfrak{l}_{2})\to\widehat{U}_{q}(\mathfrak{s}\mathfrak{l}_{2})$
is injective under this assumption.

## References

  * [Ar19+] O. Yu. Aristov, _The relation “commutator equals function” in Banach algebras_ , in Russian, Math. Notes (to appear) arXiv: math/1911.03293, (2019).
  * [Ar20a] O. Yu. Aristov, _Arens-Michael envelopes of nilpotent Lie algebras, functions of exponential type, and homological epimorphisms_ , Tr. Mosk. Mat. Obs Trans. 81:1 (2020), 117–136, Moscow Math. Soc., arXiv:1810.13213.
  * [Ar20b] O. Yu. Aristov, _Holomorphically finitely generated Hopf algebras and quantum Lie groups_ , arXiv:2006.12175 (2020).
  * [BS01] D. Beltiţă, M. Şabac. _Lie algebras of bounded operators_ , Birkhäuser, Basel, Boston, Berlin, 2001.
  * [Dr89] V. G. Drinfeld, _Almost cocommutative Hopf algebras_ , Algebra i Analiz, 1:2 (1989), 30–46; Leningrad Math. J., 1:2 (1990), 321–342.
  * [Du19] N. Dupre, _Rigid analytic quantum groups and quantum Arens-Michael envelopes_ , J. Algebra 537 (2019), 98–146.
  * [He93] A. Ya. Helemskii, _Banach and polynormed algebras: General theory, representations, homology_ , Nauka, Moscow 1989 (Russian); English transl.: Oxford University Press, 1993.
  * [Ka95] C. Kassel, _Quantum Groups_ , Springer, 1995.
  * [KS97] A. U. Klimyk, K. Schmüdgen, _Quantum groups and their representations_ , Springer, Berlin 1997.
  * [Lu93] G. Lusztig, _Introduction to quantum groups_ , Birkhäuser, Boston, 1993 .
  * [Ly13] A. Lyubinin, _$p$ -adic quantum hyperenveloping algebra for $\mathfrak{s}\mathfrak{l}_{2}$_, arXiv:1312.4372, 2013.
  * [Pi06] A. Yu. Pirkovskii, _Arens-Michael enveloping algebras and analytic smash products_ , Proc. Amer. Math. Soc. 134 (2006), 2621–2631.
  * [Pi08] A. Yu. Pirkovskii, _Arens-Michael envelopes, homological epimorphisms, and relatively quasi-free algebras_ , (Russian), Tr. Mosk. Mat. Obs. 69 (2008), 34–125; English translation in Trans. Moscow Math. Soc. (2008), 27–104.
  * [Pi11] A. Yu. Pirkovskii, _The Arens-Michael envelope of a smash product_ , arXiv:1101.0166.
  * [Pi14] A. Yu. Pirkovskii, _Noncommutative Analogues of Stein Spaces of Finite Embedding Dimension_ , Algebraic methods in functional analysis: the Victor Shulman anniversary volume, Operator theory Aadvances and applications, 233, ed. Todorov, Turowska, Birkhauser 2014, 135–153.
  * [Pi15] A. Yu. Pirkovskii, _Holomorphically finitely generated algebras_ , J. Noncommutative Geom. 9:1 (2015), 215–264.
  * [Pe15] D. Pedchenko, _Arens-Michael envelopes of Jordanian Plane and $U_{q}(\mathfrak{s}\mathfrak{l}(2))$_, Master Thesis, National Research University Higher School of Economics, Department of Mathematics, Moscow, 2015.
  * [Ro88] M. Rosso, _Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra_ , Commun. Math. Phys. 117 (1988), 581–593.
  * [Sm92] S. P. Smith, _Quantum groups: an introduction and survey for ring theorists_ in: S. Montgomery, L. Small eds. _Noncommutative Rings_ , MSRI Publ. 24, Springer, Berlin 1992, 131–178.
  * [Sm18+] C. Smith, _On analytic analogues of quantum groups_ , arXiv:1806.10502, 2018.
  * [Ta72] J. L. Taylor, _A general framework for a multi-operator functional calculus_ , Adv. Math. 9 (1972), 183–252.
  * [We52] J. Wermer, _On invariant subspaces of normal operators_ , Proc. Amer. Math. Soc. 3 (1952), 270–277.