$\displaystyle({}_{V}\phi^{e^{i}}_{e_{j}})^{(a)}$.

By Proposition 6.2, $\displaystyle\Phi_{n}$ induces a well-defined algebra
morphism
$\displaystyle{\mathcal{L}}_{0,n}\rightarrow\mathbb{U}_{q}^{{\otimes}n}$ if it
preserves the fusion and exchange relations. Using the graphical encoding
recalled in Figure 6.1, this is shown in the next two figures in the case
$\displaystyle n=2$, which generalizes immediately to any $\displaystyle n$.
The symbol $\displaystyle\stackrel{{\scriptstyle\centerdot}}{{=}}$ means
equality up to isotopy. Similarly, the relation (53) is proved by the third
figure.

$\displaystyle\stackrel{{\scriptstyle\centerdot}}{{=}}$$\displaystyle
W$$\displaystyle V$$\displaystyle W$$\displaystyle V$

$\displaystyle V\otimes W$$\displaystyle V$$\displaystyle
W$$\displaystyle\stackrel{{\scriptstyle\centerdot}}{{=}}$

$\displaystyle\stackrel{{\scriptstyle\centerdot}}{{=}}$$\displaystyle\stackrel{{\scriptstyle\centerdot}}{{=}}$VVV

Figure 6.2 : the colored braid identities representing the exchange relation
(47) (top),

the fusion relation (27) (middle), and the relation (53) (bottom).

The linear map defined by $\displaystyle\stackrel{{\scriptstyle
V}}{{M}}{}^{\\!\\!(a)}\mapsto R_{0a}R_{0a}^{\prime}$ on the
$\displaystyle\mathbb{C}(q^{1/D})$-vector space spanned by all the matrix
coefficients $\displaystyle({}_{V}\phi^{e^{i}}_{e_{j}})^{(a)}$, for every
$\displaystyle 1\leq a\leq n$ and object $\displaystyle V$ of
$\displaystyle\mathcal{C}$, is injective by Theorem 4.3. It differs from
$\displaystyle\Phi_{n}$ by a linear isomorphism (induced on the $\displaystyle
a$-th tensorand by conjugating with $\displaystyle R_{0n}\ldots R_{0a+1}$), so
$\displaystyle\Phi_{n}$ is injective.

That $\displaystyle\Phi_{n}$ maps $\displaystyle{\mathcal{L}}_{0,n}$ to
$\displaystyle\tilde{U}_{q}^{\otimes
n}\otimes\mathbb{C}(q^{1/D})\subset\mathbb{U}_{q}^{{\otimes}n}$ is an
immediate consequence of Theorem 4.3.

Finally, let us show that $\displaystyle\Phi_{n}$ is a morphism of
$\displaystyle U_{q}$-modules. Since both multiplications in
$\displaystyle{\mathcal{L}}_{0,n}$ and
$\displaystyle\mathbb{U}_{q}^{{\otimes}n}$ commute with the respective actions
of $\displaystyle U_{q}$, it is enough to check this on generators. For those
given by the matrix coefficients we then have to show that

(54) $(id\otimes\Phi_{n})\left(coad^{r}(y)\left(\stackrel{{\scriptstyle
V}}{{M}}{}^{\\!\\!(a)}\right)\right)=(id\otimes
ad^{r}(y))\left((id\otimes\Phi_{n})\left(\stackrel{{\scriptstyle
V}}{{M}}{}^{\\!\\!(a)}\right)\right)$

for every object $\displaystyle V$ in $\displaystyle\mathcal{C}$,
$\displaystyle 1\leq a\leq n$ and $\displaystyle y\in U_{q}$. In the case
$\displaystyle a=n$, $\displaystyle\Phi_{n}$ has the same expression as the
RSD map. Hence (54) follows from the statement of equivariance in Proposition
4.3. This can also be proved easily:

$\displaystyle\displaystyle(id\otimes
ad^{r}(y))\left((id\otimes\Phi_{n})\left(\stackrel{{\scriptstyle
V}}{{M}}{}^{\\!\\!(n)}\right)\right)=(id\otimes ad^{r}(y))(\pi_{V}\otimes
id)(R_{0n}R_{0n}^{\prime})$
$\displaystyle\displaystyle=\sum_{(RR^{\prime}),(y)}\pi_{V}((RR^{\prime})_{(1)})S(y_{(1)})(RR^{\prime})_{(2)}y_{(2)}$
$\displaystyle\displaystyle=\sum_{(RR^{\prime}),(y)}\pi_{V}(y_{(1)}S(y_{(2)}))\pi_{V}((RR^{\prime})_{(1)})S(y_{(3)})(RR^{\prime})_{(2)}y_{(4)}$
$\displaystyle\displaystyle=\sum_{(RR^{\prime}),(y)}\pi_{V}(y_{(1)})\pi_{V}((RR^{\prime})_{(1)})\pi_{V}(S(y_{(2)}))(RR^{\prime})_{(2)}S(y_{(3)})y_{(4)}$
$\displaystyle\displaystyle=\sum_{(RR^{\prime}),(y)}\pi_{V}(y_{(1)})\pi_{V}((RR^{\prime})_{(1)})\pi_{V}(S(y_{(2)}))(RR^{\prime})_{(2)}$
$\displaystyle\displaystyle=(id\otimes\Phi_{n})\left(coad^{r}(y)\left(\stackrel{{\scriptstyle
V}}{{M}}{}^{\\!\\!(n)}\right)\right),$

where we have used $\displaystyle[RR^{\prime},(S\otimes S)(\Delta(y))]=0.$

More generally, by writing the actions explicitly, that result implies as well
the relation

(55) $\sum_{(y)}\left((\pi_{V}(y_{(1)})\otimes
1)R_{0a}R_{0a}^{\prime}(\pi_{V}(S(y_{(2)}))\otimes
1)\right)=\sum_{(y)}(1\otimes\mathfrak{i}_{a}(S(y_{(1)})))R_{0a}R_{0a}^{\prime}(1\otimes\mathfrak{i}_{a}(y_{(2)})).$

Now set

$\displaystyle R^{(a)}=R_{0n}\ldots R_{0a+1}$

and denote by
$\displaystyle\mathfrak{i}_{an}\colon\mathbb{U}_{q}^{{\otimes}n-a}\rightarrow
1^{\otimes
a}\otimes\mathbb{U}_{q}^{{\otimes}n-a}\subset\mathbb{U}_{q}^{{\otimes}n}$ the
identification map with the final $\displaystyle n-a$ tensorands. Then

$\displaystyle\displaystyle(1\otimes\Phi_{n})$
$\displaystyle\displaystyle\textstyle\left(coad^{r}(y)\left(\stackrel{{\scriptstyle
V}}{{M}}{}^{\\!\\!(a)}\right)\right)=\sum_{(y)}(\pi_{V}(y_{(1)})\otimes
1)R^{(a)}R_{0a}R_{0a}^{\prime}R^{(a)}{}^{\\!\\!-1}(\pi_{V}(S(y_{(2)}))\otimes
1)$ $\displaystyle\displaystyle=\textstyle\sum_{(y)}(\pi_{V}(y_{(1)})\otimes
1)R^{(a)}R_{0a}R_{0a}^{\prime}$ $\displaystyle\displaystyle\hskip
99.58464pt\times\
(1\otimes\mathfrak{i}_{an}(\Delta^{(n-a-1)}(\underbrace{S(y_{(2)})y_{(3)}}_{=\varepsilon(y_{(2)})})))R^{(a)}{}^{\\!\\!-1}(\pi_{V}(S(y_{(4)}))\otimes
1)$ $\displaystyle\displaystyle=\textstyle\sum_{(y)}(\pi_{V}(y_{(1)})\otimes
1)R^{(a)}(1\otimes\mathfrak{i}_{an}(\Delta^{(n-a-1)}(S(y_{(2)}))))$
$\displaystyle\displaystyle\hskip 99.58464pt\times\
R_{0a}R_{0a}^{\prime}(1\otimes\mathfrak{i}_{an}(\Delta^{(n-a-1)}(y_{(3)})))R^{(a)}{}^{\\!\\!-1}(\pi_{V}(S(y_{(4)}))\otimes
1).$

Since $\displaystyle R\Delta=\Delta^{\prime}R$ in
$\displaystyle\mathbb{U}_{q}^{\otimes 2}$, we have
$\displaystyle\textstyle\sum_{(y)}R^{-1}(y_{(2)}\otimes
y_{(1)})=\sum_{(y)}(y_{(1)}\otimes y_{(2)})R^{-1}$ and then

$\displaystyle\displaystyle\sum_{(y)}(1\otimes S(y_{(1)}))R(y_{(2)}\otimes 1)$
$\displaystyle\displaystyle=\sum_{(y)}(y_{(1)}\otimes 1)R(1\otimes
S(y_{(2)}))$ $\displaystyle\displaystyle\sum_{(y)}(S(y_{(1)})\otimes
1)R^{-1}(1\otimes y_{(2)})$ $\displaystyle\displaystyle=\sum_{(y)}(1\otimes
y_{(1)})R^{-1}(S(y_{(2)})\otimes 1)$

by applying $\displaystyle id\otimes S$, and using $\displaystyle(id\otimes
S)(R^{-1})=R$. We deduce recursively

$\textstyle\sum_{(y)}(\pi_{V}(y_{(1)})\otimes
1)R^{(a)}(1\otimes\mathfrak{i}_{an}(\Delta^{(n-a-1)}(S(y_{(2)}))))=\\\
(1\otimes\mathfrak{i}_{an}(\Delta^{(n-a-1)}(S(y_{(1)}))))R^{(a)}(\pi_{V}(y_{(2)})\otimes
1)$

and

$\textstyle\sum_{(y)}(1\otimes\mathfrak{i}_{an}(\Delta^{(n-a-1)}(y_{(1)})))R^{(a)}{}^{\\!\\!-1}(\pi_{V}(S(y_{(2)}))\otimes
1)=\\\ (\pi_{V}(S(y_{(1)}))\otimes
1)R^{(a)}{}^{\\!\\!-1}(1\otimes\mathfrak{i}_{an}(\Delta^{(n-a-1)}(y_{(2)}))).$

So

$\displaystyle\displaystyle(1\otimes\Phi_{n})$
$\displaystyle\displaystyle\left(coad^{r}(y)\left(\stackrel{{\scriptstyle
V}}{{M}}{}^{\\!\\!(a)}\right)\right)$
$\displaystyle\displaystyle\textstyle=\sum_{(y)}(1\otimes\mathfrak{i}_{an}(\Delta^{(n-a-1)}(S(y_{(1)}))))R^{(a)}(\pi_{V}(y_{(2)})\otimes
1)$ $\displaystyle\displaystyle\hskip 113.81102pt\times\
R_{0a}R_{0a}^{\prime}(\pi_{V}(S(y_{(3)}))\otimes
1)R^{(a)}{}^{\\!\\!-1}(1\otimes\mathfrak{i}_{an}(\Delta^{(n-a-1)}(y_{(4)})))$
$\displaystyle\displaystyle\textstyle=\sum_{(y)}(1\otimes\mathfrak{i}_{an}(\Delta^{(n-a-1)}(S(y_{(1)}))))R^{(a)}(1\otimes\mathfrak{i}_{a}(S(y_{(2)})))$
$\displaystyle\displaystyle\hskip 113.81102pt\times\
R_{0a}R_{0a}^{\prime}(1\otimes\mathfrak{i}_{a}(y_{(3)}))R^{(a)}{}^{\\!\\!-1}(1\otimes\mathfrak{i}_{an}(\Delta^{(n-a-1)}(y_{(4)})))$
$\displaystyle\displaystyle\textstyle=\sum_{(y)}(1\otimes\mathfrak{i}_{an}(\Delta^{(n-a-1)}(S(y_{(1)}))))(1\otimes\mathfrak{i}_{a}(S(y_{(2)})))R^{(a)}$
$\displaystyle\displaystyle\hskip 113.81102pt\times\
R_{0a}R_{0a}^{\prime}R^{(a)}{}^{\\!\\!-1}(1\otimes\mathfrak{i}_{a}(y_{(3)}))(1\otimes\mathfrak{i}_{an}(\Delta^{(n-a-1)}(y_{(4)})))$
$\displaystyle\displaystyle\textstyle=\sum_{(y)}(1\otimes\mathfrak{i}_{a-1n}(\Delta^{(n-a)}(S(y_{(1)}))))R^{(a)}R_{0a}R_{0a}^{\prime}R^{(a)}{}^{\\!\\!-1}(1\otimes\mathfrak{i}_{a-1n}(\Delta^{(n-a)}(y_{(2)})))$
$\displaystyle\displaystyle\textstyle=\sum_{(y)}(1\otimes\Delta^{(n-1)}(S(y_{(1)})))R^{(a)}R_{0a}R_{0a}^{\prime}R^{(a)}{}^{\\!\\!-1}(1\otimes\Delta^{(n-1)}(y_{(2)}))$

where we use (55) in the second equality, and the others follow from trivial
commutations between elements lying on different tensorands, and the property
$\displaystyle(S\otimes S)\Delta^{\prime}=\Delta S$. This proves (54). The
relation (53) has been proved with a picture which encapsulates the following
simple computation:

(56) $\displaystyle\displaystyle(id\otimes\Phi_{n})(\stackrel{{\scriptstyle
V}}{{M}}{}^{\\!\\!(1)}\ldots\stackrel{{\scriptstyle V}}{{M}}{}^{\\!\\!(n)})=$
(57) $\displaystyle\displaystyle=\prod_{a=1}^{n}(\left(R_{0n}\ldots
R_{0a+1}\right)R_{0a}R_{0a}^{\prime}\left(R_{0n}\ldots R_{0a+1}\right)^{-1})$
(58) $\displaystyle\displaystyle=R_{0n}\ldots R_{01}R_{01}^{\prime}\ldots
R_{0n}^{\prime}=(\pi_{V}\otimes\Delta^{(n-1)})(RR^{\prime}).$

This concludes the proof. $\displaystyle\Box$

Denote by $\displaystyle({\tilde{U}}_{q}^{{\otimes}n})^{lf}$ the set of
invariant elements of $\displaystyle{\tilde{U}}_{q}^{{\otimes}n}$ with respect
to the action (52). Consider the subalgebras
$\displaystyle({\mathcal{L}}_{0,n})^{U_{q}}$ and
$\displaystyle({\mathbb{U}}_{q}^{\otimes n})^{U_{q}}$ of invariant elements of
the $\displaystyle U_{q}$-module algebras $\displaystyle{\mathcal{L}}_{0,n}$
and $\displaystyle{\mathbb{U}}_{q}^{\otimes n}$ respectively. We have:

######  Proposition 6.5.

The Alekseev map sends $\displaystyle{\mathcal{L}}_{0,n}$ isomorphically onto
$\displaystyle({\tilde{U}}_{q}^{{\otimes}n})^{lf}\otimes\mathbb{C}(q^{1/D})$,
and its restriction to invariant elements affords an isomorphism
$\displaystyle\Phi_{n}:{\mathcal{L}}_{0,n}^{U_{q}}\rightarrow({\tilde{U}}_{q}^{{\otimes}n})^{U_{q}}\otimes\mathbb{C}(q^{1/D})$.

Proof. Consider the first claim. By (49) the action $\displaystyle coad^{r}$
on $\displaystyle{\mathcal{L}}_{0,n}$ preserves the spaces of matrix
coefficients of each object $\displaystyle V$ of $\displaystyle\mathcal{C}$.
Since it gives $\displaystyle{\mathcal{L}}_{0,n}$ a structure of module
algebra, it follows that $\displaystyle coad^{r}$ is locally finite. The
equivariance of $\displaystyle\Phi_{n}$ then implies that its image is
contained in
$\displaystyle({\tilde{U}}_{q}^{{\otimes}n})^{lf}\otimes\mathbb{C}(q^{1/D})$.
As $\displaystyle\Phi_{n}$ is injective, it remains to prove that it maps
$\displaystyle{\mathcal{L}}_{0,n}$ surjectively onto
$\displaystyle({\tilde{U}}_{q}^{{\otimes}n})^{lf}\otimes\mathbb{C}(q^{1/D})$.
Let us denote by $\displaystyle ad_{n}^{r}$ the action (52). We can consider
another action of $\displaystyle U_{q}$ on
$\displaystyle{\mathbb{U}}_{q}^{\otimes n}$ defined as the $\displaystyle
n$-fold tensor product of the adjoint action of $\displaystyle U_{q}$ on
$\displaystyle{\mathbb{U}}_{q}^{\otimes n}$, which we denote
$\displaystyle(ad^{r})^{\otimes n}$. It is thus defined as follows:

$\displaystyle(ad^{r})^{\otimes n}(y)(a^{(1)}\otimes\ldots\otimes
a^{(n)})=\sum_{(y)}ad^{r}(y_{(1)})(a^{(1)})\otimes\ldots\otimes
ad^{r}(y_{(n)})(a^{(n)}).$

We first show that these two actions on
$\displaystyle{\mathbb{U}}_{q}^{\otimes n}$ define isomorphic modules. We give
the proof for $\displaystyle n=2$, the generalisation to any $\displaystyle n$
being straightforward. Consider the map

$\displaystyle\psi:{\mathbb{U}}_{q}^{\otimes
2}\rightarrow{\mathbb{U}}_{q}^{\otimes 2},a\otimes b\mapsto
R^{-1}_{12}(a\otimes 1)R_{12}(1\otimes b).$

We claim that $\displaystyle\psi$ intertwines $\displaystyle(ad^{r})^{\otimes
2}$ and $\displaystyle ad_{2}^{r}$. This follows from a small variation of the
previous proof. Indeed we have

$\displaystyle\displaystyle ad_{2}^{r}(y)(\psi(a\otimes b))$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{(y),(R),(R^{-1})}S(y_{(2)})(R^{-1})_{(1)}aR_{(1)}y_{(3)}\otimes
S(y_{(1)})(R^{-1})_{(2)}R_{(2)}by_{(4)}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{(y),(R),(R^{-1})}(R^{-1})_{(1)}S(y_{(1)})aR_{(1)}y_{(3)}\otimes(R^{-1})_{(2)}S(y_{(2)})R_{(2)}by_{(4)}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{(y),(R),(R^{-1})}(R^{-1})_{(1)}S(y_{(1)})ay_{(2)}R_{(1)}\otimes(R^{-1})_{(2)}R_{(2)}S(y_{(3)})by_{(4)}$
$\displaystyle\displaystyle=\psi((ad^{r})^{\otimes 2}(a\otimes b))).$

It is easy to show that $\displaystyle\psi$ is an isomorphism. Now, let
$\displaystyle
z\in({\tilde{U}}_{q}^{{\otimes}2})^{lf}\otimes\mathbb{C}(q^{1/D})$. Then
$\displaystyle\psi^{-1}(z)$ is locally finite for the action
$\displaystyle(ad^{r})^{\otimes 2}$. By the main theorem of [60], the set of
locally finite elements of $\displaystyle{\tilde{U}}_{q}^{{\otimes}n}$ for
$\displaystyle(ad^{r})^{\otimes n}$ is equal to
$\displaystyle({\tilde{U}}_{q}^{lf})^{\otimes n}$. Therefore
$\displaystyle\psi^{-1}(z)\in{\tilde{U}}_{q}^{lf}\otimes{\tilde{U}}_{q}^{lf}\otimes\mathbb{C}(q^{1/D})$,
and from the surjectivity of the map $\displaystyle\Phi_{1}$ onto
$\displaystyle{\tilde{U}}_{q}^{lf}$, we deduce that $\displaystyle
z=\psi\circ\Phi_{1}^{\otimes 2}(t)$ for some $\displaystyle
t\in{\mathcal{L}}_{0,1}^{\otimes 2}$. But because of the identity
$\displaystyle
R_{12}^{-1}R_{01}R^{\prime}_{01}R_{12}=R_{02}R_{01}R^{\prime}_{01}R_{02}^{-1}$,
which is a consequence of the Yang-Baxter equation, we obtain that
$\displaystyle z=\psi\circ\Phi_{1}^{\otimes 2}(t)=\Phi_{2}(t)$, where we have
identified the two vector spaces $\displaystyle{\mathcal{L}}_{0,2}$ and
$\displaystyle{\mathcal{L}}_{0,1}^{\otimes 2}$. This shows
$\displaystyle\Phi_{2}$ is a surjection onto
$\displaystyle({\tilde{U}}_{q}^{{\otimes}2})^{lf}\otimes\mathbb{C}(q^{1/D})$.

By the previous theorem we have an inclusion
$\displaystyle\Phi_{n}({\mathcal{L}}_{0,n}^{U_{q}})\subset({\tilde{U}}_{q}^{{\otimes}n})^{U_{q}}\otimes\mathbb{C}(q^{1/D})$.
The above argument applies in particular to $\displaystyle
ad_{n}^{r}$-invariant elements $\displaystyle
z\in({\tilde{U}}_{q}^{{\otimes}n})^{U_{q}}\otimes\mathbb{C}(q^{1/D})$.
Together with the injectivity of $\displaystyle\Phi_{n}$ it implies that this
inclusion is an equality. Therefore the second claim follows.
$\displaystyle\Box$

######  Remark 6.6.

This last result is a generalisation to $\displaystyle n\geq 1$ of Theorem 4.3
(3).

By the integrality property recalled in Section 3, the factors $\displaystyle
q^{\pm(\lambda,\mu)}$ cancel each other in the exchange relations (47), when
$\displaystyle V=V_{\lambda}$ and $\displaystyle W=V_{\mu}$,
$\displaystyle\lambda,\mu\in P_{+}$ are endowed with their canonical basis
$\displaystyle{\mathcal{B}}_{\lambda}$, $\displaystyle{\mathcal{B}}_{\mu}$.
Therefore these relations hold in $\displaystyle A$, and we can set:

######  Definition 6.7.

The $\displaystyle A$-algebra $\displaystyle{\mathcal{L}}_{0,n}^{A}$ is the
quotient of the $\displaystyle A$-algebra freely generated by the matrix
coefficients $\displaystyle({}_{V_{\lambda}}\phi^{e^{i}}_{e_{j}})^{(a)}$ by
the ideal generated by the relations (27) and (47), for every
$\displaystyle\lambda\in P_{+}$ and $\displaystyle 1\leq a\leq n$, where
$\displaystyle{\mathcal{B}}_{\lambda}=\\{e_{i}\\}$.

Clearly
$\displaystyle{\mathcal{L}}_{0,n}^{A}\otimes\mathbb{C}(q^{1/D})={\mathcal{L}}_{0,n}$.

######  Lemma 6.8.

The action $\displaystyle coad^{r}$ yields on
$\displaystyle{\mathcal{L}}_{0,n}^{A}$ a structure of right $\displaystyle
U_{A}$-module algebra, and the Alekseev map restricts to an embedding of
$\displaystyle U_{A}$-module algebras
$\displaystyle\Phi_{n}\colon{\mathcal{L}}_{0,n}^{A}\rightarrow\tilde{U}_{A}^{\otimes
n}$.

Proof. The first claim follows immediately from the case of
$\displaystyle{\mathcal{L}}_{0,1}^{A}$ (see Lemma 4.11), since
$\displaystyle{\mathcal{L}}_{0,n}^{A}=({\mathcal{L}}_{0,1}^{A})^{\otimes n}$
as an $\displaystyle A$-module. For the second claim, one simply note that
Theorem 3.1 implies that $\displaystyle R_{0n}\ldots R_{0a+1}\in
End_{A}({}_{A}V_{\lambda})\otimes\tilde{U}_{A}^{\otimes n}$, for every
$\displaystyle\lambda\in P_{+}$ and $\displaystyle 1\leq a<n$, and use that
$\displaystyle id\otimes\Phi_{n}$ is defined on generators of $\displaystyle
End_{A}({}_{A}V_{\lambda})\otimes{\mathcal{L}}_{0,n}^{A}$ as $\displaystyle
id\otimes\Phi_{1}^{\otimes n}$ followed by conjugations with matrices of the
form $\displaystyle R_{0n}\ldots R_{0a+1}$. $\displaystyle\Box$

######  Proposition 6.9.

$\displaystyle{\mathcal{L}}_{0,n}$, and therefore its subalgebras
$\displaystyle{\mathcal{L}}_{0,n}^{A}$ and
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}$, does not have non trivial
zero divisors.

Proof. Because of the injectivity of $\displaystyle\Phi_{n}$ it is sufficient
to show that $\displaystyle\tilde{U}_{q}^{\otimes n}$ and
$\displaystyle\tilde{U}_{A}^{\otimes n}$ have no zero divisors. We note that
$\displaystyle U_{A}(\mathfrak{g})^{\otimes n}=U_{A}(\mathfrak{g}^{\oplus
n}).$ Then the result for $\displaystyle{U}_{A}^{\otimes n}$ is a consequence
of Corollary 1.8 of [29] applied to $\displaystyle\mathfrak{g}^{\oplus n}.$
Note that in that paper it is assumed that the Cartan matrix is indecomposable
but their method, which consists in proving that an associated graded algebra
is quasipolynomial, does not use this assumption. It applies as well for
$\displaystyle\tilde{U}_{A}^{\otimes n},\tilde{U}_{q}^{\otimes n}$.
$\displaystyle\Box$

In the case of $\displaystyle\mathfrak{g}=sl(2)$ we can restrict the target:

######  Proposition 6.10.

The Alekseev map
$\displaystyle\Phi_{n}\colon{\mathcal{L}}_{0,n}(sl(2))\rightarrow\tilde{U}_{q}(sl(2))^{{\otimes}n}\otimes\mathbb{C}(q^{1/2})$
takes values in $\displaystyle
U_{q}(sl(2))^{{\otimes}n}\otimes\mathbb{C}(q^{1/2})$, and yields an embedding
of $\displaystyle U_{A}$-module algebras
$\displaystyle\Phi_{n}\colon{\mathcal{L}}_{0,n}^{A}(sl(2))\rightarrow
U_{A}(sl(2))^{{\otimes}n}$.

Proof. The second claim follows from the first and Lemma 6.8. Consider the
first claim. For every $\displaystyle 1\leq a\leq n$ and object $\displaystyle
V$ of $\displaystyle\mathcal{C}$ we have $\displaystyle
R_{0a}R_{0a}^{\prime}\in End(V)\otimes U_{q}^{lf}(sl(2))^{\otimes n}$ by
Proposition 5.4. We have to show that conjugating by $\displaystyle
R_{0n}\ldots R_{0a+1}\in End(V)\otimes\mathbb{U}_{q}(sl(2))^{{\otimes}n}$ maps
$\displaystyle End(V)\otimes U_{q}^{lf}(sl(2))^{\otimes n}$ to $\displaystyle
End(V)\otimes U_{q}(sl(2))^{\otimes n}$. It is enough to prove it for
$\displaystyle V=V_{2}$, and for the conjugation by $\displaystyle R_{0a+1}$
only (the general case follows from this one by an easy induction). Recall the
expression of $\displaystyle R$ in (12). Using that $\displaystyle E$,
$\displaystyle F$ act nilpotently on $\displaystyle V_{2}$ with order
$\displaystyle 2$, and $\displaystyle R^{-1}=(S\otimes id)(R)$, we get

(61) $\displaystyle\displaystyle
R_{0a+1}=(\pi_{V_{2}}\otimes\mathfrak{i}_{a+1})(R)$
$\displaystyle\displaystyle=\left(\begin{array}[]{ll}1\otimes
q^{H/2}&(q-q^{-1})1\otimes q^{H/2}F\\\ 0&1\otimes q^{-H/2}\end{array}\right)$
(64) $\displaystyle\displaystyle
R_{0a+1}^{-1}=(\pi_{V_{2}}\otimes\mathfrak{i}_{a+1})(R^{-1})$
$\displaystyle\displaystyle=\left(\begin{array}[]{ll}1\otimes
q^{-H/2}&-q(q-q^{-1})1\otimes q^{H/2}F\\\ 0&1\otimes
q^{H/2}\end{array}\right)$

where $\displaystyle q^{H/2}\in\mathbb{U}_{q}(sl(2))$ is defined in section 3,
and for each matrix entry we write only the components in the $\displaystyle
a$-th and $\displaystyle a+1$-th tensorands of
$\displaystyle\mathbb{U}_{q}(sl(2))^{{\otimes}n}$ (the others being
$\displaystyle 1$’s). With this convention, let

(65) $A=\left(\begin{array}[]{ll}u\otimes 1&v\otimes 1\\\ w\otimes 1&x\otimes
1\end{array}\right)\in End(V_{2})\otimes\mathbb{U}_{q}(sl(2))^{{\otimes}n}.$

Then

$\displaystyle R_{0a+1}AR_{0a+1}^{-1}=\left(\begin{array}[]{cc}u\otimes
1+q^{-1}(q-q^{-1})w\otimes F&-(q^{2}-1)u\otimes KF-q^{2}(q-q^{-1})w\otimes
KF^{2}\\\ &\hskip 28.45274pt+v\otimes K+q^{-1}(q-q^{-1})x\otimes KF\\\ &\\\
w\otimes K^{-1}&-q(q-q^{-1})w\otimes F+x\otimes 1\end{array}\right).$

This matrix has entries in $\displaystyle U_{q}(sl(2))^{\otimes n}$ if the
matrix entries of $\displaystyle A$ belong to $\displaystyle
U_{q}(sl(2))^{\otimes n}$. Conjugating recursively $\displaystyle
R_{0a}R_{0a}^{\prime}$ with $\displaystyle R_{0i}$, for $\displaystyle i$ from
$\displaystyle a+1$ to $\displaystyle n$, by the same computation we deduce
that $\displaystyle\Phi_{n}$ takes values in $\displaystyle
U_{q}(sl(2))^{\otimes n}$. $\displaystyle\Box$

We are now going to define a localization of
$\displaystyle{\mathcal{L}}_{0,n}(sl(2))$, which will satisfy a generalization
of Proposition 5.6. We need the following lemma. For every $\displaystyle u\in
U_{q}(sl(2))$, denote by $\displaystyle u^{(i)}$ the element of $\displaystyle
U_{q}(sl(2))^{\otimes n}$ with $\displaystyle u$ in the $\displaystyle i$-th
tensorand and $\displaystyle 1$’s elsewhere. Analogously to (35), for every
$\displaystyle 1\leq i\leq n$, put

(66) $\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(i)}=\begin{pmatrix}a^{(i)}&b^{(i)}\\\
c^{(i)}&d^{(i)}\end{pmatrix}\ \in
End(V_{2})\otimes{\mathcal{L}}_{0,n}(sl(2))^{(i)}.$

######  Lemma 6.11.

For every $\displaystyle 1\leq i\leq n$, $\displaystyle\Phi_{1}^{\otimes
n}({\mathcal{L}}_{0,n}^{(i)}(sl(2)))$ is contained in the subalgebra of
$\displaystyle U_{q}(sl(2))^{\otimes n}$ generated by
Im$\displaystyle(\Phi_{n})$ and the elements $\displaystyle
K^{(i+1)},\ldots,K^{(n)}$.

Proof. The case $\displaystyle i=n$ is clear, as
$\displaystyle\Phi_{1}^{\otimes
n}({\mathcal{L}}_{0,n}^{(n)}(sl(2)))=\Phi_{n}({\mathcal{L}}_{0,n}^{(n)}(sl(2)))$
by definition. We argue by decreasing induction on $\displaystyle
i\in\\{1,\ldots,n\\}$. Take

$\displaystyle A=(id\otimes\Phi_{1}^{\otimes n})(\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(n-1)})$

in (65). The entries of $\displaystyle A$ generate the algebra
$\displaystyle\Phi_{1}^{\otimes n}({\mathcal{L}}_{0,n}^{(n-1)}(sl(2)))$. By
the formula of $\displaystyle R_{0n}AR_{0n}^{-1}$ we have
$\displaystyle\Phi_{n}(c^{(n-1)})=w\otimes K^{-1}$. Hence $\displaystyle
w\otimes 1=(w\otimes K^{-1})(1\otimes K)$ belongs to the algebra generated by
$\displaystyle\Phi_{n}({\mathcal{L}}_{0,n}(sl(2)))$ and $\displaystyle
K^{(n)}$. Since $\displaystyle q^{-1}(q-q^{-1})(1\otimes F)=\Phi_{1}^{\otimes
n}(b^{(n)})=\Phi_{n}(b^{(n)})$, the same is true of
$\displaystyle-q(q-q^{-1})w\otimes F$, and hence eventually also of
$\displaystyle x\otimes 1$, $\displaystyle u\otimes 1$ and $\displaystyle
v\otimes 1$ by using again the formula of $\displaystyle R_{0n}AR_{0n}^{-1}$.
This proves the statement for $\displaystyle i=n-1$. Inducting on
$\displaystyle i$, using the matrix $\displaystyle A$ of generators of
$\displaystyle{\mathcal{L}}_{0,n}^{(i)}(sl(2))$, the same reasoning proves the
result for all values of $\displaystyle i$. $\displaystyle\Box$

######  Lemma 6.12.

Define elements $\displaystyle\xi^{(i)}\in{\mathcal{L}}_{0,n}(sl(2))$,
$\displaystyle i=1,...,n$ by

(67) $\xi^{(i)}=(\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(i)}\cdots\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(n)})_{22}$

where $\displaystyle 22$ denotes the lower right matrix element. The elements
$\displaystyle\xi^{(i)}$ are commuting and satisfy:

(68) $\Phi_{n}(\xi^{(i)})=(K^{-1})^{(i)}\cdots(K^{-1})^{(n)}.$

Proof. As in (53) we have

$\displaystyle\displaystyle(id\otimes\Phi_{n})(\stackrel{{\scriptstyle
V}}{{M}}{}^{\\!\\!(i)}\cdots\stackrel{{\scriptstyle V}}{{M}}{}^{\\!\\!(n)})$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle R_{0n}\ldots
R_{0i}R_{0i}^{\prime}\ldots R_{0n}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle(\pi_{V_{2}}\otimes
1^{\otimes(i-1)}\otimes\Delta^{(n-i)})(RR^{\prime}).$

The lower right matrix element of $\displaystyle(\pi_{V_{2}}\otimes
id)(RR^{\prime})$ is equal to $\displaystyle K^{-1}$. As a result, by applying
$\displaystyle\Delta^{(n-i)}$ we obtain

(69) $\Phi_{n}((\stackrel{{\scriptstyle
V}}{{M}}{}^{\\!\\!(i)}\cdots\stackrel{{\scriptstyle
V}}{{M}}{}^{\\!\\!(n)})_{22})=1^{\otimes(i-1)}\otimes\Delta^{(n-i)}(K^{-1})=(K^{-1})^{(i)}\cdots(K^{-1})^{(n)}.$

By injectivity of $\displaystyle\Phi_{n}$ this proves that the elements
$\displaystyle\xi^{(i)}$ are commuting. $\displaystyle\Box$

Although the elements $\displaystyle\xi^{(i)}$ commute, although the
multiplicative set $\displaystyle\\{(\xi^{(1)}{}^{k}\\}_{k\in{\mathbb{N}}}$ is
an Ore set of $\displaystyle{\mathcal{L}}_{0,n}(sl(2))$, the multiplicative
sets $\displaystyle\\{(\xi^{(i)}{}^{k}\\}_{k\in{\mathbb{N}}}$ are not Ore sets
of $\displaystyle{\mathcal{L}}_{0,n}(sl(2))$ for $\displaystyle i\geq 2.$ We
therefore cannot localize with respect to the elements
$\displaystyle\xi^{(i)}$ as easily as in the case where $\displaystyle n=1.$
Let $\displaystyle{\mathcal{L}}_{0,n}^{(i\leq)}(sl(2))$ be the subalgebra of
$\displaystyle{\mathcal{L}}_{0,n}(sl(2))$ generated by the subalgebras
$\displaystyle{\mathcal{L}}_{0,n}^{(a)}(sl(2)),$ $\displaystyle a\geq i.$ The
elements $\displaystyle\xi^{(i)}$ are Ore elements of
$\displaystyle{\mathcal{L}}_{0,n}^{(i\leq)}(sl(2)).$ Indeed because
$\displaystyle{\mathcal{L}}_{0,n}$ has no non trivial zero divisors,
$\displaystyle\xi^{(i)}$ is a regular element. For all $\displaystyle
x\in{\mathcal{L}}_{0,n}^{(i\leq)}(sl(2))$ there exists elements $\displaystyle
y,y^{\prime}\in{\mathcal{L}}_{0,n}^{(i\leq)}(sl(2))$ such that $\displaystyle
x\xi^{(i)}=\xi^{(i)}y$ and $\displaystyle\xi^{(i)}x=y^{\prime}\xi^{(i)}.$ This
is shown as follows:
$\displaystyle\Phi_{n}(x)\Phi_{n}(\xi^{(i)})=\Phi_{n}(x)(K^{-1})^{(i)}\cdots(K^{-1})^{(n)}=(K^{-1})^{(i)}\cdots(K^{-1})^{(n)}ad^{r}(K)(\Phi_{n}(x))$
where we have used the fact that $\displaystyle\Phi_{n}(x)\in 1^{\otimes
i-1}\otimes U_{q}^{\otimes(n-i+1)}$. But $\displaystyle
ad^{r}(K)(\Phi_{n}(x))=\Phi_{1}(coad^{r}(K)(x)),$ therefore the Ore condition
is satisfied with $\displaystyle y=coad^{r}(K)(x).$

In order to proceed we first explain the case $\displaystyle n=2$. The element
$\displaystyle\xi^{(1)}=(\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(1)}\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(2)})_{22}=d^{(1)}d^{(2)}+c^{(1)}b^{(2)}$ is a regular
Ore element of $\displaystyle{\mathcal{L}}_{0,2}$, so we can define the
localisation $\displaystyle{\mathcal{L}}_{0,2}[\xi^{(1)}{}^{-1}].$ We want to
define the inverse of the element $\displaystyle\xi^{(2)}$, and a new algebra
$\displaystyle{\mathcal{L}}_{0,2}[\xi^{(1)}{}^{-1}][\xi^{(2)}{}^{-1}]$ such
that
$\displaystyle{\mathcal{L}}_{0,2}[\xi^{(1)}{}^{-1}]\subset{\mathcal{L}}_{0,2}[\xi^{(1)}{}^{-1}][\xi^{(2)}{}^{-1}]$
and $\displaystyle\Phi_{2}$ extends naturally to an homomorphism of algebras
$\displaystyle\Phi_{2}:{\mathcal{L}}_{0,2}[\xi^{(1)}{}^{-1}][\xi^{(2)}{}^{-1}]\rightarrow
U_{q}(sl(2))^{\otimes 2}.$

This can be done by writing explicitely the exchange relations between
$\displaystyle\stackrel{{\scriptstyle V}}{{M}}{}^{\\!\\!(1)}$ and
$\displaystyle\stackrel{{\scriptstyle V}}{{M}}{}^{\\!\\!(2)}$ involving
$\displaystyle d^{(2)}=\xi^{(2)}:$

$\displaystyle\displaystyle c^{(1)}d^{(2)}=d^{(2)}c^{(1)},$
$\displaystyle\displaystyle
d^{(2)}a^{(1)}=a^{(1)}d^{(2)}+(1-q^{-2})c^{(1)}b^{(2)},$
$\displaystyle\displaystyle
d^{(2)}d^{(1)}=d^{(1)}d^{(2)}+(1-q^{2})c^{(1)}b^{(2)},$
$\displaystyle\displaystyle
d^{(2)}b^{(1)}=b^{(1)}d^{(2)}+(1-q^{2})(a^{(1)}-d^{(1)})b^{(2)}.$

We define
$\displaystyle{\mathcal{L}}_{0,2}[\xi^{(1)}{}^{-1}][\xi^{(2)}{}^{-1}]$ to be
the algebra generated by the elements $\displaystyle
a^{(1)},b^{(1)},c^{(1)},d^{(1)}$ $\displaystyle
a^{(2)},b^{(2)},c^{(2)},d^{(2)},\xi^{(1)}{}^{-1},\xi^{(2)}{}^{-1},$ where
$\displaystyle a^{(1)},\cdots,d^{(2)}$ satisfy the exchange relation (36) and
(47), $\displaystyle\xi^{(1)}{}^{-1}$ is the inverse in the sense of Ore of
the element $\displaystyle\xi^{(1)}=d^{(1)}d^{(2)}+c^{(1)}b^{(2)}$ and the
following relations for $\displaystyle\xi^{(2)}{}^{-1}$, the inverse of
$\displaystyle\xi^{(2)}=d^{(2)},$ hold true:

$\displaystyle\displaystyle\xi^{(2)}\xi^{(2)}{}^{-1}=\xi^{(2)}{}^{-1}\xi^{(2)}=1,$
$\displaystyle\displaystyle\xi^{(2)}{}^{-1}c^{(1)}=c^{(1)}\xi^{(2)}{}^{-1},$
$\displaystyle\displaystyle\xi^{(2)}{}^{-1}a^{(1)}=a^{(1)}\xi^{(2)}{}^{-1}-q^{-2}(1-q^{-2})c^{(1)}b^{(2)}\xi^{(2)}{}^{-2},$
$\displaystyle\displaystyle\xi^{(2)}{}^{-1}d^{(1)}=d^{(1)}\xi^{(2)}{}^{-1}-q^{-2}(1-q^{2})c^{(1)}b^{(2)}\xi^{(2)}{}^{-2},$
$\displaystyle\displaystyle\xi^{(2)}{}^{-1}b^{(1)}=b^{(1)}\xi^{(2)}{}^{-1}+q^{-1}(1-q^{-2})(a^{(1)}-d^{(1)})b^{(2)}\xi^{(2)}{}^{-2}+$
$\displaystyle\displaystyle\hskip
56.9055pt+q^{-5}(1-q^{2})(1-q^{-4})c^{(1)}(b^{(2)})^{2}\xi^{(2)}{}^{-3}.$

The last relations are chosen in order that
$\displaystyle{\mathcal{L}}_{0,2}[\xi^{(1)}{}^{-1}]\subset{\mathcal{L}}_{0,2}[\xi^{(1)}{}^{-1}][\xi^{(2)}{}^{-1}]$
as an algebra and are a direct consequence of the exchange relations (47) and
the invertibility of $\displaystyle\xi^{(2)}.$ As a result
$\displaystyle\Phi_{2}$ extends to a morphism of algebra
$\displaystyle\Phi_{2}:{\mathcal{L}}_{0,2}[\xi^{(1)}{}^{-1}][\xi^{(2)}{}^{-1}]\rightarrow
U_{q}(sl(2))^{\otimes 2}.$ This morphism of algebra will be shown to be an
isomorphism.

The construction of the localisation for
$\displaystyle{\mathcal{L}}_{0,n}(sl(2))$ is defined by the same procedure. We
want to define a localisation
$\displaystyle{\mathcal{L}}_{0,n}(sl(2))[\xi^{(n)}{}^{-1},...,\xi^{(1)}{}^{-1}].$
The element $\displaystyle\xi^{(n)}$ is a regular Ore element of
$\displaystyle{\mathcal{L}}_{0,n}(sl(2)),$ therefore we can define the
standard localisation
$\displaystyle{\mathcal{L}}_{0,n}(sl(2))[\xi^{(n)}{}^{-1}]$. We proceed as in
the $\displaystyle n=2$ case. We note that the matrix
$\displaystyle\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(2)}\cdots\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(n)}$ has the same reflection equation with
$\displaystyle\stackrel{{\scriptstyle V_{2}}}{{M}}{}^{\\!\\!(1)}$ as
$\displaystyle\stackrel{{\scriptstyle V_{2}}}{{M}}{}^{\\!\\!(2)}.$ Therefore
we can define, by the same method as $\displaystyle n=2,$
$\displaystyle{\mathcal{L}}_{0,n}(sl(2))[\xi^{(n)}{}^{-1}][\xi^{(n-1)}{}^{-1}].$

######  Definition 6.13.

By iterating the preceeding construction we define:

$\displaystyle{}_{loc}{\mathcal{L}}_{0,n}(sl(2))={\mathcal{L}}_{0,n}(sl(2))[\xi^{(n)}{}^{-1}][\xi^{(n-1)}{}^{-1}]\cdots[\xi^{(1)}{}^{-1}].$

In the sequel it will be convenient to define invertible elements
$\displaystyle\delta^{(i)}\in{}_{loc}{\mathcal{L}}_{0,n}(sl(2)),i=1,...,n,$
satisfying $\displaystyle\xi^{(i)}=\delta^{(i)}\cdots\delta^{(n)},$ i.e
$\displaystyle\delta^{(i)}=\xi^{(i)}\xi^{(i+1)}{}^{-1}$.

The elements $\displaystyle\delta^{(i)},i=1,...,n,$ are invertible, commute
and satisfy $\displaystyle\Phi_{n}(\delta^{(i)})=(K^{-1})^{(i)}$ .

We can define the localization
$\displaystyle{}_{loc}{\mathcal{L}}_{0,n}^{A}(sl(2))$ of the integral form
$\displaystyle{\mathcal{L}}_{0,n}^{A}(sl(2))$ in the very same way. Let

$\displaystyle\Phi_{n}\colon{}_{loc}{\mathcal{L}}_{0,n}(sl(2))\rightarrow
U_{q}(sl(2))^{{\otimes}n}$

be the unique morphism of module algebras extending the Alekseev map.

######  Proposition 6.14.

The extended Alekseev map
$\displaystyle\Phi_{n}\colon{}_{loc}{\mathcal{L}}_{0,n}(sl(2))\rightarrow
U_{q}(sl(2))^{{\otimes}n}$ is an isomorphism of $\displaystyle U_{q}$-module
algebras. It restricts to an embedding of $\displaystyle U_{A}$-module
algebras
$\displaystyle\Phi_{n}\colon{}_{loc}{\mathcal{L}}_{0,n}^{A}(sl(2))\rightarrow
U_{A}(sl(2))^{{\otimes}n}$, which becomes an isomorphism by adjoining to
$\displaystyle A$ the inverse of $\displaystyle q-q^{-1}$.

Proof. All claims are clear by the previous results, except surjectivity in
the first and third claims. For the first, it follows from Lemma 6.11, since
the algebra generated by $\displaystyle\Phi_{1}^{\otimes
n}({\mathcal{L}}_{0,n}^{(i)}(sl(2)))$ and $\displaystyle K^{(i)}$ is
$\displaystyle U_{q}(sl(2))^{(i)}$ (the $\displaystyle i$-th tensorand). For
the third, it follows from Lemma 5.7 (3). $\displaystyle\Box$

######  Remark 6.15.

One can check that for $\displaystyle n\geq 2$ none of the elements
$\displaystyle d^{(1)},...,d^{(n)}$ are Ore elements of
$\displaystyle{\mathcal{L}}_{0,n}(sl(2))$ and moreover they do not commute. So
it is not clear how to define an algebra extension of
$\displaystyle{\mathcal{L}}_{0,n}(sl(2))$ containing inverses of the elements
$\displaystyle d^{(i)}$, replacing
$\displaystyle{}_{loc}{\mathcal{L}}_{0,n}(sl(2))$. This is the reason why we
are led to define the elements $\displaystyle\xi^{(i)}$ and localize with
respect to them.

Next we go back to the general situation of
$\displaystyle{\mathcal{L}}_{0,n}^{A}={\mathcal{L}}_{0,n}^{A}(\mathfrak{g})$
for an arbitrary $\displaystyle\mathfrak{g}$. We need the following fact,
proved in [35] for $\displaystyle{\mathcal{O}}_{q}$, but which extends
immediately to $\displaystyle\mathcal{O}_{A}$:

######  Proposition 6.16.

The iterated coproduct
$\displaystyle\Delta^{(n-1)}\colon\mathcal{O}_{A}\rightarrow\mathcal{O}_{A}^{\otimes
n}$, considered as a linear map
$\displaystyle{\mathcal{L}}_{0,1}^{A}\rightarrow{\mathcal{L}}_{0,n}^{A}$, is
an algebra morphism and satisfies the following commutative diagram:

$\displaystyle\textstyle{{\mathcal{L}}_{0,1}^{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{\Delta^{(n-1)}}$$\displaystyle\scriptstyle{\Phi_{1}}$$\displaystyle\textstyle{{\mathcal{L}}_{0,n}^{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{\Phi_{n}}$$\displaystyle\textstyle{\tilde{U}_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{\Delta^{(n-1)}}$$\displaystyle\textstyle{\tilde{U}_{A}^{\otimes
n}.}$

Proof. The main point of the proof is that

(70) $(id\otimes\Delta^{(n-1)})\left(\stackrel{{\scriptstyle
V}}{{M}}\right)=\stackrel{{\scriptstyle
V}}{{M}}{}^{\\!\\!(1)}\ldots\stackrel{{\scriptstyle V}}{{M}}{}^{\\!\\!(n)},$

as can be seen by a direct application of the definitions. But Proposition 4.6
implies that
$\displaystyle\Delta^{(n-1)}\colon{\mathcal{L}}_{0,1}^{A}\rightarrow{\mathcal{L}}_{0,n}^{A}$
is an algebra morphism if and only if the matrix on the left hand side of (70)
satisfies the fusion relation, which is easily shown to be true of the right
hand side by using the exchange relation (47) recursively. This proves the
proposition. It shows as well that
$\displaystyle\Delta^{(n-1)}({\mathcal{L}}_{0,1}^{A})$ is the algebra
generated by the set of matrix elements of (70) in the basis of $\displaystyle
V$ extracted from the canonical basis $\displaystyle\mathcal{B}_{\lambda}$,
$\displaystyle\lambda\in P_{+}$. The commutativity diagram is a reformulation
of (53). $\displaystyle\Box$

Consider now the algebra of invariant elements,

$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}:=\\{\alpha\in{\mathcal{L}}_{0,n}^{A}|\forall
y\in U_{A},coad^{r}(y)(\alpha)=\varepsilon(y)\alpha\\}.$
$\displaystyle{\mathcal{L}}_{0,n}^{U_{q}}:=\\{\alpha\in{\mathcal{L}}_{0,n}|\forall
y\in U_{q},coad^{r}(y)(\alpha)=\varepsilon(y)\alpha\\}.$

Note that
$\displaystyle{\mathcal{L}}_{0,n}^{U_{q}}=({\mathcal{L}}_{0,n}^{A})^{U_{A}}\otimes_{A}\mathbb{C}(q^{1/D})$.

######  Proposition 6.17.

The algebra $\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}$ is the
centralizer of $\displaystyle\Delta^{(n-1)}({\mathcal{L}}_{0,1}^{A})$ in
$\displaystyle{\mathcal{L}}_{0,n}^{A}$. As a corollary we have
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{A})^{U_{A}}=\mathcal{Z}({\mathcal{L}}_{0,n}^{A})$.

Proof. Clearing denominators it is enough to give the details for
$\displaystyle{\mathcal{L}}_{0,n}^{U_{q}}$. First note that an element
$\displaystyle z$ of $\displaystyle\tilde{U}_{q}^{\otimes n}$ is invariant
under the right adjoint action of $\displaystyle U_{q}$ if and only if it
commutes with $\displaystyle\Delta^{(n-1)}(x)$ for every $\displaystyle x\in
U_{q}$. Indeed, we have

$\displaystyle\displaystyle z\Delta^{(n-1)}(x)$
$\displaystyle\displaystyle=\sum_{(x)}z\varepsilon(x_{(1)})\Delta^{(n-1)}(x_{(2)})=\sum_{(x)}\varepsilon(x_{(1)})z\Delta^{(n-1)}(x_{(2)})$
$\displaystyle\displaystyle=\sum_{(x)}\Delta^{(n-1)}(x_{(1)})\Delta^{(n-1)}(S(x_{(2)}))z\Delta^{(n-1)}(x_{(3)})=\sum_{(x)}\Delta^{(n-1)}(x_{(1)})ad^{r}(x_{(2)})(z).$

Hence $\displaystyle\textstyle
z\Delta^{(n-1)}(x)=\sum_{(x)}\Delta^{(n-1)}(x_{1})\varepsilon(x_{(2)})z=\Delta^{(n-1)}(x)z$
if $\displaystyle z$ is an invariant element. Conversely, if $\displaystyle z$
commutes with $\displaystyle\Delta^{(n-1)}(x)$ for every $\displaystyle x\in
U_{q}$, then

$\displaystyle
ad^{r}(x)(z)=\sum_{(x)}\Delta^{(n-1)}(S(x_{(1)}))\Delta^{(n-1)}(x_{(2)})z=\Delta^{(n-1)}(\varepsilon(x))z=\varepsilon(x)z.$

This proves our claim. Now, let
$\displaystyle\alpha\in{\mathcal{L}}_{0,n}^{U_{q}}$. Then
$\displaystyle\Phi_{n}(\alpha)$ is an $\displaystyle ad^{r}(U_{q})$-invariant
element of $\displaystyle\tilde{U}_{q}^{\otimes n}$, by Theorem 6.4. By the
claim above, and the fact that commuting with elements of
$\displaystyle\tilde{U}_{q}$ or $\displaystyle U_{q}$ is the same,
$\displaystyle\Phi_{n}(\alpha)$ commutes with the matrix coefficients of
$\displaystyle(\pi_{V}\otimes\Delta^{(n-1)})(RR^{\prime})$, for every object
$\displaystyle V$ of $\displaystyle\mathcal{C}$. By the injectivity of
$\displaystyle\Phi_{n}$ and the relation (53), $\displaystyle\alpha$ commutes
with the matrix coefficients of (70) for all objects $\displaystyle V$ of
$\displaystyle\mathcal{C}$. These generate the algebra
$\displaystyle\Delta^{(n-1)}({\mathcal{L}}_{0,1})$, so
$\displaystyle{\mathcal{L}}_{0,n}^{U_{q}}$ lies in the centralizer of the
latter.

Conversely, the same reasoning shows that if $\displaystyle\alpha$ lies in the
centralizer of $\displaystyle\Delta^{(n-1)}({\mathcal{L}}_{0,1})$, then
$\displaystyle\Phi_{n}(\alpha)$ commutes with the matrix coefficients of
$\displaystyle(\pi_{V}\otimes\Delta^{(n-1)})(RR^{\prime})$, for every object
$\displaystyle V$ of $\displaystyle\mathcal{C}$. From Theorem 4.3 we deduce
that $\displaystyle\Phi_{n}(\alpha)$ lies in the centralizer of
$\displaystyle\Delta^{(n-1)}(\tilde{U}_{q}^{lf})$. This is the same as the
centralizer of $\displaystyle\Delta^{(n-1)}(\tilde{U}_{q})$ by Remark 4.5 (1),
so as above we deduce that $\displaystyle\Phi_{n}(\alpha)$ is an invariant
element of $\displaystyle\tilde{U}_{q}^{\otimes n}$, and by injectivity and
equivariance of $\displaystyle\Phi_{n}$ that $\displaystyle\alpha$ is an
invariant element of $\displaystyle{\mathcal{L}}_{0,n}$.

The corollary is immediate, because a central element of
$\displaystyle{\mathcal{L}}_{0,n}$ necessarily commutes with
$\displaystyle\Delta^{(n-1)}({\mathcal{L}}_{0,1}^{A})$, so it is invariant.
$\displaystyle\Box$

We now give an explicit basis of the algebras of invariant elements in term of
intertwiners. Let $\displaystyle n$ be an integer greater than $\displaystyle
1$. For every $\displaystyle 2\leq k\leq n$ we denote by $\displaystyle
S(k)\in{\mathbb{U}_{q}}^{{\otimes}n}$ the element defined by

$\displaystyle
S(k)=id^{\otimes(k-2)}\otimes\left(id\otimes\Delta^{(n-k)}\right)(R).$

Let $\displaystyle\lambda_{1},...,\lambda_{n}\in P_{+}$, and $\displaystyle
V_{\lambda_{1}},\ldots,V_{\lambda_{n}}$ the type $\displaystyle 1$ simple
$\displaystyle U_{A}^{res}$-modules of highest weights
$\displaystyle\lambda_{1},...,\lambda_{n}$ respectively. Put
$\displaystyle[\lambda]=(\lambda_{1},...,\lambda_{n})$, and consider the
$\displaystyle U_{A}^{res}$-module

$\displaystyle V_{[\lambda]}=\bigotimes_{j=1}^{n}V_{\lambda_{j}}.$

Define $\displaystyle\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}\in
End(V_{[\lambda]})\otimes{\mathcal{L}}_{0,n}^{A}$ by
$\displaystyle\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}=\stackrel{{\scriptstyle
V_{\lambda}}}{{M}}$ if $\displaystyle n=1$, and if $\displaystyle n\geq 2$ by

(71) $\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}=\stackrel{{\scriptstyle
V_{\lambda_{1}}}}{{M}}_{1}{}^{\\!\\!\\!(1)}\prod_{k=2}^{n}\left(S(k)^{-1}_{V_{\lambda_{1}},...,V_{\lambda_{n}}}\stackrel{{\scriptstyle
V_{\lambda_{k}}}}{{M}}_{k}{}^{\\!\\!\\!(k)}\right)\prod_{k=n}^{2}S(k)_{V_{\lambda_{1}},...,V_{\lambda_{n}}}.$

For example for $\displaystyle n=2$ we have

$\displaystyle\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}=\stackrel{{\scriptstyle
V_{\lambda_{1}}}}{{M}}_{1}{}^{\\!\\!\\!(1)}R_{12}^{-1}\stackrel{{\scriptstyle
V_{\lambda_{2}}}}{{M}}_{2}{}^{\\!\\!\\!(2)}R_{12}$

and for $\displaystyle n=3$,

$\displaystyle\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}=\stackrel{{\scriptstyle
V_{\lambda_{1}}}}{{M}}_{1}{}^{\\!\\!\\!(1)}(id\otimes\Delta)(R^{-1})\stackrel{{\scriptstyle
V_{\lambda_{2}}}}{{M}}_{2}{}^{\\!\\!\\!(2)}(R^{-1})_{23}\stackrel{{\scriptstyle
V_{\lambda_{3}}}}{{M}}_{3}{}^{(3)}R_{23}(id\otimes\Delta)(R).$

The reason for considering the matrices
$\displaystyle\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}$ comes from the
two following propositions. The first generalizes the identity (29).

######  Proposition 6.18.

The right coadjoint action of $\displaystyle U_{A}$ on the matrix elements of
$\displaystyle\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}$ can be written
in matrix form as

(72)
$coad^{r}(y)\left(\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}\right)=\sum_{(y)}\left((\pi_{V_{[\lambda]}}(y_{(1)})\otimes
id)\stackrel{{\scriptstyle[\lambda]}}{{{\mathbb{M}}}}(\pi_{V_{[\lambda]}}(S(y_{(2)}))\otimes
id)\right).$

Proof. We show it for $\displaystyle n=2$, since the general proof follows
from it by an easy induction on $\displaystyle n$. We have

$\displaystyle\displaystyle
coad^{r}(y)\left(\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}\right)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
coad^{r}(y)\left(\stackrel{{\scriptstyle
V_{\lambda_{1}}}}{{M}}_{1}{}^{\\!\\!\\!(1)}R_{12}^{-1}\stackrel{{\scriptstyle
V_{\lambda_{2}}}}{{M}}_{2}{}^{\\!\\!\\!(2)}R_{12}\right)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{(y)}(y_{(1)})_{1}\stackrel{{\scriptstyle
V_{\lambda_{1}}}}{{M}}_{1}{}^{\\!\\!\\!(1)}S(y_{(2)})_{1}R_{12}^{-1}(y_{(3)})_{2}\stackrel{{\scriptstyle
V_{\lambda_{2}}}}{{M}}_{2}{}^{\\!\\!\\!(2)}S(y_{(4)})_{2}R_{12}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{(y)}(y_{(1)})_{1}\stackrel{{\scriptstyle
V_{\lambda_{1}}}}{{M}}_{1}{}^{\\!\\!\\!(1)}(y_{(2)})_{2}R_{12}^{-1}(S(y_{(3)}))_{1}\stackrel{{\scriptstyle
V_{\lambda_{2}}}}{{M}}_{2}{}^{\\!\\!\\!(2)}S(y_{(4)})_{2}R_{12}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{(y)}(y_{(1)}\otimes
y_{(2)})\stackrel{{\scriptstyle
V_{\lambda_{1}}}}{{M}}_{1}{}^{\\!\\!\\!(1)}R_{12}^{-1}\stackrel{{\scriptstyle
V_{\lambda_{2}}}}{{M}}_{2}{}^{\\!\\!\\!(2)}(S(y_{(3)})\otimes
S(y_{(4)}))R_{12}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{(y)}(y_{(1)}\otimes
y_{(2)})\stackrel{{\scriptstyle
V_{\lambda_{1}}}}{{M}}_{1}{}^{\\!\\!\\!(1)}R_{12}^{-1}\stackrel{{\scriptstyle
V_{\lambda_{2}}}}{{M}}_{2}{}^{\\!\\!\\!(2)}R_{12}(S(y_{(4)})\otimes
S(y_{(3)}))$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{(y)}\left((\pi_{V_{[\lambda]}}(y_{(1)})\otimes
id)\stackrel{{\scriptstyle[\lambda]}}{{{\mathbb{M}}}}(\pi_{V_{[\lambda]}}(S(y_{(2)}))\otimes
id)\right).$

All equalities are clear but the third and fifth, which follow from
$\displaystyle(id\otimes S^{-1})(R)=R^{-1}$, $\displaystyle R\Delta
R^{-1}=\Delta^{cop}$, $\displaystyle\Delta\circ S=(S\otimes S)\circ\Delta$,
and the fact that $\displaystyle S$ is an algebra antimorphism. The result
follows. $\displaystyle\Box$

Denote by $\displaystyle(\stackrel{{\scriptstyle[\lambda]}}{{e}}_{i})$ and
$\displaystyle(\stackrel{{\scriptstyle[\lambda]}}{{e}}{}^{\\!\\!i})$ the
canonical basis of $\displaystyle V_{[\lambda]}$ (the tensor product of the
canonical basis $\displaystyle\mathcal{B}_{\lambda_{j}}$) and the dual basis.

######  Proposition 6.19.

The elements
$\displaystyle(\stackrel{{\scriptstyle[\lambda]}}{{e}}{}^{\\!\\!\\!j}\otimes
id)\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}(\stackrel{{\scriptstyle[\lambda]}}{{e}}{}_{\\!\\!\\!i}\otimes
id)$ of $\displaystyle{\mathcal{L}}_{0,n}^{A}$, where
$\displaystyle[\lambda]\in P_{+}^{n}$, $\displaystyle i$ labels the basis of
$\displaystyle V_{[\lambda]}$ and $\displaystyle j$ the dual basis, form an
$\displaystyle A$-basis of $\displaystyle{\mathcal{L}}_{0,n}^{A}$.
Equivalently, for every $\displaystyle x\in{\mathcal{L}}_{0,n}^{A}$ there
exists a unique family of endomorphisms $\displaystyle a_{[\lambda]}(x)\in
End_{A}(V_{[\lambda]})$, $\displaystyle[\lambda]\in P_{+}^{n}$, which is zero
except possibly for a finite number of terms, such that

(73) $x=\sum_{[\lambda]}(Tr_{{V_{[\lambda]}}}\otimes
id)\left((\pi_{V_{\lambda}}(\ell)a_{[\lambda]}(x)\otimes
id)\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}\right)=\sum_{[\lambda]}qTr_{{V_{[\lambda]}}}\left((a_{[\lambda]}(x)\otimes
id)\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}\right)$

where as usual $\displaystyle\ell$ is the pivotal element and
$\displaystyle\pi_{V_{\lambda}}(\ell)$ is the endomorphism of $\displaystyle
V_{[\lambda]}$ given by the action of $\displaystyle\ell$.

Note that $\displaystyle\pi_{V_{\lambda}}(\ell)$ is introduced in (73) in
order to simplify the statement of the next proposition.

Proof. We show the first claim of the proposition for $\displaystyle n=2$; the
general proof is similar. In this case it is sufficient to prove that the set
of matrix elements of the tensors

$\displaystyle\stackrel{{\scriptstyle
V_{\lambda_{1}}}}{{M}}_{1}{}^{\\!\\!\\!(1)}R_{12}^{-1}\stackrel{{\scriptstyle
V_{\lambda_{2}}}}{{M}}_{2}{}^{\\!\\!\\!(2)}=\sum_{(R)}(R_{(2)})_{2}\stackrel{{\scriptstyle
V_{\lambda_{1}}}}{{M}}_{1}{}^{\\!\\!\\!(1)}\stackrel{{\scriptstyle
V_{\lambda_{2}}}}{{M}}_{2}{}^{\\!\\!\\!(2)}S(R_{(1)})_{1}\quad,[\lambda]\in
P_{+}^{n},$

is a basis of $\displaystyle{\mathcal{L}}_{0,2}^{A}$. Note that we use
$\displaystyle\textstyle R^{-1}=(S\otimes id)(R)=\sum_{(R)}S(R_{(1)})\otimes
R_{(2)}$ in the above equality. Let $\displaystyle T=(S\otimes
S^{-1})(R^{-1})$, that we denote as usual by $\displaystyle\textstyle
T=\sum_{(T)}T_{(1)}\otimes T_{(2)}$. From the identity
$\displaystyle\textstyle 1\otimes 1=(S\otimes
id)(R^{-1}R)=\sum_{(R),(T)}S(R_{(1)})T_{(1^{\prime})}\otimes
S(T_{(2^{\prime})})R_{(2)}$, we obtain

$\displaystyle\stackrel{{\scriptstyle
V_{\lambda_{1}}}}{{M}}_{1}{}^{\\!\\!\\!(1)}\stackrel{{\scriptstyle
V_{\lambda_{2}}}}{{M}}_{2}{}^{\\!\\!\\!(2)}=\sum_{(T)}S(T_{(2)})_{2}\stackrel{{\scriptstyle
V_{\lambda_{1}}}}{{M}}_{1}{}^{\\!\\!\\!(1)}R_{12}^{-1}\stackrel{{\scriptstyle
V_{\lambda_{2}}}}{{M}}_{2}{}^{\\!\\!\\!(2)}(T_{(1)})_{1}.$

The conclusion follows from this, since the set of matrix elements of
$\displaystyle\stackrel{{\scriptstyle
V_{\lambda_{1}}}}{{M}}_{1}{}^{\\!\\!\\!(1)}\stackrel{{\scriptstyle
V_{\lambda_{2}}}}{{M}}_{2}{}^{\\!\\!\\!(2)}$, $\displaystyle[\lambda]\in
P_{+}^{2}$, is a basis of $\displaystyle{\mathcal{L}}_{0,2}^{A}$.

The second claim is a trivial consequence of the first one.
$\displaystyle\Box$

By combining the two previous propositions we obtain the following
characterisation of the invariants elements.

######  Proposition 6.20.

An element $\displaystyle x\in{\mathcal{L}}_{0,n}^{A}$ belongs to
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}$ if and only if for every
$\displaystyle[\lambda]\in P_{+}^{n}$ we have $\displaystyle
a_{[\lambda]}(x)\in End_{U_{A}^{res}}(V_{[\lambda]})$. Therefore, the elements

(74)
$v_{[\lambda]}(a_{[\lambda]}^{(k)}):=qTr_{V_{[\lambda]}}\left(a_{[\lambda]}^{(k)}\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}\right),$

where $\displaystyle\left\\{a_{[\lambda]}^{(k)}\right\\}_{k}$ is any
$\displaystyle A$-basis of $\displaystyle End_{U_{A}^{res}}(V_{[\lambda]})$,
make an $\displaystyle A$-basis of
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}$.

Proof. We have $\displaystyle x\in({\mathcal{L}}_{0,n}^{A})^{U_{A}}$ if and
only if $\displaystyle coad^{r}(y)(x)=\epsilon(y)x$ for every $\displaystyle
y\in U_{A}$, or equivalently for every $\displaystyle y\in U_{A}^{res}$. From
Proposition 6.18, Proposition 6.19, and the fact that $\displaystyle
S^{2}(y)=\ell y\ell^{-1}$, this is equivalent to
$\displaystyle\textstyle\sum_{(y)}S^{-1}(y_{(2)})_{V_{[\lambda]}}a_{[\lambda]}(x)(y_{(1)})_{V_{[\lambda]}}=\epsilon(y)a_{[\lambda]}(x)$
for every $\displaystyle[\lambda]\in P_{+}^{n}$, which is also equivalent to
$\displaystyle
y_{V_{[\lambda]}}a_{[\lambda]}(x)=a_{[\lambda]}(x)y_{V_{[\lambda]}}$.
$\displaystyle\Box$

In the case of $\displaystyle\mathfrak{g}=sl(2)$, let us state the following
result. It is a direct consequence of Theorem 8.4 and the fact that the skein
algebra $\displaystyle K_{\zeta}(\Sigma)$ is finitely generated and
Noetherian, which is proved in [70], Theorem 3, by topological means. The
method relies on the fact that $\displaystyle{\mathcal{L}}_{0,n}^{A}$ has a
natural filtration, and consists in proving that the graded algebra associated
to $\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}$ by this filtration is
finitely generated and Noetherian. In [16] we prove that the same result holds
true for the invariant algebras
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}$ associated to any
$\displaystyle{\mathfrak{g}}$.

######  Theorem 6.21.

The algebra $\displaystyle({\mathcal{L}}_{0,n}^{A}(sl(2)))^{U_{A}(sl(2))}$ is
finitely generated and Noetherian.

Next we turn to the centers $\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n})$
and $\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{U_{q}})$ of
$\displaystyle{\mathcal{L}}_{0,n}$ and
$\displaystyle{\mathcal{L}}_{0,n}^{U_{q}}$. The basis we find clearly form
$\displaystyle A$-basis of $\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{A})$
and $\displaystyle(\mathcal{Z}({\mathcal{L}}_{0,n}^{A})^{U_{A}})$ too.

######  Proposition 6.22.

The Alekseev map affords an isomorphism from
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n})$ to
$\displaystyle\mathcal{Z}(\tilde{U}_{q}^{\otimes n})$. Moreover, the elements

$\displaystyle{}_{\lambda}\omega^{(i)}:=qTr_{V_{\lambda}}\left(\stackrel{{\scriptstyle
V_{\lambda}}}{{M}}{}^{\\!\\!(i)}\right)\quad,\lambda\in
P_{+},i\in\\{1,\ldots,n\\},$

belong to the center $\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n})$ and the
family of elements
$\displaystyle\textstyle\prod_{i=1}^{n}{}_{\lambda_{i}}\omega^{(i)}$, where
$\displaystyle\lambda_{1},\ldots,\lambda_{n}\in P_{+}$, form a basis of
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n})$.

Proof. We have
$\displaystyle\Phi_{n}(\mathcal{Z}({\mathcal{L}}_{0,n}))=\mathcal{Z}(\Phi_{n}({\mathcal{L}}_{0,n}))$
by Theorem 6.4. We claim that if $\displaystyle x\in\tilde{U}_{q}^{\otimes n}$
commutes with $\displaystyle\Phi_{n}({\mathcal{L}}_{0,n})$, it is central.
Indeed, $\displaystyle x$ commutes with
$\displaystyle\Phi_{n}({\mathcal{L}}_{0,n}^{(n)})=1^{\otimes(n-1)}\otimes\tilde{U}_{q}^{lf}$,
and hence with $\displaystyle 1^{\otimes(n-1)}\otimes\tilde{U}_{q}$, and the
matrix coefficients of $\displaystyle R_{0n}^{\pm 1}$ by Remark 4.5. (For
instance, when $\displaystyle\mathfrak{g}=sl(2)$, $\displaystyle K=q^{H}$ and
$\displaystyle q^{H/2}$ commute and are diagonalizable on objects of
$\displaystyle\mathcal{C}$, and since $\displaystyle x$ commutes with
$\displaystyle 1^{\otimes(n-1)}\otimes K$, it commutes with $\displaystyle
1^{\otimes(n-1)}\otimes q^{H/2}$ too, and the conclusion follows from (61) and
(64)). Using that $\displaystyle x$ commutes also with
$\displaystyle\Phi_{n}({\mathcal{L}}_{0,n}^{(n-1)})$, and hence with the
matrix coefficients of $\displaystyle
R_{0n}R_{0n-1}R_{0n-1}^{\prime}R_{0n}^{-1}$, we deduce that it commutes with
the matrix coefficients of $\displaystyle R_{0n-1}R_{0n-1}^{\prime}$, and
hence with every element of $\displaystyle
1^{\otimes(n-2)}\otimes\tilde{U}_{q}\otimes 1$. Continuing in this way
recursively, we get that $\displaystyle x\in\mathcal{Z}(\tilde{U}_{q}^{\otimes
n})$, which proves our claim. It implies that
$\displaystyle\Phi_{n}(\mathcal{Z}({\mathcal{L}}_{0,n}))\subset\mathcal{Z}(\tilde{U}_{q}^{\otimes
n})$.

Let us prove the converse inclusion. Put $\displaystyle R^{(a)}:=R_{0n}\ldots
R_{0a+1}$. For every $\displaystyle 1\leq a\leq n$ we have

(75)
$\displaystyle\displaystyle\Phi_{n}\left(qTr_{V_{\lambda}}\left(\stackrel{{\scriptstyle
V_{\lambda}}}{{M}}{}^{\\!\\!(a)}\right)\right)$
$\displaystyle\displaystyle=(Tr_{V_{\lambda}}\otimes
id)\left(\left(\pi_{V_{\lambda}}(\ell)\otimes
id\right)R^{(a)}R_{0a}R_{0a}^{\prime}R^{(a)}{}^{\\!\\!-1}\right)$
$\displaystyle\displaystyle=(Tr_{V_{\lambda}}\otimes
id)\left(\left(\pi_{V_{\lambda}}(\ell)\otimes
id\right)R_{0a}R_{0a}^{\prime}\right)$
$\displaystyle\displaystyle=\mathfrak{i}_{a}\left(\Phi_{1}\left(qTr_{V_{\lambda}}\left(\stackrel{{\scriptstyle
V_{\lambda}}}{{M}}\right)\right)\right)$

where $\displaystyle\Phi_{1}$ in the last equality is the RSD map, and the
second equality follows from $\displaystyle R^{-1}=(S\otimes id)(R)$ and
$\displaystyle S^{2}(a)=\ell a\ell^{-1}$ for every $\displaystyle
a\in\mathbb{U}_{q}$. By Theorem 4.3 and Proposition 4.8, the family of these
elements, when $\displaystyle\lambda$ spans $\displaystyle P_{+}$, forms a
basis of the center of the $\displaystyle a$-th tensorand of
$\displaystyle\tilde{U}_{q}^{\otimes n}$. By injectivity of
$\displaystyle\Phi_{n}$ this proves the second claim of the statement, as well
as the inclusion $\displaystyle\mathcal{Z}(\tilde{U}_{q}^{\otimes
n})\subset\Phi_{n}(\mathcal{Z}({\mathcal{L}}_{0,n}))$. By our first claim they
eventually coincide. The conclusion follows. $\displaystyle\Box$

In particular, recall from (66) that for $\displaystyle\mathfrak{g}=sl(2)$ and
every $\displaystyle 1\leq i\leq n$ we put

$\displaystyle\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(i)}=\begin{pmatrix}a^{(i)}&b^{(i)}\\\
c^{(i)}&d^{(i)}\end{pmatrix}\ \in
End(V_{2})\otimes{\mathcal{L}}_{0,n}(sl(2))^{(i)}.$

Set

$\displaystyle\omega^{(i)}=qa^{(i)}+q^{-1}d^{(i)}=qTr_{V_{2}}\left(\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(i)}\right).$

By (75) we have

$\displaystyle\Phi_{n}(\omega^{(i)})=\Omega^{(i)},$

that is, $\displaystyle(q-q^{-1})^{2}$ times the Casimir element of the
$\displaystyle i$-th tensorand of $\displaystyle U_{q}(sl(2))^{\otimes n}$,
and $\displaystyle\mathcal{Z}(U_{q}(sl(2))^{\otimes
n})=\mathbb{C}(q^{1/2})[\Omega^{(1)},\ldots,\Omega^{(n)}]$. Then the
proposition implies

(76)
$\mathcal{Z}({\mathcal{L}}_{0,n}(sl(2)))=\mathbb{C}(q^{1/2})[\omega^{(1)},\ldots,\omega^{(n)}].$

More generally, by the same arguments and the results recalled in Remark 4.9,
there is an analogous description of
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}(\mathfrak{g}))$ as a polynomial
algebra for an arbitrary finite dimensional complex simple Lie algebra
$\displaystyle\mathfrak{g}$ of type $\displaystyle A,B,C$ or $\displaystyle
D$.

For every $\displaystyle\lambda\in P_{+}$ denote

$\displaystyle{}_{\lambda}\eta=qTr_{V_{\lambda}}\left(\stackrel{{\scriptstyle
V_{\lambda}}}{{M}}{}^{\\!\\!(1)}\ldots\stackrel{{\scriptstyle
V_{\lambda}}}{{M}}{}^{\\!\\!(n)}\right).$

In the specific case of $\displaystyle\mathfrak{g}=sl(2)$, let us put

(77) $\eta:=qTr_{V_{2}}\left(\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(1)}\cdots\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(n)}\right).$

######  Lemma 6.23.

$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{U_{q}})$ contains the
commutative algebra generated over $\displaystyle\mathbb{C}(q^{1/D})$ by the
elements
$\displaystyle{}_{\lambda_{1}}\omega^{(1)},\ldots,{}_{\lambda_{n}}\omega^{(n)}$
and $\displaystyle{}_{\lambda}\eta$ for all
$\displaystyle\lambda,\lambda_{1},\ldots,\lambda_{n}\in P_{+}$.

Proof. Clearly $\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{U_{q}})$
contains $\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n})^{U_{q}}$, whence the
elements
$\displaystyle{}_{\lambda_{1}}\omega^{(1)},\ldots,{}_{\lambda_{n}}\omega^{(n)}$
for every $\displaystyle\lambda_{1},\ldots,\lambda_{n}\in P_{+}$, by
Proposition 6.22 and Proposition 6.17. Moreover, by this result
$\displaystyle{\mathcal{L}}_{0,n}^{U_{q}}$ is the centralizer of
$\displaystyle\Delta^{(n-1)}({\mathcal{L}}_{0,1})$ in
$\displaystyle{\mathcal{L}}_{0,n}$. Therefore
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{U_{q}})\supset\Delta^{(n-1)}(\mathcal{Z}({\mathcal{L}}_{0,1}))$.
By Proposition 4.8 and the relation (70), the elements
$\displaystyle{}_{\lambda}\eta$, $\displaystyle\lambda\in P_{+}$, form a basis
of $\displaystyle\Delta^{(n-1)}(\mathcal{Z}({\mathcal{L}}_{0,1}))$. The result
follows. $\displaystyle\Box$

Note that the $\displaystyle\mathbb{C}(q^{1/D})$-algebra generated by the
elements
$\displaystyle{}_{\lambda_{1}}\omega^{(1)},\ldots,{}_{\lambda_{n}}\omega^{(n)}$
and $\displaystyle{}_{\lambda}\eta$ for all
$\displaystyle\lambda,\lambda_{1},\ldots,\lambda_{n}\in P_{+}$ is just the
polynomial algebra generated by these elements when
$\displaystyle\lambda,\lambda_{1},\ldots,\lambda_{n}$ are fundamental weights.
We have even a much better result:

######  Theorem 6.24.

$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{U_{q}})$ is equal to the
polynomial algebra generated over $\displaystyle\mathbb{C}(q^{1/D})$ by the
elements
$\displaystyle{}_{\lambda_{1}}\omega^{(1)},\ldots,{}_{\lambda_{n}}\omega^{(n)}$
and $\displaystyle{}_{\lambda}\eta$ for all fundamental weights
$\displaystyle\lambda,\lambda_{1},\ldots,\lambda_{n}$. In particular, when
$\displaystyle\mathfrak{g}=sl(2)$ we have

$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}(sl(2))^{U_{q}(sl(2))})=\mathbb{C}(q^{1/2})[\omega^{(1)},\ldots,\omega^{(n)},\eta].$

This is a consequence of the following lemmas, which are interesting by
themselves. Denote by $\displaystyle U_{h}:=U_{h}(\mathfrak{g})$ the Hopf
algebra over $\displaystyle{\mathbb{C}}[[h]]$ generated topologically (in the
$\displaystyle h$-adic sense) by $\displaystyle E_{i},F_{i},H_{i}$, where
$\displaystyle i=1,...,m$, satisfying the relations
$\displaystyle[H_{i},H_{j}]=0$, $\displaystyle[H_{i},E_{j}]=a_{ij}E_{j}$,
$\displaystyle[H_{i},F_{j}]=-a_{ij}F_{j}$ and (4), (7) and (10), where
$\displaystyle K_{i}$ is replaced by $\displaystyle q_{i}^{H_{i}}$ and
$\displaystyle q_{i}$ by $\displaystyle e^{d_{i}h}$. By the same formula as
(52) with $\displaystyle U_{h}$ in place of $\displaystyle U_{q}$,
$\displaystyle U_{h}$ acts on $\displaystyle U_{h}^{\otimes n}$ by the right
adjoint action. Let $\displaystyle\mathcal{Z}(U_{h})$ be the center of
$\displaystyle U_{h}$.

######  Lemma 6.25.

The $\displaystyle h$-adic completion of
$\displaystyle\Delta^{(n-1)}(\mathcal{Z}(U_{h}))\otimes_{{\mathbb{C}}[[h]]}\mathcal{Z}(U_{h})^{\otimes
n}$, considered as a subalgebra of $\displaystyle U_{h}^{\otimes n}$, is the
center of $\displaystyle(U_{h}^{\otimes n})^{U_{h}}$.

Proof. Denote by $\displaystyle U=U(\mathfrak{g})$ the envelopping algebra of
$\displaystyle{\mathfrak{g}}$, and by $\displaystyle\Delta_{0}$ its canonical
coproduct. We have $\displaystyle U_{h}/hU_{h}=U.$ As shown in [72], Lemma
3.10, a direct application of Theorem 10.1 of [58] proves the result for
$\displaystyle h=0$, i.e for $\displaystyle U$. Because
$\displaystyle{\mathfrak{g}}$ is finite dimensional and semisimple, Drinfeld’s
results in [36] show that there exists an isomorphism of algebras
$\displaystyle\phi:U_{h}\rightarrow U[[h]]$, equal to the identity on the
quotient $\displaystyle U_{h}/hU_{h}$, and there exists an invertible element
$\displaystyle J\in U^{\otimes 2}[[h]]$ such that $\displaystyle\forall x\in
U_{h},(\phi\otimes\phi)(\Delta(x))=J\Delta_{0}(\phi(x))J^{-1}$. By using
$\displaystyle J$ one can easily define an invertible element $\displaystyle
J_{n}\in U^{\otimes n}[[h]]$ such that $\displaystyle\forall x\in U_{h}$,
$\displaystyle(\phi^{\otimes
n})(\Delta^{(n-1)}(x))=J_{n}\Delta_{0}^{(n-1)}(\phi(x))J_{n}^{-1}$. As a
result the map $\displaystyle\psi:(U_{h}^{\otimes
n})^{U_{h}}\rightarrow(U^{\otimes n})^{U}[[h]]$,
$\displaystyle\psi(x)=J_{n}^{-1}(\phi^{\otimes n}(x))J_{n}$, is an isomorphism
of algebras. We obviously have
$\displaystyle\phi(\mathcal{Z}(U_{h}))=\mathcal{Z}(U)[[h]]$. The lemma follows
after having checked that $\displaystyle\psi^{-1}(\mathcal{Z}(U)^{\otimes
n}[[h]])=\mathcal{Z}(U_{h})^{\otimes n}$ and
$\displaystyle\psi^{-1}(\Delta_{0}^{(n-1)}(\mathcal{Z}(U)[[h]])=\Delta^{(n-1)}(\mathcal{Z}(U_{h})).$
$\displaystyle\Box$

######  Lemma 6.26.

The center of $\displaystyle(U_{q}^{\otimes n})^{U_{q}}$ is
$\displaystyle\mathcal{Z}((U_{q}^{\otimes
n})^{U_{q}})=\Delta^{(n-1)}(\mathcal{Z}(U_{q}))\otimes_{{\mathbb{C}}(q)}\mathcal{Z}(U_{q})^{\otimes
n}$.

Proof. By Lemma 6.23 it only remains to prove the inclusion
“$\displaystyle\subset$”. Let
$\displaystyle\varphi:U_{A}\otimes_{A}{\mathbb{C}}[[h]]\rightarrow U_{h}$ be
the morphism of algebras defined by
$\displaystyle\varphi(K_{i})=e^{hd_{i}H_{i}},\varphi(E_{i})=E_{i},\varphi(F_{i})=F_{i}$.
Let $\displaystyle z\in\mathcal{Z}((U_{q}^{\otimes n})^{U_{q}})$. Up to
multiplication by an element of $\displaystyle A$ we can assume that
$\displaystyle z\in\mathcal{Z}((U_{A}^{\otimes n})^{U_{A}})$. Because
$\displaystyle\varphi^{\otimes n}(z)$ centralizes
$\displaystyle\varphi^{\otimes n}((U_{A}^{\otimes
n})^{U_{A}}\otimes_{A}{\mathbb{C}}[[h]])$, it centralizes its $\displaystyle
h$-completion. Combining Proposition 6.5 and Proposition 6.20 we get that the
elements $\displaystyle\Phi_{n}(v_{[\lambda]}(a_{[\lambda]}^{(k)})$ make an
$\displaystyle A$-basis of $\displaystyle(U_{A}^{\otimes n})^{U_{A}}$. These
results still hold by working with $\displaystyle U_{h}$ and the topological
version of $\displaystyle{\mathcal{L}}_{0,n}$ defined over
$\displaystyle\mathbb{C}[[h]]$, so $\displaystyle\varphi^{\otimes
n}(\Phi_{n}(v_{[\lambda]}(a_{[\lambda]}^{(k)}))$ is a
$\displaystyle{\mathbb{C}}[[h]]$-topological basis of
$\displaystyle(U_{h}^{\otimes n})^{U_{h}}$. We deduce that
$\displaystyle\varphi^{\otimes n}(z)$ centralizes
$\displaystyle(U_{h}^{\otimes n})^{U_{h}}$, and therefore
$\displaystyle\varphi^{\otimes
n}(z)\in\Delta^{(n-1)}(\mathcal{Z}(U_{h}))\otimes_{{\mathbb{C}}[[h]]}\mathcal{Z}(U_{h})^{\otimes
n}$ by Lemma 6.25. This proves $\displaystyle
z\in\Delta^{(n-1)}(\mathcal{Z}(U_{q}))\otimes_{{\mathbb{C}}(q)}\mathcal{Z}(U_{q})^{\otimes
n}$. $\displaystyle\Box$

Proof of Theorem 6.24. We have an isomorphism
$\displaystyle{\mathcal{L}}_{0,n}^{U_{q}}\cong(\tilde{U}_{q}^{\otimes
n})^{U_{q}}\otimes{\mathbb{C}}(q^{1/D})$ by Proposition 6.5, and
$\displaystyle\mathcal{Z}((\tilde{U}_{q}^{\otimes
n})^{U_{q}})=\Delta^{(n-1)}(\mathcal{Z}(\tilde{U}_{q}))\otimes_{{\mathbb{C}}(q)}\mathcal{Z}(\tilde{U}_{q})^{\otimes
n}$ by the last lemma applied to $\displaystyle\tilde{U}_{q}$ instead of
$\displaystyle U_{q}$. But
$\displaystyle\Phi_{n}^{-1}(\mathcal{Z}(\tilde{U}_{q})^{\otimes n})$ is the
vector space generated by
$\displaystyle{}_{\lambda_{1}}\omega^{(1)},\ldots,{}_{\lambda_{n}}\omega^{(n)}$
for $\displaystyle\lambda_{1},\ldots,\lambda_{n}\in P_{+}$ (by Proposition
4.8), and
$\displaystyle\Phi_{n}^{-1}(\Delta^{(n-1)}(\mathcal{Z}(\tilde{U}_{q})))$ is
the vector space generated by $\displaystyle{}_{\lambda}\eta$ for all
$\displaystyle\lambda\in P_{+}$ (by Proposition 4.8 and the relation (70)).
$\displaystyle\Box$

##  7\. Center of unrestricted specializations at roots of unity

Let $\displaystyle\epsilon\in{\mathbb{C}}^{\times}$. We defined the
unrestricted specialization $\displaystyle U_{\epsilon}$ of $\displaystyle
U_{q}$ in (15). Similarly, the unrestricted specialisation of
$\displaystyle{\mathcal{L}}_{0,n}^{A}$ at $\displaystyle\epsilon$ is the
$\displaystyle U_{\epsilon}$-module algebra

(78)
${\mathcal{L}}_{0,n}^{\epsilon}={\mathcal{L}}_{0,n}^{A}\otimes_{A}{\mathbb{C}}_{\epsilon}$

where $\displaystyle{\mathbb{C}}_{\epsilon}={\mathbb{C}}$ as a vector space,
and as an $\displaystyle A$-module, $\displaystyle q$ is evaluated on
$\displaystyle{\mathbb{C}}_{\epsilon}$ by $\displaystyle\epsilon$. By Lemma
6.8 the Alekseev map yields an embedding of $\displaystyle
U_{\epsilon}$-module algebras

$\displaystyle\Phi_{n}:{\mathcal{L}}_{0,n}^{\epsilon}\to\tilde{U}_{\epsilon}^{\otimes
n}.$

By Lemma 4.11, when $\displaystyle n=1$ we have an isomorphism

(79)
$\Phi_{1}:{\mathcal{L}}_{0,1}^{\epsilon}\to\tilde{U}_{A}^{lf}\otimes_{A}{\mathbb{C}}_{\epsilon}.$

When $\displaystyle\epsilon$ is a root of unity,
$\displaystyle\tilde{U}_{\epsilon}$ is a free module of finite rank over its
center (see eg. [27], Section 9.2). Hence
$\displaystyle\tilde{U}_{\epsilon}^{lf}=\tilde{U}_{\epsilon}$. On another
hand, $\displaystyle\tilde{U}_{A}^{lf}\otimes_{A}{\mathbb{C}}_{\epsilon}$ is
strictly contained in $\displaystyle\tilde{U}_{\epsilon}$; for instance it
does not contain the elements $\displaystyle K_{i}$.

Finally, recall that $\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}$ is the
centralizer of $\displaystyle\Delta^{(n-1)}({\mathcal{L}}_{0,1}^{A})$, see
Proposition 6.17. By the same arguments
$\displaystyle({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}}$ is the
centralizer of $\displaystyle\Delta^{(n-1)}({\mathcal{L}}_{0,1}^{\epsilon})$.
We have obvious inclusions

$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})\subset({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}}\
,\
({\mathcal{L}}_{0,n}^{A})^{U_{A}}\otimes_{A}\mathbb{C}_{\epsilon}\subset({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}}$

and the multiplication map

$\displaystyle(({\mathcal{L}}_{0,n}^{A})^{U_{A}}\otimes_{A}\mathbb{C}_{\epsilon})\otimes\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})\rightarrow({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}}$

is a morphism of algebras. In the rest of this paper we will simplify
notations by setting

$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}_{\epsilon}:=({\mathcal{L}}_{0,n}^{A})^{U_{A}}\otimes_{A}\mathbb{C}_{\epsilon}.$

Since $\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}$ is a free
$\displaystyle A$-module,
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}\otimes_{\mathbb{Z}}\mathbb{C}$
is isomorphic to
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}_{\epsilon}\otimes_{\mathbb{C}}\mathbb{C}[q,q^{-1}]$
by mapping the elements $\displaystyle v_{[\lambda]}(a_{[\lambda]}^{(k)})$
defined in (74) to a basis of
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}_{\epsilon}$. Moreover, by the
same arguments as of Proposition 6.9, taking $\displaystyle q=\epsilon$, we
have:

######  Proposition 7.1.

The algebra $\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$ does not have non
trivial zero divisors, and therefore the subalgebra
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}_{\epsilon}$ too.

###  7.1. Center of $\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}(sl(2))$

From now, we stipulate that $\displaystyle\mathfrak{g}=sl(2)$ and
$\displaystyle\epsilon$ is a primitive $\displaystyle l$-th root of unity,
with $\displaystyle l\geq 3$ is odd. We omit $\displaystyle\mathfrak{g}$ from
the notations of the various algebras, and denote $\displaystyle
U_{\epsilon}(sl(2))$ by $\displaystyle U_{\epsilon}$, etc.

As above define

(80)
${}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}={}_{loc}{\mathcal{L}}_{0,n}^{A}\otimes_{A}{\mathbb{C}}_{\epsilon}.$

By Lemma 5.7 (3), Proposition 6.10 and Proposition 6.14, the Alekseev map
yields embeddings of $\displaystyle U_{\epsilon}$-module algebras

$\displaystyle\Phi_{1}:{\mathcal{L}}_{0,1}^{\epsilon}\to U_{\epsilon},\
\Phi_{n}:{\mathcal{L}}_{0,n}^{\epsilon}\to U_{\epsilon}^{\otimes n}$

and isomorphisms of $\displaystyle U_{\epsilon}$-module algebras

$\displaystyle\Phi_{1}:{}_{loc}{\mathcal{L}}_{0,1}^{\epsilon}\to
U_{\epsilon},\ \Phi_{n}:{}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}\to
U_{\epsilon}^{\otimes n}.$

Arguing as in the proof of Proposition 6.5, working with
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}$ instead of
$\displaystyle({\mathcal{L}}_{0,n})^{U_{q}}$, one obtains that the restriction
of $\displaystyle\Phi_{n}$ to the invariant subalgebras is an isomorphism,

(81)
$\Phi_{n}:({\mathcal{L}}_{0,n}^{\epsilon})^{U_{\epsilon}}\to(U_{\epsilon}^{\otimes
n})^{U_{\epsilon}}.$

By the relations (37) and (38),
$\displaystyle\Phi_{1}({\mathcal{L}}_{0,1}^{\epsilon})=\tilde{U}_{A}^{lf}\otimes_{A}{\mathbb{C}}_{\epsilon}=U_{A}^{lf}\otimes_{A}{\mathbb{C}}_{\epsilon}$
is the algebra generated over $\displaystyle\mathbb{C}$ by
$\displaystyle\Omega$, $\displaystyle EK^{-1}$, $\displaystyle F$ and
$\displaystyle K^{-1}$. Adding the generator $\displaystyle K$ gives
$\displaystyle U_{\epsilon}$. By results of [29], the center
$\displaystyle\mathcal{Z}(U_{\epsilon})$ of $\displaystyle U_{\epsilon}$ is
the $\displaystyle\mathbb{C}$-algebra generated by $\displaystyle E^{l}$,
$\displaystyle F^{l}$, $\displaystyle K^{\pm l}$ and $\displaystyle\Omega$
satisfying the relation

(82) $\prod_{j=1}^{l}(\Omega-
c_{j})=(\epsilon-\epsilon^{-1})^{2l}E^{l}F^{l}+K^{l}+K^{-l}-2,$

where $\displaystyle\textstyle c_{j}=\epsilon^{j}+\epsilon^{-j}$. Let
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ be the subalgebra of
$\displaystyle\mathcal{Z}(U_{\epsilon})$ generated by $\displaystyle E^{l}$,
$\displaystyle F^{l}$ and $\displaystyle K^{\pm l}$. It is a sub-Hopf algebra
of $\displaystyle U_{\epsilon}$, with

$\begin{array}[]{c}\Delta(K^{\pm l})=K^{\pm l}\otimes K^{\pm l}\ ,\
\Delta(E^{l})=E^{l}\otimes K^{l}+1\otimes E^{l}\ ,\
\Delta(F^{l})=K^{-l}\otimes F^{l}+F^{l}\otimes 1\\\ S(E^{l})=-E^{l}K^{-l}\ ,\
S(F^{l})=-K^{l}F^{l}\ ,\ S(K^{\pm l})=K^{\mp l}\\\
\varepsilon(E^{l})=\varepsilon(F^{l})=0,\ \varepsilon(K^{l})=1.\end{array}$

Consider the sequence of polynomials $\displaystyle T_{k}$, $\displaystyle
k\in\mathbb{N}$, defined recursively by

(83) $T_{0}(x)=2,\ T_{1}(x)=x,\ T_{k}(x)=xT_{k-1}(x)-T_{k-2}(x)\quad{\rm for}\
k\geq 2.$

Note that $\displaystyle T_{k}(x)/2$ is the $\displaystyle k$-th Chebyshev
polynomial of the first type in the variable $\displaystyle x/2$. One has
$\displaystyle T_{k}(u+u^{-1})=u^{k}+u^{-k}$, from which one derives easily
that $\displaystyle\textstyle T_{l}(x)-2=\prod_{j=1}^{l}(x-c_{j})$. Therefore,
the relation (82) can be written as:

(84) $T_{l}(\Omega)=(\epsilon-\epsilon^{-1})^{2l}E^{l}F^{l}+K^{l}+K^{-l}.$

Since $\displaystyle l$ is odd, it is also equivalent to

(85)
$\prod_{j=1}^{l}(\Omega+c_{j})=(\epsilon-\epsilon^{-1})^{2l}E^{l}F^{l}+K^{l}+K^{-l}+2.$

By (37) we have

(86) $\Phi_{1}(\omega)=\Omega,\
\Phi_{1}(b^{l})=(\epsilon-\epsilon^{-1})^{l}F^{l},\
\Phi_{1}(c^{l})=(\epsilon-\epsilon^{-1})^{l}(EK^{-1})^{l},\
\Phi_{1}(d^{l})=K^{-l}.$

Hence

(87) $T_{l}(\omega)=b^{l}c^{l}d^{-l}+d^{-l}+d^{l}.$

Using that $\displaystyle\Phi_{1}$ is equivariant, injective, and surjective
when extended to the localization, we deduce

$\displaystyle\displaystyle\mathcal{Z}({}_{loc}{\mathcal{L}}_{0,1}^{\epsilon})$
$\displaystyle\displaystyle=\mathbb{C}[\omega,b^{l},c^{l},d^{\pm
l}]/\mathcal{I}$
$\displaystyle\displaystyle\mathcal{Z}({\mathcal{L}}_{0,1}^{\epsilon})$
$\displaystyle\displaystyle=\mathbb{C}[\omega,b^{l},c^{l},d^{l}]/\mathcal{I}$

where $\displaystyle\mathcal{I}$ is the ideal of
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,1}^{\epsilon})$ generated by
$\displaystyle(T_{l}(\omega)-d^{l})d^{l}-b^{l}c^{l}-1$. Moreover, we have
$\displaystyle(U_{A}^{lf}\otimes_{A}{\mathbb{C}}_{\epsilon})^{U_{\epsilon}}=\mathcal{Z}(U_{A}^{lf}\otimes_{A}{\mathbb{C}}_{\epsilon})$
by the arguments of Corollary 6.17 for $\displaystyle n=1$. Therefore

$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,1}^{\epsilon})=\Phi_{1}^{-1}((U_{A}^{lf}\otimes_{A}{\mathbb{C}}_{\epsilon})^{U_{\epsilon}})=({\mathcal{L}}_{0,1}^{\epsilon})^{U_{\epsilon}}.$

We can now define a notion of quantum Frobenius homomorphism for
$\displaystyle{\mathcal{L}}_{0,1}^{\epsilon},$ similar in spirit to the one
defined for $\displaystyle\mathcal{O}_{\epsilon}$ in [69], which is a map
$\displaystyle\mathcal{O}_{1}\rightarrow\mathcal{Z}(\mathcal{O}_{\epsilon})$.
Consider the specialization $\displaystyle{\mathcal{L}}_{0,1}^{1}$ of
$\displaystyle{\mathcal{L}}_{0,1}$ at $\displaystyle q=1$. We have
$\displaystyle{\mathcal{L}}_{0,1}^{1}=\mathcal{O}_{1}=\mathcal{O}(G)$ as
commutative algebras (with $\displaystyle G=SL(2,\mathbb{C}))$). Denote by
$\displaystyle\underline{a},\underline{b},\underline{c},\underline{d}$ the
images of the generators $\displaystyle a,b,c,d$ of
$\displaystyle{\mathcal{L}}_{0,1}$ under the specialization map
$\displaystyle{\mathcal{L}}_{0,1}\rightarrow{\mathcal{L}}_{0,1}^{1}$. They
satisfy
$\displaystyle\underline{a}\underline{d}-\underline{b}\underline{c}=1.$ Let us
define $\displaystyle Q_{l}\in{\mathbb{C}}[X,Y]$ by $\displaystyle
Q_{l}(X,Y)=T_{l}(\epsilon X+\epsilon^{-1}Y)-X^{l}-Y^{l}$. Recall that
$\displaystyle a$ and $\displaystyle d$ commute. So we put:

######  Definition 7.2.

The Frobenius map $\displaystyle
Fr:{\mathcal{L}}_{0,1}^{1}\rightarrow\mathcal{Z}({\mathcal{L}}_{0,1}^{\epsilon})$
is the homomorphism of algebras given by

$\displaystyle Fr(\underline{a})=a^{l}+Q_{l}(a,d)=T_{l}(\omega)-d^{l},\
Fr(\underline{b})=b^{l},\ Fr(\underline{c})=c^{l},\ Fr(\underline{d})=d^{l}.$

We shall denote

(88) $\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}=\begin{pmatrix}\underline{a}&\underline{b}\\\
\underline{c}&\underline{d}\end{pmatrix}\quad{\rm and}\quad
Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}=\begin{pmatrix}a^{l}+Q_{l}(a,d)&b^{l}\\\
c^{l}&d^{l}\end{pmatrix}.$

Note that :

  * •

$\displaystyle det\left(Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}\right)-1=(T_{l}(\omega)-d^{l})d^{l}-b^{l}c^{l}-1$, ie. the
generator of the ideal $\displaystyle\mathcal{I}$.

  * •

$\displaystyle T_{l}\left(qTr\left(\stackrel{{\scriptstyle
V_{2}}}{{M}}\right)\right)=Tr\left(Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}\right).$

The notions above can be developed similarly for every
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$, $\displaystyle n\geq 1$. First,
recall the additional generators $\displaystyle\delta^{(i)-1}$ of the
localization $\displaystyle{}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}$ introduced
in Definition 6.13.

######  Proposition 7.3.

We have

$\displaystyle\begin{array}[]{l}\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})=\mathbb{C}[\omega^{(i)},b^{(i)l},c^{(i)l},d^{(i)l},i=1,\ldots,n]/(\mathcal{I}^{(i)},i=1,\ldots,n),\\\
\mathcal{Z}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})=\mathbb{C}[\omega^{(i)},b^{(i)l},c^{(i)l},d^{(i)l},(\delta^{(i)l})^{-1},i=1,\ldots,n]/(\mathcal{I}^{(i)},i=1,\ldots,n)\end{array}$

where $\displaystyle\mathcal{I}^{(i)}$ is the ideal generated by the element
$\displaystyle(T_{l}(\omega^{(i)})-d^{(i)l})d^{(i)l}-b^{(i)l}c^{(i)l}-1$.

Proof. First we prove that $\displaystyle\Phi_{n}(b^{(i)l})$,
$\displaystyle\Phi_{n}(c^{(i)l})$,
$\displaystyle\Phi_{n}(d^{(i)l})\in\mathcal{Z}(U_{\epsilon}^{\otimes n})$ for
every $\displaystyle i=1,\ldots,n$. By injectivity of $\displaystyle\Phi_{n}$
it will follow that $\displaystyle b^{(i)l}$, $\displaystyle c^{(i)l}$,
$\displaystyle d^{(i)l}\in\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})$. By
(86) the claim is true for $\displaystyle i=n$. Let $\displaystyle 1\leq i\leq
n-1$, and denote $\displaystyle R^{(i)}:=R_{0n}\ldots R_{0i+1}\in
End(V_{2})\otimes\tilde{U}_{q}^{\otimes n}$, with $\displaystyle q$ an
indeterminate, as in the previous sections. Define $\displaystyle
r_{11}^{(i)}$, $\displaystyle r_{12}^{(i)}$, $\displaystyle r_{21}^{(i)}$,
$\displaystyle r_{22}^{(i)}\in\tilde{U}_{q}^{\otimes n}$ by

$\displaystyle R^{(i)}=\begin{pmatrix}r_{11}^{(i)}&r_{12}^{(i)}\\\
r_{21}^{(i)}&r_{22}^{(i)}\end{pmatrix}$

and $\displaystyle m_{11}^{(i)}$, $\displaystyle m_{12}^{(i)}$, $\displaystyle
m_{21}^{(i)}$, $\displaystyle m_{22}^{(i)}\in 1^{\otimes(i-1)}\otimes
U_{q}^{lf}\otimes 1^{\otimes(n-i)}$ by

$\displaystyle id_{V_{2}}\otimes\left(1^{\otimes(i-1)}\otimes\Phi_{1}\otimes
1^{\otimes(n-i)}\right)\begin{pmatrix}a^{(i)}&b^{(i)}\\\
c^{(i)}&d^{(i)}\end{pmatrix}=\begin{pmatrix}m_{11}^{(i)}&m_{12}^{(i)}\\\
m_{21}^{(i)}&m_{22}^{(i)}\end{pmatrix}$

where as in (66) we put

$\displaystyle\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(i)}=\begin{pmatrix}a^{(i)}&b^{(i)}\\\
c^{(i)}&d^{(i)}\end{pmatrix}\ \in End(V_{2})\otimes{\mathcal{L}}_{0,n}^{(i)}.$

By Proposition 5.4, $\displaystyle m_{11}^{(i)}$, $\displaystyle
m_{12}^{(i)}$, $\displaystyle m_{21}^{(i)}$, $\displaystyle m_{22}^{(i)}$
generate the subalgebra $\displaystyle 1^{\otimes(i-1)}\otimes
U_{q}^{lf}\otimes 1^{\otimes(n-i)}$ of $\displaystyle U_{q}^{\otimes n}$.
These elements satisfy the relations (36), with $\displaystyle m_{11}^{(i)}$,
$\displaystyle m_{12}^{(i)}$, $\displaystyle m_{21}^{(i)}$, $\displaystyle
m_{22}^{(i)}$ replacing $\displaystyle a$, $\displaystyle b$, $\displaystyle
c$, $\displaystyle d$ respectively, and they commute with $\displaystyle
r_{11}^{(i)}$, $\displaystyle r_{12}^{(i)}$, $\displaystyle r_{21}^{(i)}$,
$\displaystyle r_{22}^{(i)}$, since the non trivial tensor components of the
latters do not lie on the $\displaystyle i$-th tensorand of $\displaystyle
U_{q}^{\otimes n}$. Denote
$\displaystyle\mathfrak{i}_{in}\colon\mathbb{U}_{q}^{\otimes(n-i)}\rightarrow\mathbb{U}_{q}^{\otimes
n}$ the identification map with the last $\displaystyle n-i$ tensorands. We
have $\displaystyle
R^{(i)}=(\pi_{V_{2}}\otimes\mathfrak{i}_{in}\circ\Delta^{(n-i-1)})(R)$ by the
relation (1). So the formulas (61)-(64) yield

$\displaystyle r_{21}^{(i)}=0\ ,\ r_{22}^{(i)}=r_{11}^{(i)-1}\ ,\
r_{12}^{(i)}r_{11}^{(i)}=qr_{11}^{(i)}r_{12}^{(i)}\ ,\
r_{12}^{(i)}=r_{11}^{(i)}f_{12}^{(i)}\quad{\rm and}\quad
R^{(i)-1}=\begin{pmatrix}r_{11}^{(i)-1}&-qr_{12}^{(i)}\\\
0&r_{11}^{(i)}\end{pmatrix}$

where

(89) $r_{11}^{(i)}=\mathfrak{i}_{in}\circ\Delta^{(n-i-1)}(q^{H/2})\ ,\
f_{12}^{(i)}=\mathfrak{i}_{in}\circ\Delta^{(n-i-1)}((q-q^{-1})F).$

Therefore

$R^{(i)}\left(id_{V_{2}}\otimes\left(1^{\otimes(i-1)}\otimes\Phi_{1}\otimes
1^{\otimes(n-i)}\right)\right)\begin{pmatrix}a^{(i)}&b^{(i)}\\\
c^{(i)}&d^{(i)}\end{pmatrix}R^{(i)-1}=\\\
\begin{pmatrix}m_{11}^{(i)}+r_{12}^{(i)}r_{11}^{(i)-1}m_{21}^{(i)}&*\\\
r_{11}^{(i)-2}m_{21}^{(i)}&-qr_{11}^{(i)-1}r_{12}^{(i)}m_{21}^{(i)}+m_{22}^{(i)}\end{pmatrix}.$

Note also that, by definition,

$\displaystyle\left(id_{V_{2}}\otimes\Phi_{n}\right)\begin{pmatrix}a^{(i)}&b^{(i)}\\\
c^{(i)}&d^{(i)}\end{pmatrix}=R^{(i)}\left(id_{V_{2}}\otimes\left(1^{\otimes(i-1)}\otimes\Phi_{1}\otimes
1^{\otimes(n-i)}\right)\right)\begin{pmatrix}a^{(i)}&b^{(i)}\\\
c^{(i)}&d^{(i)}\end{pmatrix}R^{(i)-1}.$

Hence

$\displaystyle\displaystyle\Phi_{n}(c^{(i)})$
$\displaystyle\displaystyle=r_{11}^{(i)-2}m_{21}^{(i)},$
$\displaystyle\displaystyle\Phi_{n}(d^{(i)})$
$\displaystyle\displaystyle=-qr_{11}^{(i)-1}r_{12}^{(i)}m_{21}^{(i)}+m_{22}^{(i)}.$

Let now take the specialization $\displaystyle q=\epsilon$ as above, a
primitive $\displaystyle l$-th root of unity where $\displaystyle l\geq 3$ is
odd. We have

(90) $\displaystyle\displaystyle\Phi_{n}(c^{(i)l})$
$\displaystyle\displaystyle=(r_{11}^{(i)-2}m_{21}^{(i)})^{l}=r_{11}^{(i)-2l}m_{21}^{(i)l},$
(91) $\displaystyle\displaystyle\Phi_{n}(d^{(i)l})$
$\displaystyle\displaystyle=(-\epsilon
r_{11}^{(i)-1}r_{12}^{(i)}m_{21}^{(i)}+m_{22}^{(i)})^{l}=-r_{11}^{(i)-l}r_{12}^{(i)l}m_{21}^{(i)l}+m_{22}^{(i)l},$

by using $\displaystyle
m_{21}^{(i)}m_{22}^{(i)}=\epsilon^{2}m_{22}^{(i)}m_{21}^{(i)}$, the
$\displaystyle q$-binomial formula (see eg. [56], Proposition IV.2.2), and the
vanishing at $\displaystyle q=\epsilon$ of the $\displaystyle q$-Gauss
polynomials $\displaystyle[l]_{q}/[k]_{q}[l-k]_{q}$, $\displaystyle 0<k<l$.
Now

$\displaystyle
m_{21}^{(i)l}=(\epsilon-\epsilon^{-1})^{l}1^{\otimes(i-1)}\otimes(K^{-1}E)^{l}\otimes
1^{\otimes(n-i)},\
r_{11}^{(i)-2l}=\mathfrak{i}_{in}\circ\Delta^{(n-i-1)}(K^{-l}).$

These are central elements of $\displaystyle U_{\epsilon}^{\otimes n}$, so
$\displaystyle\Phi_{n}(c^{(i)l})$ is central. As

$\displaystyle
r_{11}^{(i)-l}r_{12}^{(i)l}=f_{12}^{(i)l}=\mathfrak{i}_{in}\circ\Delta^{(n-i-1)}((\epsilon-\epsilon^{-1})^{l}F^{l})$

and $\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ is a Hopf algebra,
$\displaystyle r_{11}^{(i)-l}r_{12}^{(i)l}$ is a central element of
$\displaystyle U_{\epsilon}^{\otimes n}.$ Again, $\displaystyle m_{21}^{(i)l}$
and $\displaystyle m_{22}^{(i)l}$ being central in $\displaystyle
U_{\epsilon}^{\otimes n}$, $\displaystyle\Phi_{n}(d^{(i)l})$ is central.

Finally, recalling that $\displaystyle{\mathcal{L}}_{0,n}^{(i)}$ is isomorphic
to $\displaystyle{\mathcal{L}}_{0,1}$, by specializing $\displaystyle q$ to
$\displaystyle\epsilon$ we get

(92) $d^{(i)l}T_{l}(\omega^{(i)})-d^{(i)2l}-1=b^{(i)l}c^{(i)l}.$

We know that
$\displaystyle\omega^{(i)}\in\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})$ and
we just proved that $\displaystyle d^{(i)l}$, $\displaystyle
c^{(i)l}\in\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})$. Therefore, for every
$\displaystyle x\in{\mathcal{L}}_{0,n}$ we get $\displaystyle
b^{(i)l}c^{(i)l}x=xb^{(i)l}c^{(i)l}=b^{(i)l}xc^{(i)l}$ i.e
$\displaystyle(b^{(i)l}x-xb^{(i)l})c^{(i)l}=0,$ $\displaystyle c^{(i)l}$ is
not a zero divisor, we deduce $\displaystyle b^{(i)l}x-xb^{(i)l}=0$. Hence
$\displaystyle b^{(i)l}$ is central in
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$.

A formula of $\displaystyle\Phi_{n}(b^{(i)l})$ can be obtained as follows.
Recall that (75) implies

(93) $\Phi_{n}(\omega^{(i)})=\Omega^{(i)}.$

This can also be checked by using the above formulas:

$\displaystyle\displaystyle\Phi_{n}(\omega^{(i)})$
$\displaystyle\displaystyle=q\Phi_{n}(a^{(i)})+q^{-1}\Phi_{n}(d^{(i)})=q(m_{11}^{(i)}+q^{-1}f_{12}^{(i)}m_{21}^{(i)})+q^{-1}(m_{22}^{(i)}-qf_{12}^{(i)}m_{21}^{(i)})$
$\displaystyle\displaystyle=qm_{11}^{(i)}+q^{-1}m_{22}^{(i)}.$

Then, by applying $\displaystyle\Phi_{n}$ to the relation (92) and using (90),
(91) and (93), one finds

(94)
$\Phi_{n}(b^{(i)l})=-r_{11}^{(i)l}r_{12}^{(i)l}(T_{l}(\Omega^{(i)})-2m_{22}^{(i)l})+r_{11}^{(i)2l}m_{12}^{(i)l}-r_{12}^{(i)2l}m_{21}^{(i)l}.$

We can now achieve the proof. Note that
$\displaystyle\mathbb{C}[\omega^{(1)},\ldots,\omega^{(n)}]=\mathcal{Z}({\mathcal{L}}_{0,n}^{A})\otimes_{A}{\mathbb{C}}_{\epsilon}$
by (7.1). The inclusion
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{A})\otimes_{A}{\mathbb{C}}_{\epsilon}\subset\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})$
is clear, and the natural embedding
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}\rightarrow{}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}$
maps $\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})$ into
$\displaystyle\mathcal{Z}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})$. By the
case $\displaystyle n=1$ the elements
$\displaystyle\omega^{(i)},b^{(i)l},c^{(i)l},d^{(i)l}$ generate
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon}{}^{(i)})$, with ideal
of relations $\displaystyle\mathcal{I}^{(i)}$. The set of these ideals for
$\displaystyle i=1,\ldots,n$ provide all the relations in
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})$, for there are no
others in Im$\displaystyle(\Phi_{n})$ (as shows eg. Corollary 7.5 below).
Therefore one has inclusions

$\displaystyle\mathbb{C}[\omega^{(i)},b^{(i)l},c^{(i)l},d^{(i)l},i=1,\ldots,n]/(\mathcal{I}^{(i)},i=1,\ldots,n)\subset\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})\subset\mathcal{Z}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}).$

The conclusion follows at once, since by their very definition the elements
$\displaystyle\delta^{(i)\pm l}$ are central in
$\displaystyle{}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}{}$, and
$\displaystyle\Phi_{n}$ maps the algebra generated by them and the left-hand
side isomorphically to $\displaystyle U_{\epsilon}^{\otimes n}$.
$\displaystyle\Box$

Analogously to the case $\displaystyle n=1$, recalling that
$\displaystyle{\mathcal{L}}_{0,1}^{1}=\mathcal{O}(G)$ we put:

######  Definition 7.4.

The Frobenius map $\displaystyle Fr:({\mathcal{L}}_{0,1}^{1})^{\otimes
n}\rightarrow\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})$ is the homomorphism
of algebras given by

$\displaystyle Fr(\underline{a}^{(i)})=a^{(i)l}+Q_{l}(a^{(i)},d^{(i)}),\
Fr(\underline{b}^{(i)})=b^{(i)l},\ Fr(\underline{c}^{(i)})=c^{(i)l},\
Fr(\underline{d}^{(i)})=d^{(i)l}.$

We shall denote

(95) $\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(i)}=\begin{pmatrix}\underline{a}^{(i)}&\underline{b}^{(i)}\\\
\underline{c}^{(i)}&\underline{d}^{(i)}\end{pmatrix}\quad{\rm and}\quad
Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(i)}=\begin{pmatrix}a^{(i)l}+Q_{l}(a^{(i)},d^{(i)})&b^{(i)l}\\\
c^{(i)l}&d^{(i)l}\end{pmatrix}$

where we recall that $\displaystyle
a^{(i)l}+Q_{l}(a^{(i)},d^{(i)})=T_{l}(\omega^{(i)})-d^{(i)l}$. We can express
$\displaystyle\Phi_{n}\circ(\Phi_{1}^{\otimes n})^{-1}$ on the center as
follows. Set

(96) $\displaystyle\displaystyle\mathcal{R}^{(i)}$
$\displaystyle\displaystyle=\begin{pmatrix}r_{11}^{(i)l}&r_{12}^{(i)l}\\\
0&r_{11}^{(i)-l}\end{pmatrix}$ $\displaystyle\displaystyle\mathcal{M}^{(i)}$
$\displaystyle\displaystyle=\left(1^{\otimes(i-1)}\otimes\Phi_{1}\otimes
1^{\otimes(n-i)}\right)\left(Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(i)}\right)$
$\displaystyle\displaystyle=\begin{pmatrix}T_{l}(\Omega^{(i)})-m_{22}^{(i)l}&m_{12}^{(i)l}\\\
m_{21}^{(i)l}&m_{22}^{(i)l}\end{pmatrix}$

where we use the notations in the proof of Proposition 7.3. These matrices
belong to $\displaystyle
End(V_{2})\otimes\mathcal{Z}_{0}(\tilde{U}_{\epsilon})^{\otimes n}$ and
$\displaystyle End(V_{2})\otimes\mathcal{Z}_{0}(U_{\epsilon})^{\otimes n}$
respectively. Here we note that
$\displaystyle\mathcal{Z}(\tilde{U}_{\epsilon})$ is generated by
$\displaystyle\Omega$, $\displaystyle E^{l}$, $\displaystyle F^{l}$ and
$\displaystyle K^{\pm\frac{l}{2}}$ satisfying the relation (82) (or (85)), and
we define $\displaystyle\mathcal{Z}_{0}(\tilde{U}_{\epsilon})$ as the
subalgebra generated by $\displaystyle E^{l}$, $\displaystyle F^{l}$ and
$\displaystyle K^{\pm\frac{l}{2}}$.

By using (90), (91), (93) and (94) it is easy to check that:

######  Corollary 7.5.

The map $\displaystyle\Phi_{n}\circ(\Phi_{1}^{\otimes
n})^{-1}\colon\mathcal{Z}_{0}(U_{\epsilon})^{\otimes
n}\rightarrow\mathcal{Z}_{0}(U_{\epsilon})^{\otimes n}$ is given by

$\displaystyle\left(id_{V_{2}}\otimes(\Phi_{n}\circ(\Phi_{1}^{\otimes
n})^{-1})\right)(\mathcal{M}^{(i)})=\mathcal{R}^{(i)}\mathcal{M}^{(i)}\mathcal{R}^{(i)-1}.$

We will find useful later to have explicit formulas. Let us introduce the
following generators of $\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$:

(97) $x=-(\epsilon-\epsilon^{-1})^{l}E^{l}K^{-l}\ ,\
y=(\epsilon-\epsilon^{-1})^{l}F^{l}\ ,\ z^{\pm 1}=K^{\pm l}.$

Similarly, denote by
$\displaystyle\mathcal{Z}_{0}(\tilde{U}_{\epsilon})\subset\tilde{U}_{\epsilon}$
the subalgebra generated by $\displaystyle x$, $\displaystyle y$,
$\displaystyle z$ and

$\displaystyle z^{\prime}{}^{\pm 1}=K^{\pm\frac{l}{2}}.$

For every $\displaystyle a\in\\{x,y,z^{\pm 1}\\}$ put $\displaystyle
a^{(i)}=1^{\otimes(i-1)}\otimes a\otimes 1^{\otimes(n-i)}$. We can view
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})^{\otimes n}$ as a polynomial
algebra in the variables $\displaystyle x^{(i)},y^{(i)},(z^{\pm 1})^{(i)}$,
and $\displaystyle\mathcal{Z}_{0}(\tilde{U}_{\epsilon})^{\otimes n}$ as a
polynomial algebra in $\displaystyle x^{(i)},y^{(i)},(z^{\prime}{}^{\pm
1})^{(i)}$. Then

(98)
$\mathcal{M}^{(i)}=\begin{pmatrix}T_{l}(\Omega^{(i)})-(z^{-1})^{(i)}&y^{(i)}\\\
-x^{(i)}&(z^{-1})^{(i)}\end{pmatrix}=\begin{pmatrix}z^{(i)}(1-x^{(i)}y^{(i)})&y^{(i)}\\\
-x^{(i)}&(z^{-1})^{(i)}\end{pmatrix}.$

Also,

(99) $\displaystyle\displaystyle r_{11}^{(i)l}$
$\displaystyle\displaystyle=(z^{\prime}{}^{-1})^{(i+1)}\ldots(z^{\prime}{}^{-1})^{(n)}$
(100) $\displaystyle\displaystyle r_{12}^{(i)l}$
$\displaystyle\displaystyle=(z^{\prime}{}^{-1})^{(i+1)}\ldots(z^{\prime}{}^{-1})^{(n)}\left(y^{(i+1)}+\sum_{j=1}^{n-i-1}(z^{-1})^{(i+1)}\ldots(z^{-1})^{(i+j)}y^{(i+j+1)}\right)$

yields

(101) $\displaystyle\displaystyle\Phi_{n}(c^{(i)l})$
$\displaystyle\displaystyle=r_{11}^{(i)-2l}m_{21}^{(i)l}=-x^{(i)}(z^{-1})^{(i+1)}\ldots(z^{-1})^{(n)}$
(102) $\displaystyle\displaystyle\Phi_{n}(d^{(i)l})$
$\displaystyle\displaystyle=m_{22}^{(i)l}-m_{21}^{(i)l}r_{11}^{(i)-l}r_{12}^{(i)l}$
$\displaystyle\displaystyle=(z^{-1})^{(i)}+x^{(i)}\left(y^{(i+1)}+\sum_{j=1}^{n-i-1}(z^{-1})^{(i+1)}\ldots(z^{-1})^{(i+j)}y^{(i+j+1)}\right).$

One can readily express
$\displaystyle\Phi_{n}(b^{(i)l})=-r_{11}^{(i)l}r_{12}^{(i)l}(T_{l}(\Omega^{(i)})-2m_{22}^{(i)l})+r_{11}^{(i)2l}m_{12}^{(i)l}-r_{12}^{(i)2l}m_{21}^{(i)l}$
as well as a polynomial in the variables $\displaystyle
x^{(j)},y^{(j)},(z^{\pm 1})^{(j)}$.

In [69], Parshall-Wang showed that the quantum Frobenius homomorphism
$\displaystyle\mathcal{O}_{1}\rightarrow\mathcal{Z}(\mathcal{O}_{\epsilon})$
they defined is a morphism of coalgebra. As we now explain, our quantum
Frobenius homomorphism $\displaystyle
Fr:{\mathcal{L}}_{0,1}^{1}\rightarrow\mathcal{Z}({\mathcal{L}}_{0,1}^{\epsilon})$
satisfies a similar property.

Recall the algebra morphism
$\displaystyle\Delta^{(n-1)}:{\mathcal{L}}_{0,1}^{A}\rightarrow{\mathcal{L}}_{0,n}^{A}$
(see Proposition 6.16). Denote again by
$\displaystyle\Delta^{(n-1)}:{\mathcal{L}}_{0,1}^{\epsilon}\rightarrow{\mathcal{L}}_{0,n}^{\epsilon}$
its evaluation at $\displaystyle\epsilon$. For every $\displaystyle 1\leq
i_{1}<\ldots<i_{k}\leq n$ we define
$\displaystyle\iota_{i_{1},\ldots,i_{k}}\colon{\mathcal{L}}_{0,k}^{\epsilon}\rightarrow{\mathcal{L}}_{0,n}^{\epsilon}$
to be the identification map of the $\displaystyle j$-th tensorand of
$\displaystyle{\mathcal{L}}_{0,k}^{\epsilon}$ with the $\displaystyle
i_{j}$-th tensorand of $\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$,
$\displaystyle 1\leq j\leq k$.

######  Lemma 7.6.

For every $\displaystyle 1\leq i_{1}<\ldots<i_{k}\leq n$ we have

(103)
$\iota_{i_{1},\ldots,i_{k}}\circ\Delta^{(k-1)}(Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}})=Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(i_{1})}\ldots Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(i_{k})}.$

Proof. First we show (103) in the case $\displaystyle k=n=2$, that is

(104) $\Delta(Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}})=Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(1)}Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(2)}.$

The commutation relations of $\displaystyle\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!\\!(1)}$ and $\displaystyle\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!\\!(2)}$ being complicated, we cannot compute the matrix
components directly (which is the way used in [69] for their quantum Frobenius
homomorphism
$\displaystyle\mathcal{O}_{1}\rightarrow\mathcal{Z}(\mathcal{O}_{\epsilon})$,
where such computations reduce to the $\displaystyle q$-binomial identity).
Instead, we first use the Alekseev map. Indeed, because
$\displaystyle\Phi_{2}$ is an algebra embedding, it is sufficient to show that
(104) holds after having been composed with $\displaystyle\Phi_{2}$. We have :

$\displaystyle\Phi_{1}(Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}})=\begin{pmatrix}T_{l}(\Omega)-K^{-l}&(\epsilon-\epsilon^{-1})^{l}F^{l}\\\
(\epsilon-\epsilon^{-1})^{l}(EK^{-1})^{l}&K^{-l}\end{pmatrix}.$

Therefore (104) is a consequence of the following four equations in
$\displaystyle U_{\epsilon}^{\otimes 2}:$

$\displaystyle\displaystyle\Delta(K^{-l})=\Phi_{2}(c^{(1)l}b^{(2)l}+d^{(1)l}d^{(2)l})$
$\displaystyle\displaystyle\Delta((\epsilon-\epsilon^{-1})^{l}(EK^{-1})^{l})=\Phi_{2}(c^{(1)l}(a^{(2)l}+Q_{l}(a^{(2)},d^{(2)}))+d^{(1)l}c^{(2)l})$
$\displaystyle\displaystyle\Delta((\epsilon-\epsilon^{-1})^{l}F^{l})=\Phi_{2}((a^{(1)l}+Q_{l}(a^{(1)},d^{(1)}))b^{(2)l}+b^{(1)l}d^{(2)l})$
$\displaystyle\displaystyle\Delta(T_{l}(\Omega)-K^{-l})=\Phi_{2}((a^{(1)l}+Q_{l}(a^{(1)},d^{(1)}))(a^{(2)l}+Q_{l}(a^{(2)},d^{(2)}))+b^{(1)l}c^{(2)l}).$

We listed them in order of complexity. The first equation, using the explicit
expression of $\displaystyle\Phi_{2}$ on components, is rewritten as:

$\displaystyle\Delta(K^{-l})=(\epsilon-\epsilon^{-1})^{2l}(K^{-1}E\otimes
K^{-1})^{l}(1\otimes F^{l})+(K^{-1}\otimes
1-\epsilon(\epsilon-\epsilon^{-1})^{2}K^{-1}E\otimes F)^{l}(1\otimes K^{-l}).$

This relation holds thanks to the $\displaystyle q-$binomial identity. The
other three relations, although more complicated, are shown similarly by a
direct computation and using the expressions (90), (91), (93) and (94).

Because the relative commutations relations of
$\displaystyle{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(1)},\ldots,{\stackrel{{\scriptstyle V_{2}}}{{M}}}{}^{(k)}$
and $\displaystyle{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(i_{1})},\ldots,{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(i_{k})}$ for general sequences $\displaystyle
i_{1}<...<i_{k}$ are the same, the proof of (103) follows immediately from the
case where $\displaystyle i_{j}=j$, for $\displaystyle 1\leq j\leq k$. This,
in turn, follows from (104) by induction on $\displaystyle k$. This concludes
the proof. $\displaystyle\Box$

The following consequence of the Lemma will be a key tool in Section 9.

######  Proposition 7.7.

For every $\displaystyle 1\leq i_{1}<\ldots<i_{k}\leq n$ we have

$\displaystyle T_{l}\left(qTr\left(\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(i_{1})}\ldots\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(i_{k})}\right)\right)=Tr\left(Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(i_{1})}\ldots Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(i_{k})}\right).$

In particular, this element is central in
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$.

Proof. Let
$\displaystyle\iota_{i_{1},\ldots,i_{k}}\colon{\mathcal{L}}_{0,k}^{\epsilon}\rightarrow{\mathcal{L}}_{0,n}^{\epsilon}$
be the identification map of the $\displaystyle j$-th tensorand of
$\displaystyle{\mathcal{L}}_{0,k}^{\epsilon}$ with the $\displaystyle
i_{j}$-th tensorand of $\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$,
$\displaystyle 1\leq j\leq k$. We have

$\displaystyle\displaystyle T_{l}\left(qTr\left(\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(i_{1})}\ldots\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(i_{k})}\right)\right)$
$\displaystyle\displaystyle=T_{l}\left(qTr\left(\left(id\otimes\left(\iota_{i_{1},\ldots,i_{k}}\circ\Delta^{(k-1)}\right)\right)\left(\stackrel{{\scriptstyle
V_{2}}}{{M}}\right)\right)\right)$
$\displaystyle\displaystyle=T_{l}\left(\left(\iota_{i_{1},\ldots,i_{k}}\circ\Delta^{(k-1)}\right)\left(qTr\left(\stackrel{{\scriptstyle
V_{2}}}{{M}}\right)\right)\right)$
$\displaystyle\displaystyle=\left(\iota_{i_{1},\ldots,i_{k}}\circ\Delta^{(k-1)}\right)\left(T_{l}\left(qTr\left(\stackrel{{\scriptstyle
V_{2}}}{{M}}\right)\right)\right)$
$\displaystyle\displaystyle=\left(\iota_{i_{1},\ldots,i_{k}}\circ\Delta^{(k-1)}\right)\left(T_{l}(\omega)\right).$

In the first equality we used the formula (70), the second and fourth
equalities follow from definitions, and the third comes from the fact that
$\displaystyle\Delta^{(k-1)}:{\mathcal{L}}_{0,1}^{A}\rightarrow{\mathcal{L}}_{0,k}^{A}$
is a homomorphism of algebras. By the identity (87) and the fact that
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})\cong\mathbb{C}[b^{l},c^{l},d^{\pm
l}]$ is a Hopf subalgebra of $\displaystyle U_{\epsilon}$, we have
$\displaystyle\Delta^{(k-1)}\left(T_{l}(\omega)\right)\in\mathcal{Z}({}_{loc}{\mathcal{L}}_{0,1}^{\epsilon})^{\otimes
k}$. This and Proposition 7.3 imply that the above element is central in
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$. Moreover

$\displaystyle\displaystyle\iota_{i_{1},\ldots,i_{k}}\circ\Delta^{(k-1)}\left(T_{l}(\omega)\right)$
$\displaystyle\displaystyle=\iota_{i_{1},\ldots,i_{k}}\circ\Delta^{(k-1)}\left(Tr\left(Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}\right)\right)$
$\displaystyle\displaystyle=Tr\left(\left(id\otimes\iota_{i_{1},\ldots,i_{k}}\circ\Delta^{(k-1)}\right)\left(Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}\right)\right)$
$\displaystyle\displaystyle=Tr\left(Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(i_{1})}\ldots Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(i_{k})}\right)$

where the first equality follows from the observations we made before
Proposition 7.3, and the others from the previous Lemma. $\displaystyle\Box$

###  7.2. $\displaystyle{\mathcal{G}}$-invariant central elements and
$\displaystyle SL(2,\mathbb{C})$-characters

We are going to relate
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})$ with the algebra of
regular functions on the variety of $\displaystyle
SL(2,\mathbb{C})$-characters of the sphere with $\displaystyle n+1$ punctures,
endowed with the Atiyah-Bott-Goldman Poisson structure. This is achieved in
Section 7.2.2. To this aim we recall a few preliminary results in the next
section.

####  7.2.1. The quantum coadjoint action for $\displaystyle U_{\epsilon}$

We refer to [29, 30, 31] for details about the material discussed in this
section. It can be formulated for any of the quantum groups $\displaystyle
U_{\epsilon}(\mathfrak{g})$, but we restrict to $\displaystyle
U_{\epsilon}(sl(2))$ as we shall need Proposition 7.3 in Section 7.2.2.

Consider the sets Spec$\displaystyle(\mathcal{Z}(U_{\epsilon}))$ and
Spec$\displaystyle(\mathcal{Z}_{0}(U_{\epsilon}))$ of algebra homomorphisms
from $\displaystyle\mathcal{Z}(U_{\epsilon})$ and
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ to $\displaystyle\mathbb{C}$,
respectively. They are affine algebraic sets. An element of
Spec$\displaystyle(\mathcal{Z}(U_{\epsilon}))$ is called a central character
of $\displaystyle U_{\epsilon}$. The inclusion
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})\subset\mathcal{Z}(U_{\epsilon})$
induces a regular (restriction) map

(105) $\tau\colon{\rm Spec}(\mathcal{Z}(U_{\epsilon}))\longrightarrow{\rm
Spec}(\mathcal{Z}_{0}(U_{\epsilon})).$

Since $\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ is a polynomial algebra,
any $\displaystyle\chi\in{\rm Spec}(\mathcal{Z}_{0}(U_{\epsilon}))$ is
entirely defined by its values
$\displaystyle(x_{\chi},y_{\chi},z_{\chi})\in\mathbb{C}^{2}\times\mathbb{C}^{*}$
on the tuple $\displaystyle(x,y,z)$ of generators of
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ defined in (97). By (82)-(85),
any $\displaystyle\chi\in{\rm Spec}(\mathcal{Z}(U_{\epsilon}))$ is entirely
defined by its values
$\displaystyle(x_{\chi},y_{\chi},z_{\chi},\Omega_{\chi})\in\mathbb{C}^{2}\times\mathbb{C}^{*}\times\mathbb{C}$
on the tuple $\displaystyle(x,y,z,\Omega)$ of generators of
$\displaystyle\mathcal{Z}(U_{\epsilon})$, solutions to one of the equivalent
equations

$\displaystyle\prod_{j=1}^{l}(\Omega_{\chi}\mp
c_{j})=-x_{\chi}y_{\chi}z_{\chi}+z_{\chi}+z_{\chi}^{-1}\mp 2$

where $\displaystyle c_{j}=\epsilon^{j}+\epsilon^{-j}$. Hence
$\displaystyle\tau$ has degree $\displaystyle l$, and
Spec$\displaystyle(\mathcal{Z}(U_{\epsilon}))$ is a branched covering of
$\displaystyle{\rm
Spec}(\mathcal{Z}_{0}(U_{\epsilon}))=\mathbb{C}^{2}\times\mathbb{C}^{*}$ of
degree $\displaystyle l$, a hypersurface in
$\displaystyle\mathbb{C}^{2}\times\mathbb{C}^{*}\times\mathbb{C}$ with
quadratic singularities at the points $\displaystyle(0,0,\pm 1,\pm c_{j})$,
$\displaystyle j=1,\ldots,(l-1)/2$. Because
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ is a commutative Hopf algebra,
Spec$\displaystyle(\mathcal{Z}_{0}(U_{\epsilon}))$ has a canonical group
structure defined dually by

(106) $\chi_{1}\chi_{2}(u):=(\chi_{1}\otimes\chi_{2})\Delta(u)\ ,\
\chi_{1}^{-1}(u):=\chi_{1}(S(u))\ ,\ e(u):=\varepsilon(u)$

for any $\displaystyle u\in\mathcal{Z}_{0}(U_{\epsilon})$ and
$\displaystyle\chi_{1}$, $\displaystyle\chi_{2}\in{\rm
Spec}(\mathcal{Z}_{0}(U_{\epsilon}))$, where $\displaystyle e\in{\rm
Spec}(\mathcal{Z}_{0}(U_{\epsilon}))$ is the identity element. In formulas:

(107)
$\begin{array}[]{c}x_{\chi_{1}\chi_{2}}=x_{\chi_{1}}+z_{\chi_{1}}^{-1}x_{\chi_{2}}\
,\ y_{\chi_{1}\chi_{2}}=y_{\chi_{1}}+y_{\chi_{2}}z_{\chi_{1}}^{-1}\ ,\
z_{\chi_{1}\chi_{2}}=z_{\chi_{1}}z_{\chi_{2}}\\\
x_{\chi^{-1}}=-z_{\chi}x_{\chi}\ ,\ y_{\chi^{-1}}=-y_{\chi}z_{\chi}\ ,\
z_{\chi^{-1}}=z_{\chi}^{-1}\\\ x_{e}=0\ ,\ y_{e}=0\ ,\ z_{e}=1.\end{array}$

This can be formulated as follows. Put $\displaystyle G=SL(2,\mathbb{C})$, and
let $\displaystyle G^{*}$ be the group formed by the pairs of matrices

$\displaystyle\left(\begin{pmatrix}a&b\\\
0&a^{-1}\end{pmatrix},\begin{pmatrix}a^{-1}&0\\\ c&a\end{pmatrix}\right)\in
SL(2,\mathbb{C})^{op}\times SL(2,\mathbb{C})^{op}$

where $\displaystyle SL(2,\mathbb{C})^{op}$ is $\displaystyle
SL(2,\mathbb{C})$ endowed with the opposite multiplication. Set

$\displaystyle\psi\left(\begin{pmatrix}z^{\prime}&z^{\prime}y\\\
0&z^{\prime}{}^{-1}\end{pmatrix},\begin{pmatrix}z^{\prime}{}^{-1}&0\\\
z^{\prime}x&z^{\prime}\end{pmatrix}\right)=(x,y,z^{\prime}{}^{2}).$

Identifying $\displaystyle{\rm Spec}(\mathcal{Z}_{0}(U_{\epsilon})))$ with
$\displaystyle\mathbb{C}^{2}\times\mathbb{C}^{*}$ by mapping
$\displaystyle\chi$ to $\displaystyle(x_{\chi},y_{\chi},z_{\chi})$ defined as
above, it is readily checked that this defines a surjective morphism of
algebraic groups

$\displaystyle\psi\colon G^{*}\rightarrow{\rm
Spec}(\mathcal{Z}_{0}(U_{\epsilon})))$

with kernel the subgroup generated by $\displaystyle-(I,I)$, where
$\displaystyle I$ is the $\displaystyle 2$-by-$\displaystyle 2$ identity
matrix. Put

$\displaystyle\bar{G}^{*}=G^{*}/\\{\pm(I,I)\\}.$

We will denote the quotient isomorphism by

$\displaystyle\bar{\psi}\colon\bar{G}^{*}\rightarrow{\rm
Spec}(\mathcal{Z}_{0}(U_{\epsilon}))).$

Let us endow the ring of regular functions $\displaystyle\mathcal{O}(G)$ with
the Sklyanin-Drinfeld Poisson bracket $\displaystyle\\{\ ,\ \\}$, associated
to the classical $\displaystyle r$-matrix

$\displaystyle\mathfrak{r}=\frac{1}{4}H\otimes H+E\otimes F\
\in\mathfrak{g}\otimes\mathfrak{g}.$

Recall that it can be given the following expression (see eg. [73], or [2],
[27] for a setup close to ours). First note that it is entirely determined by
its values on the matrix coefficients (coordinate functions) $\displaystyle
l_{11},l_{12},l_{21},l_{22}$ of the fundamental representation of
$\displaystyle G$ on $\displaystyle\mathbb{C}^{2}$. Put

(108) $L=\begin{pmatrix}l_{11}&l_{12}\\\
l_{21}&l_{22}\end{pmatrix}=\sum_{r,s=1}^{2}E_{r}^{s}\otimes l_{rs}\ \in
End(\mathbb{C}^{2})\otimes\mathcal{O}(G).$

Denote by $\displaystyle\\{\stackrel{{\scriptstyle
1}}{{L}},\stackrel{{\scriptstyle 2}}{{L}}\\}$ and
$\displaystyle\stackrel{{\scriptstyle 1}}{{L}}\stackrel{{\scriptstyle
2}}{{L}}$ the tensors in $\displaystyle End(\mathbb{C}^{2})\otimes
End(\mathbb{C}^{2})\otimes\mathcal{O}(G)$ defined by

$\displaystyle\\{\stackrel{{\scriptstyle 1}}{{L}},\stackrel{{\scriptstyle
2}}{{L}}\\}=\sum_{r,s,t,u=1}^{2}E_{r}^{s}\otimes
E_{t}^{u}\otimes\\{l_{rs},l_{tu}\\}\ ,\ \stackrel{{\scriptstyle
1}}{{L}}\stackrel{{\scriptstyle 2}}{{L}}=\sum_{r,s,t,u=1}^{2}E_{r}^{s}\otimes
E_{t}^{u}\otimes l_{rs}l_{tu}$

and as usual put $\displaystyle\textstyle\stackrel{{\scriptstyle
1}}{{L}}=\sum_{r,s=1}^{2}E_{r}^{s}\otimes 1\otimes l_{rs}$,
$\displaystyle\textstyle\stackrel{{\scriptstyle
2}}{{L}}=\sum_{t,u=1}^{2}1\otimes E_{t}^{u}\otimes l_{tu}$. The $\displaystyle
r$-matrix in the fundamental representation is

$\displaystyle\mathfrak{r}=\frac{1}{4}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\otimes\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}+\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\otimes\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}\ \in
End(\mathbb{C}^{2})\otimes End(\mathbb{C}^{2})$

which we identify with $\displaystyle\mathfrak{r}\otimes 1\in
End(\mathbb{C}^{2})\otimes End(\mathbb{C}^{2})\otimes\mathcal{O}(G)$. Then the
bracket $\displaystyle\\{\ ,\ \\}$ on $\displaystyle\mathcal{O}(G)$ is defined
by the identity

(109) $\\{\stackrel{{\scriptstyle 1}}{{L}},\stackrel{{\scriptstyle
2}}{{L}}\\}=[\mathfrak{r},\stackrel{{\scriptstyle
1}}{{L}}\stackrel{{\scriptstyle 2}}{{L}}]=\mathfrak{r}\stackrel{{\scriptstyle
1}}{{L}}\stackrel{{\scriptstyle 2}}{{L}}-\stackrel{{\scriptstyle
1}}{{L}}\stackrel{{\scriptstyle 2}}{{L}}\mathfrak{r}.$

The group $\displaystyle G$ with the bracket $\displaystyle\\{\ ,\ \\}$ is a
Poisson-Lie group. As such, it admits a dual Poisson-Lie group, which is
precisely the group $\displaystyle G^{*}$ defined as above. Its Poisson
bracket $\displaystyle\\{\ ,\ \\}_{G^{*}}$ is defined by

(110) $\\{\stackrel{{\scriptstyle 1}}{{L}}_{\pm},\stackrel{{\scriptstyle
2}}{{L}}_{\pm}\\}_{G^{*}}=[\mathfrak{r},\stackrel{{\scriptstyle
1}}{{L}}_{\pm}\stackrel{{\scriptstyle 2}}{{L}}_{\pm}]\ ,\
\\{\stackrel{{\scriptstyle 1}}{{L}}_{+},\stackrel{{\scriptstyle
2}}{{L}}_{-}\\}_{G^{*}}=[\mathfrak{r},\stackrel{{\scriptstyle
1}}{{L}}_{+}\stackrel{{\scriptstyle 2}}{{L}}_{-}]$

where $\displaystyle(L_{+},L_{-})$ is the pair of matrices of coordinate
functions on $\displaystyle G^{*}\subset SL(2,\mathbb{C})^{op}\times
SL(2,\mathbb{C})^{op}$. We will denote by $\displaystyle\\{,\\}_{\bar{G}^{*}}$
the induced bracket on $\displaystyle\bar{G}^{*}$. The push-forward
$\displaystyle\psi_{*}\\{\ ,\ \\}_{G^{*}}$ is a Poisson bracket on
$\displaystyle{\rm Spec}(\mathcal{Z}_{0}(U_{\epsilon}))$ that can be defined
directly via the quantum coadjoint action, that we now recall.

The specialization morphism $\displaystyle ev_{\epsilon}\colon
U_{A}\otimes\mathbb{C}[q,q^{-1}]\rightarrow U_{\epsilon}$, $\displaystyle
q\mapsto\epsilon$, is surjective, with kernel the subalgebra
$\displaystyle(q^{l}-q^{-l})U_{A}\otimes\mathbb{C}[q,q^{-1}]$. Given an
element $\displaystyle x\in U_{\epsilon}$, let us denote by
$\displaystyle\tilde{x}\in ev_{\epsilon}^{-1}(x)$ any preimage of
$\displaystyle x$. For every $\displaystyle a\in\mathcal{Z}(U_{\epsilon})$,
$\displaystyle u\in U_{\epsilon}$ we have
$\displaystyle[\tilde{a},\tilde{u}]=\tilde{a}\tilde{u}-\tilde{u}\tilde{a}\in(q^{l}-q^{-l})U_{A}\otimes\mathbb{C}[q,q^{-1}]$.
So, let us put

(111)
$D_{a}(u)=-\lim_{q\rightarrow\epsilon}\frac{\left[\tilde{a},\tilde{u}\right]}{l(q^{l}-q^{-l})},\quad
a\in\mathcal{Z}_{0}(U_{\epsilon}).$

It is easy to check that the maps $\displaystyle D_{a}\colon
U_{\epsilon}\rightarrow U_{\epsilon}$ are well-defined (they do not depend on
the choices of $\displaystyle\tilde{a}$ and $\displaystyle\tilde{u}$), and
that they are derivations of $\displaystyle U_{\epsilon}$ preserving
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ and
$\displaystyle\mathcal{Z}(U_{\epsilon})$. Hence they define algebraic vector
fields on Spec$\displaystyle(\mathcal{Z}_{0}(U_{\epsilon}))$. Since
$\displaystyle\Omega$ is central in $\displaystyle U_{q}(sl(2))$,
$\displaystyle D_{a}$ is the zero map when $\displaystyle
a\in\mathbb{C}[\Omega]$.

Let us introduce the elements $\displaystyle e,f$ defined in [29] by

(112)
$e=-xz=(\epsilon-\epsilon^{-1})^{l}E^{l},f=-yz=-(\epsilon-\epsilon^{-1})^{l}F^{l}K^{l}.$

Direct computations from the definition and the fact that $\displaystyle
D_{e}\Omega=0$ imply the formulas:

(113) $\displaystyle\displaystyle D_{z}(K)=0,\ D_{z}(E)=-\frac{1}{l}zE\ ,\
D_{z}(F)=\frac{1}{l}zF,$ (114) $\displaystyle\displaystyle
D_{e}(K)=\frac{1}{l}eK,D_{e}(E)=0,D_{e}(F)=-\frac{1}{l}(\epsilon-\epsilon^{-1})^{l-1}[K;1]E^{l-1},$
(115) $\displaystyle\displaystyle
D_{y}(K)=-\frac{1}{l}yK,D_{y}(F)=0,D_{y}(E)=\frac{1}{l}(\epsilon-\epsilon^{-1})^{l-1}[K;-1]F^{l-1},$

where
$\displaystyle[K;r]=\frac{K\epsilon^{r}-K^{-1}\epsilon^{-r}}{\epsilon-\epsilon^{-1}},r\in{\mathbb{Z}}.$

A Poisson bracket $\displaystyle\\{\ ,\ \\}_{\tt QCA}$ is defined on
$\displaystyle\mathcal{Z}(U_{\epsilon})$ by

$\displaystyle\\{a,b\\}_{\tt QCA}=D_{a}(b).$

With this Poisson bracket $\displaystyle\mathcal{Z}(U_{\epsilon})$ is a
Poisson algebra, $\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ being a Poisson
ideal. The Poisson structure on $\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$
is completely defined by the formulas

(116) $\\{y,x\\}_{\tt QCA}=-1+xy+z^{-2}\ ,\ \\{z,x\\}_{\tt QCA}=-zx\ ,\
\\{z,y\\}_{\tt QCA}=yz$

which can be straithforwardy checked using the definition. From them it is
easy to show that the Poisson center of
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ is the commutative algebra
generated by $\displaystyle-xyz+z+z^{-1}.$

Let us introduce the family of automorphisms of $\displaystyle U_{q}(sl(2))$
defined as follows. Let $\displaystyle r\in\mathbb{Z},$ and
$\displaystyle\tau_{r},T_{r}$ the automorphisms of $\displaystyle
U_{q}(sl(2))$ defined by:

(117)
$\displaystyle\displaystyle\tau_{r}(K)=K,\tau_{r}(E)=K^{r}E,\tau_{r}(F)=FK^{-r}$
(118) $\displaystyle\displaystyle
T_{r}(K)=K^{-1},T_{r}(E)=-FK^{-r},T_{r}(F)=-K^{r}E.$

Note that $\displaystyle T_{r}=\tau_{r}\circ T_{0}$ and $\displaystyle T_{1}$
is the braid group automorphism. We have

$\displaystyle T_{1}(x)=y,T_{1}(y)=z^{2}x.$

These automorphisms of $\displaystyle U_{q}(sl(2))$ define automorphisms of
the specialisation $\displaystyle U_{\epsilon}$ and we will keep the same
notation for them. If $\displaystyle T$ is any automorphism of the type
$\displaystyle T_{r}$ or $\displaystyle\tau_{r}$, it is easy to show from the
definition of $\displaystyle D_{a}$ for $\displaystyle
a\in{\mathcal{Z}}(U_{\epsilon})$ that

$\displaystyle D_{T(a)}\circ T=T\circ D_{a}.$

Denote by $\displaystyle G^{0}$ the big cell of $\displaystyle
G=SL(2,\mathbb{C})$. It consists of the matrices with non vanishing lower
right entry, and satisfies $\displaystyle G^{0}=B_{+}B_{-}$, where
$\displaystyle B_{+}$ and $\displaystyle B_{-}$ are the subgroups of
$\displaystyle G$ of upper and lower triangular matrices. We have an
unramified $\displaystyle 2$-fold covering

(119) $\begin{array}[]{ccll}\sigma:&G^{*}&\longrightarrow&G^{0}\\\
&(b_{+},b_{-})&\longmapsto&b_{+}b_{-}^{-1}.\end{array}$

It induces a diffeomorphism

$\displaystyle\bar{\sigma}\colon\bar{G}^{*}\rightarrow G^{0}.$

Setting $\displaystyle z:=z^{\prime}{}^{2}$ and

(120) $\mathcal{M}=\sigma\left(\begin{pmatrix}z^{\prime}&z^{\prime}y\\\
0&z^{\prime}{}^{-1}\end{pmatrix},\begin{pmatrix}z^{\prime}{}^{-1}&0\\\
z^{\prime}x&z^{\prime}\end{pmatrix}\right)$

we have

$\displaystyle\mathcal{M}=\begin{pmatrix}z-zxy&y\\\ -x&z^{-1}\end{pmatrix}.$

Consider the Poisson bracket $\displaystyle\\{\ ,\ \\}_{FR}$ on
$\displaystyle\mathcal{O}(G)$ defined by

$\displaystyle\\{\stackrel{{\scriptstyle 1}}{{L}},\stackrel{{\scriptstyle
2}}{{L}}\\}_{FR}=\mathfrak{r}\stackrel{{\scriptstyle
1}}{{L}}\stackrel{{\scriptstyle 2}}{{L}}-\stackrel{{\scriptstyle
1}}{{L}}\stackrel{{\scriptstyle
2}}{{L}}\mathfrak{r}^{\prime}+\stackrel{{\scriptstyle
2}}{{L}}\mathfrak{r}^{\prime}\stackrel{{\scriptstyle
1}}{{L}}-\stackrel{{\scriptstyle 1}}{{L}}\mathfrak{r}\stackrel{{\scriptstyle
2}}{{L}}$

where $\displaystyle\mathfrak{r}^{\prime}$ is $\displaystyle\mathfrak{r}$
post-composed with the flip map $\displaystyle a\otimes b\mapsto b\otimes a$.
The bracket $\displaystyle\\{\ ,\ \\}_{FR}$ has been introduced in [73] and
generalized in the work of Fock-Rosly [49] (which explains our notation). Note
that $\displaystyle\mathcal{O}(G^{0})$ is the localization of the algebra
$\displaystyle\mathcal{O}(G)$ with respect to the matrix coefficient
$\displaystyle l_{22}$ in (108). As a consequence, the Poisson bracket
$\displaystyle\\{,\\}_{FR}$ being quadratic, it can be extended to
$\displaystyle\mathcal{O}(G^{0})$.

The next result sums up the relationships between the brackets
$\displaystyle\\{\ ,\ \\}_{FR}$, $\displaystyle\\{\ ,\ \\}_{G^{*}}$, and
$\displaystyle\\{\ ,\ \\}_{\tt QCA}$:

######  Theorem 7.8.

(1) ([30, 31]) The map $\displaystyle\bar{\psi}\colon(\bar{G}^{*},\\{\ ,\
\\}_{\bar{G}^{*}})\rightarrow({\rm Spec}((\mathcal{Z}_{0}(U_{\epsilon})),\\{\
,\ \\}_{\tt QCA})$ is an isomorphism of complex Poisson-Lie groups.

(2) ([73]) The map $\displaystyle\bar{\sigma}\colon(\bar{G}^{*},\\{\ ,\
\\}_{\bar{G}^{*}})\rightarrow(G^{0},\\{\ ,\ \\}_{FR})$ is a diffeomorphism of
complex Poisson manifolds.

The first claim is proved in [30, 31] for the simply-connected quantum groups
$\displaystyle\tilde{U}_{\epsilon}(\mathfrak{g})$, where
$\displaystyle\mathfrak{g}$ is a complex finite dimensional simple Lie algebra
(see Remark 7.10 below). We state it rather in the case of $\displaystyle
U_{\epsilon}(sl(2))$, the adjoint quantum group for
$\displaystyle\mathfrak{g}=sl(2)$, where it follows from straighforward
computations using the formulas (110) and (116). The second statement follows
as well from straighforward computations. Namely, by using the identities
(110) and $\displaystyle\sigma^{*}(L)=L_{+}L_{-}^{-1}$, and the fact that
$\displaystyle\mathfrak{r}+\mathfrak{r}^{\prime}$ is $\displaystyle
ad$-invariant, one can identify the formulas of $\displaystyle\\{\ ,\
\\}_{FR}$ and

$\displaystyle\sigma_{*}(\\{\ ,\ \\}_{G^{*}})(L\otimes
L)=\\{\sigma^{*}(\stackrel{{\scriptstyle
1}}{{L}}),\sigma^{*}(\stackrel{{\scriptstyle 2}}{{L}})\\}_{G^{*}}.$

We leave the verifications to the reader.

The next statement summarizes the results of De Concini-Kac-Procesi on the
quantum coadjoint action. Recall the elements $\displaystyle e=-xz,f=-yz$ of
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ (see (112)). Let us identify
$\displaystyle{\rm Spec}(\mathcal{Z}_{0}(U_{\epsilon}))$ with $\displaystyle
G^{0}$ using the map $\displaystyle\bar{\sigma}\circ\bar{\psi}^{-1}$ of
Theorem 7.8, and hence the derivations $\displaystyle D_{e}$, $\displaystyle
D_{f}$, $\displaystyle D_{z}$ of $\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$
with algebraic vector fields on $\displaystyle G^{0}$ (and hence on
$\displaystyle G$). Denote by $\displaystyle\underline{H}$,
$\displaystyle\underline{X}$, $\displaystyle\underline{Y}$ the left-invariant
vector fields on $\displaystyle G$ associated to the generators $\displaystyle
H$, $\displaystyle X$, $\displaystyle Y$ of $\displaystyle sl(2)$, where

$\displaystyle H=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},\
X=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix},\ Y=\begin{pmatrix}0&0\\\
1&0\end{pmatrix}.$

Denote by $\displaystyle\hat{\mathcal{Z}}_{0}(U_{\epsilon})$ the vector space
of power series in the generators $\displaystyle x$, $\displaystyle y$, and
$\displaystyle z^{\pm 1}$ of $\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$
whose sums converge when evaluated on any $\displaystyle\chi\in{\rm
Spec}(\mathcal{Z}_{0}(U_{\epsilon}))\cong\mathbb{C}^{2}\times\mathbb{C}^{*}$,
$\displaystyle\chi=(x_{\chi},y_{\chi},z_{\chi})$ (thus defining holomorphic
functions). Set

(121)
$\hat{U}_{\epsilon}=U_{\epsilon}\otimes_{\mathcal{Z}_{0}(U_{\epsilon})}\hat{\mathcal{Z}}_{0}(U_{\epsilon})\
,\
\hat{\mathcal{Z}}(U_{\epsilon})=\mathcal{Z}(U_{\epsilon})\otimes_{\mathcal{Z}_{0}(U_{\epsilon})}\hat{\mathcal{Z}}_{0}(U_{\epsilon}).$

######  Theorem 7.9.

([30, 31]) (a) We have $\displaystyle D_{z}=z\underline{H}/2$, $\displaystyle
D_{e}=-z\underline{Y}$, $\displaystyle D_{f}=z\underline{X}$.

(b) For every $\displaystyle t\in\mathbb{C}$ the power series
$\displaystyle\exp(tD_{e})$, $\displaystyle\exp(tD_{f})$ converge to
automorphisms of $\displaystyle\hat{U}_{\epsilon}$ preserving
$\displaystyle\hat{\mathcal{Z}}_{0}(U_{\epsilon})$ and
$\displaystyle\hat{\mathcal{Z}}(U_{\epsilon})$, and fixing
$\displaystyle\mathbb{C}[\Omega]$.

Denote by $\displaystyle\mathcal{G}_{DCK}$ the subgroup of $\displaystyle{\rm
Aut}(\hat{U}_{\epsilon})$ generated by the $\displaystyle 1$-parameter groups
$\displaystyle(\exp(tD_{e}))_{t\in\mathbb{C}}$ and
$\displaystyle(\exp(tD_{f}))_{t\in\mathbb{C}}$. It acts dually by holomorphic
transformations on Spec$\displaystyle(\mathcal{Z}_{0}(U_{\epsilon}))$ and
Spec$\displaystyle(\mathcal{Z}(U_{\epsilon}))$ by defining $\displaystyle
g.\chi$ on $\displaystyle u\in\hat{\mathcal{Z}}_{0}(U_{\epsilon})$, for every
$\displaystyle g\in{\mathcal{G}}$ and $\displaystyle\chi\in{\rm
Spec}(\mathcal{Z}(U_{\epsilon}))$, by $\displaystyle
u(g.\chi)=(g^{-1}.u)(\chi)$. Then:

(c) The diffeomorphism
$\displaystyle\bar{\sigma}\circ\bar{\psi}^{-1}\colon{\rm
Spec}(\mathcal{Z}_{0}(U_{\epsilon}))\rightarrow G^{0}$ maps the action of
$\displaystyle{\mathcal{G}}_{DCK}$ on the tangent spaces of
$\displaystyle(0,0,\pm 1)\in{\rm Spec}(\mathcal{Z}_{0}(U_{\epsilon}))$ to the
coadjoint action of $\displaystyle G$ on $\displaystyle sl(2,\mathbb{C})^{*}$,
the tangent spaces of $\displaystyle{\pm I}\in G^{0}$.

(d) For any conjugacy class $\displaystyle\Gamma$ in $\displaystyle
SL(2,\mathbb{C})$, $\displaystyle(\bar{\psi}\circ\bar{\sigma}^{-1})(\Gamma\cap
G^{0})$ is a (non empty) $\displaystyle{\mathcal{G}}_{DCK}$-orbit in
Spec$\displaystyle(\mathcal{Z}_{0}(U_{\epsilon}))$, and
$\displaystyle\tau^{-1}$ of this orbit has $\displaystyle l$ connected
components, all of whose are $\displaystyle{\mathcal{G}}_{DCK}$-orbits in
Spec$\displaystyle(\mathcal{Z}(U_{\epsilon}))$.

(e) An element $\displaystyle a\in{\mathcal{Z}}(U_{\epsilon})$ is
$\displaystyle\mathcal{G}_{DCK}$-invariant if and only if $\displaystyle
a\in\mathbb{C}[\Omega]$. Dually, the sets of fixed points of the action of
$\displaystyle\mathcal{G}_{DCK}$ on
Spec$\displaystyle(\mathcal{Z}_{0}(U_{\epsilon}))$ and
Spec$\displaystyle(\mathcal{Z}(U_{\epsilon}))$ are respectively
$\displaystyle(0,0,\pm 1):=(\bar{\psi}\circ\bar{\sigma}^{-1})(\\{{\rm\pm
I}\\})$ and

$\displaystyle\mathcal{D}:=\tau^{-1}((0,0,\pm 1))=\\{(0,0,\pm 1,\pm c_{j}),\
j=1,\ldots,(l-1)/2\\}.$

(f) The $\displaystyle{\mathcal{G}}_{DCK}$-orbits on
Spec$\displaystyle(\mathcal{Z}_{0}(U_{\epsilon}))$ and
Spec$\displaystyle(\mathcal{Z}(U_{\epsilon}))$ are the symplectic leaves of
$\displaystyle\\{\ ,\ \\}_{\tt QCA}$.

######  Remark 7.10.

In the statements of Theorem 7.8 and Theorem 7.9 we use our conventions, which
differ from those in [30, 31] in the following ways:

(i) Theorem 7.8 (1) for the simply connected quantum group
$\displaystyle\tilde{U}_{\epsilon}(sl(2))$ instead of $\displaystyle
U_{\epsilon}(sl(2))$, as in [30, 31], states an isomorphism of
$\displaystyle(G^{*},\\{\ ,\ \\}_{G^{*}})$ with $\displaystyle({\rm
Spec}((\mathcal{Z}_{0}(\tilde{U}_{\epsilon})),\\{\ ,\ \\}_{\tt QCA})$; as we
take opposite comultiplications on $\displaystyle\tilde{U}_{\epsilon}(sl(2))$,
we get opposite multiplications of $\displaystyle G^{*}$ (whence
$\displaystyle\bar{G}^{*}$ for $\displaystyle U_{\epsilon}(sl(2))$).

(ii) Our derivations $\displaystyle D_{a}$ differ by a sign, which we
introduce in order to get the equality of $\displaystyle\bar{\sigma}_{*}\\{\
,\ \\}_{\bar{G}^{*}}$ with $\displaystyle\\{\ ,\ \\}_{FR}$ in Theorem 7.8 (2);

(iii) In [29, 30, 31], the matrix $\displaystyle{\mathcal{M}}$ in (120) is
different because they use the opposite coproduct on
$\displaystyle\tilde{U}_{\epsilon}(sl(2))$; their braid group automorphism
$\displaystyle T$ is in our notation $\displaystyle T_{-1}$, which satisfies
$\displaystyle T_{-1}(e)=f.$

It is easily checked that the identities (116) imply

(122) $D_{z}\mathcal{M}=\left[\frac{z}{2}\underline{H},\mathcal{M}\right]\ ,\
D_{x}\mathcal{M}=\left[-z^{-1}\underline{X},\mathcal{M}\right]\ ,\
D_{y}\mathcal{M}=\left[z^{-1}\underline{Y},\mathcal{M}\right].$

where $\displaystyle\mathcal{M}$ is defined by (120). Hence we get

(123) $D_{z}=\frac{z}{2}\underline{H}\ ,\ D_{x}=-z^{-1}\underline{X}\ ,\
D_{y}=z^{-1}\underline{Y}.$

This is the content of Theorem 7.9 (a) expressed in terms of the derivations
$\displaystyle D_{z}$, $\displaystyle D_{x}$, $\displaystyle D_{y}$. In
Theorem 7.9 (b), $\displaystyle U_{\epsilon}$ is enlarged to
$\displaystyle\hat{U}_{\epsilon}$ in order to define the automorphisms
$\displaystyle\exp(tD_{e})$, $\displaystyle\exp(tD_{f})$ because the
derivations $\displaystyle D_{e}$, $\displaystyle D_{f}$ are not nilpotent.

For reasons that will be explained in Remark 7.16, we will need a result
analogous to Theorem 7.9 but based on different derivations. This leads to
substantial differences in the details, so we give all proofs.

Consider the following derivations on $\displaystyle U_{\epsilon}$:

(124) $\mathcal{E}=zD_{x},\mathcal{F}=-zD_{y},\mathcal{H}=-2z^{-1}D_{z}.$

######  Lemma 7.11.

The derivations $\displaystyle\mathcal{E}$, $\displaystyle\mathcal{F}$ and
$\displaystyle\mathcal{H}$ satisfy the following commutation relations,
generating therefore the Lie algebra $\displaystyle sl(2)$:

(125)
$[\mathcal{H},\mathcal{E}]=2\mathcal{E},[\mathcal{H},\mathcal{F}]=-2\mathcal{F},[\mathcal{E},\mathcal{F}]=\mathcal{H}.$

Proof. Straightforward computation using (116). $\displaystyle\Box$

For any $\displaystyle{\mathbb{C}}$-algebra $\displaystyle A$ and endomorphism
$\displaystyle{\mathcal{V}}\in{End}(A)$ we can define

$\displaystyle\begin{array}[]{lcll}\exp(t{\mathcal{V}}):&A&\longrightarrow&A[[t]]\\\
&a&\longmapsto&\sum_{n\geq 0}\frac{t^{n}}{n!}{\mathcal{V}}^{n}(a).\end{array}$

If $\displaystyle{\mathcal{V}}$ is a derivation,
$\displaystyle\exp(t{\mathcal{V}})$ is a morphism of algebras. It admits a
unique extension (by continuity for the $\displaystyle t$-adic topology)
$\displaystyle\exp(t{\mathcal{V}}):A[[t]]\rightarrow A[[t]]$ which is an
automorphism with inverse $\displaystyle\exp(-t{\mathcal{V}}).$

We now give expressions of $\displaystyle\exp(t{\mathcal{F}})$,
$\displaystyle\exp(t{\mathcal{E}})\colon U_{\epsilon}\rightarrow
U_{\epsilon}[[t]]$. For every $\displaystyle\alpha\in{\mathbb{C}}$ consider
the following elements of $\displaystyle{\mathbb{C}}[[t]]$:

$\displaystyle\displaystyle(1+t)^{\alpha}$
$\displaystyle\displaystyle=\sum_{n\geq
0}\frac{1}{n!}t^{n}\alpha(\alpha-1)\cdots(\alpha-n+1),$
$\displaystyle\displaystyle\psi_{\alpha}(t)$
$\displaystyle\displaystyle=\sum_{n\geq
1}\frac{(-1)^{n}}{n!}t^{n-1}\alpha(\alpha-1)\cdots(\alpha-n+1)=\frac{(1-t)^{\alpha}-1}{t}.$

For any $\displaystyle a\in\mathcal{Z}_{0}(U_{\epsilon})$, we can similarly
define elements $\displaystyle(1+ta)^{\alpha}$,
$\displaystyle\psi_{\alpha}(ta)\in\mathcal{Z}_{0}(U_{\epsilon})[[t]]$.

######  Lemma 7.12.

The action of $\displaystyle\exp(t{\mathcal{F}})$ is given by:

$\displaystyle\displaystyle\exp(t{\mathcal{F}})(K)$
$\displaystyle\displaystyle=(1-tyz)^{-1/l}K,\exp(t{\mathcal{F}})(K^{-1})=(1-tyz)^{1/l}K^{-1}$
$\displaystyle\displaystyle\exp(t{\mathcal{F}})(F)$
$\displaystyle\displaystyle=F$
$\displaystyle\displaystyle\exp(t{\mathcal{F}})(E)$
$\displaystyle\displaystyle=E-(\epsilon-\epsilon^{-1})^{l-2}(K\epsilon^{-1}tz\psi_{-1/l}(tyz)+K^{-1}\epsilon
tz\psi_{1/l}(tyz))F^{l-1}.$

Proof. From (115) we have

(126)
${\mathcal{F}}(K)=\frac{zy}{l}K,{\mathcal{F}}(F)=0,{\mathcal{F}}(E)=-(\epsilon-\epsilon^{-1})^{l-1}\frac{z}{l}[K,-1]F^{l-1}.$

Iterating this, a straightforward computation proves the lemma.
$\displaystyle\Box$

######  Lemma 7.13.

The action of $\displaystyle\exp(t{\mathcal{E}})$ is given by:

$\displaystyle\displaystyle\exp(t{\mathcal{E}})(K)$
$\displaystyle\displaystyle=(1-txz)^{-1/l}K,\exp(t{\mathcal{E}})(K^{-1})=(1-txz)^{1/l}K^{-1}$
$\displaystyle\displaystyle\exp(t{\mathcal{E}})(K^{-1}E)$
$\displaystyle\displaystyle=K^{-1}E$
$\displaystyle\displaystyle\exp(t{\mathcal{E}})(FK)$
$\displaystyle\displaystyle=FK+(\epsilon-\epsilon^{-1})^{l-2}(K\epsilon
tz\psi_{-1/l}(txz)+K^{-1}\epsilon^{-1}tz\psi_{1/l}(txz))(K^{-1}E)^{l-1}.$

Proof. This is a little trickier than the previous proof, because it is
$\displaystyle D_{y}$ which appears in (115) and $\displaystyle{\mathcal{E}}$
involves $\displaystyle D_{x}.$ We therefore use the automorphism
$\displaystyle T_{-1}$ which is such that $\displaystyle T_{-1}(y)=-x$. As a
result, by applying it to (126), we obtain

(127) $D_{x}(K^{-1})=-\frac{x}{l}K^{-1},\ D_{x}(K^{-1}E)=0,\
D_{x}(FK)=(\epsilon-\epsilon^{-1})^{l-1}\frac{1}{l}[K,1](K^{-1}E)^{l-1}$

which imply

(128) ${\mathcal{E}}(K)=\frac{zx}{l}K,{\mathcal{E}}(K^{-1}E)=0,\
{\mathcal{E}}(FK)=(\epsilon-\epsilon^{-1})^{l-1}\frac{z}{l}[K,1](K^{-1}E)^{l-1}.$

Iterating this, a straightforward computation proves the lemma.
$\displaystyle\Box$

Because $\displaystyle{\mathcal{E}}$ and $\displaystyle{\mathcal{F}}$ leave
invariant $\displaystyle\mathcal{Z}(U_{\epsilon})$ and
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ they define maps
$\displaystyle\exp(t{\mathcal{E}}),\exp(t{\mathcal{F}}):{\mathcal{Z}}(U_{\epsilon})\rightarrow{\mathcal{Z}}(U_{\epsilon})[[t]]$
and
$\displaystyle{\mathcal{Z}}_{0}(U_{\epsilon})\rightarrow{\mathcal{Z}}_{0}(U_{\epsilon})[[t]]$.

######  Proposition 7.14.

We have

$\displaystyle\displaystyle\exp(t{\mathcal{E}})(x)=x\ ,\
\exp(t{\mathcal{F}})(y)=y\ ,\
\exp(t{\mathcal{E}})(\Omega)=\exp(t{\mathcal{F}})(\Omega)=\Omega,$
$\displaystyle\displaystyle\exp(t{\mathcal{E}})(z)=(1-tzx)^{-1}z\ ,\
\exp(t{\mathcal{E}})(y)=y+t(-xyz+z-z^{-1})+t^{2}x,$
$\displaystyle\displaystyle\exp(t{\mathcal{F}})(z)=(1-tzy)^{-1}z\ ,\
\exp(t{\mathcal{F}})(x)=x+t(-xyz+z-z^{-1})+t^{2}y.$

Proof. Rather than using the explicit action of
$\displaystyle\exp(t{\mathcal{E}})$and $\displaystyle\exp(t{\mathcal{F}})$ on
$\displaystyle U_{\epsilon}$, we remark that from (122) we have

(129) $\exp(t{\mathcal{E}})({\mathcal{M}})=\exp(-tX){\mathcal{M}}\exp(tX)\
,\exp(t{\mathcal{F}})({\mathcal{M}})=\exp(-tY){\mathcal{M}}\exp(tY).$

The announced expressions follow by writing the matrix elements of these
equations. $\displaystyle\Box$

Let $\displaystyle{\mathcal{G}}$ be the free product of
$\displaystyle({\mathbb{C}},+)$ with itself; it can be equivalently seen as
the group generated by elements $\displaystyle\phi_{s},\psi_{s}$, where
$\displaystyle s\in{\mathbb{C}}$, with the relations
$\displaystyle\phi_{s}\phi_{s^{\prime}}=\phi_{s+s^{\prime}},\psi_{s}\psi_{s^{\prime}}=\psi_{s+s^{\prime}}$
for every $\displaystyle s,s^{\prime}\in{\mathbb{C}}$.

We can define a partial action of $\displaystyle{\mathcal{G}}$ on
$\displaystyle{\rm Spec}(\mathcal{Z}(U_{\epsilon}))$ and on $\displaystyle{\rm
Spec}(\mathcal{Z}_{0}(U_{\epsilon}))$, in the sense of [37]. Let
$\displaystyle s\in{\mathbb{C}}$, and denote by $\displaystyle D(\phi_{s})$
the set of $\displaystyle\chi\in{\rm Spec}(\mathcal{Z}_{0}(U_{\epsilon}))$
such that for all $\displaystyle u\in\mathcal{Z}_{0}(U_{\epsilon})$ the series
$\displaystyle(\exp(s{\mathcal{E}})(u))(\chi)$ is normally convergent in a
small neighborhood of $\displaystyle s$. Equivalently $\displaystyle
D(\phi_{s})=\\{\chi\in{\rm
Spec}(\mathcal{Z}_{0}(U_{\epsilon})),|sx_{\chi}z_{\chi}|<1\\}$. We define an
action of $\displaystyle\phi_{s}$ on $\displaystyle D(\phi_{s})$ by

(130) $u(\phi_{s}.\chi)=(\exp(-s{\mathcal{E}})(u))(\chi).$

Similarly, put $\displaystyle D(\psi_{s})=\\{\chi\in{\rm
Spec}(\mathcal{Z}_{0}(U_{\epsilon})),|sy_{\chi}z_{\chi}|<1\\}$ and define an
action of $\displaystyle\psi_{s}$ on $\displaystyle D(\psi_{s})$ by

(131) $u(\psi_{s}.\chi)=(\exp(-s{\mathcal{F}})(u))(\chi).$

The domains $\displaystyle D(\phi_{s})$ (resp. $\displaystyle D(\psi_{s})$)
cover $\displaystyle{\rm Spec}(\mathcal{Z}_{0}(U_{\epsilon}))$ as
$\displaystyle s$ varies in $\displaystyle{\mathbb{C}}$. By results of Exel
([37] and [38], Examples (1) and (4)), the set formed by the local actions
(130) (resp. (131)) on the domains $\displaystyle D(\phi_{s})$ (resp.
$\displaystyle D(\psi_{s})$) defines a partial action of the one-parameter
group $\displaystyle(\phi_{s})_{s\in\mathbb{C}}$ (resp.
$\displaystyle(\psi_{s})_{s\in\mathbb{C}}$) on $\displaystyle{\rm
Spec}(\mathcal{Z}(U_{\epsilon}))$, and combining the two we get a partial
action of their free product $\displaystyle{\mathcal{G}}$ on
$\displaystyle{\rm Spec}(\mathcal{Z}(U_{\epsilon}))$. Similarly we get a
partial action of $\displaystyle{\mathcal{G}}$ on $\displaystyle{\rm
Spec}(\mathcal{Z}(U_{\epsilon}))$ by replacing
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ by
$\displaystyle\mathcal{Z}(U_{\epsilon})$.

We call them the partial quantum coadjoint actions of
$\displaystyle\mathcal{G}$. Orbits (called partial orbits in [37]) are defined
as for group actions: the $\displaystyle{\mathcal{G}}$-orbit of a point
$\displaystyle\chi\in{\rm Spec}(\mathcal{Z}(U_{\epsilon}))$ is the set of
points $\displaystyle g.\chi$, for every possible $\displaystyle
g\in\mathcal{G}$ such that $\displaystyle g.\chi$ is defined.

The next result states the analogs of Theorem 7.9 (c)–(f) obtained by
replacing $\displaystyle\mathcal{G}_{DCK}$ by $\displaystyle\mathcal{G}$. In
particular, it describes the $\displaystyle\mathcal{G}$-orbits.

######  Theorem 7.15.

(a) The diffeomorphism $\displaystyle\bar{\sigma}\circ\bar{\psi}^{-1}$ maps
the action of $\displaystyle{\mathcal{G}}$ on the tangent spaces of
$\displaystyle(0,0,\pm 1)\in{\rm Spec}(\mathcal{Z}_{0}(U_{\epsilon}))$ to the
coadjoint action of $\displaystyle G$ on $\displaystyle
sl(2,\mathbb{C})^{*}=T_{\pm I}(G^{0})$.

(b) For any conjugacy class $\displaystyle\Gamma$ in $\displaystyle G$,