$\displaystyle(\bar{\psi}\circ\bar{\sigma}^{-1})(\Gamma\cap G^{0})$ is a (non
empty) $\displaystyle{\mathcal{G}}$-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 $\displaystyle{\mathcal{G}}$-orbits in
Spec$\displaystyle(\mathcal{Z}(U_{\epsilon}))$.

(c) An element $\displaystyle a\in{\mathcal{Z}}(U_{\epsilon})$ is
$\displaystyle\mathcal{G}$-invariant if and only if $\displaystyle
a\in\mathbb{C}[\Omega]$. Dually, the sets of fixed points of the partial
$\displaystyle{\mathcal{G}}$-action 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

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

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

Proof. (a) Using $\displaystyle\bar{\sigma}^{-1*}\circ\bar{\psi}^{*}$ to
identify $\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ with
$\displaystyle\mathcal{O}(G^{0})$, this is a direct consequence of (129) and
the fact that $\displaystyle{\mathcal{M}}=\pm I$ at the fixed point.

(b) That $\displaystyle\Gamma\cap G^{0}$ is non-empty and connected is
classical. Then so is
$\displaystyle(\bar{\psi}\circ\bar{\sigma}^{-1})(\Gamma\cap G^{0})$. Take a
point $\displaystyle\chi\in(\bar{\psi}\circ\bar{\sigma}^{-1})(\Gamma\cap
G^{0})$. Evaluating (129) at $\displaystyle\chi$ shows that
$\displaystyle\mathcal{G}\cdot\chi$ is contained in
$\displaystyle(\bar{\psi}\circ\bar{\sigma}^{-1})(\Gamma\cap G^{0})$. We claim
that $\displaystyle\mathcal{G}\cdot\chi$ is an open and closed subset, so by
connectedness of $\displaystyle(\bar{\psi}\circ\bar{\sigma}^{-1})(\Gamma\cap
G^{0})$ it coincides with it. Indeed, it is an open subset because any point
of $\displaystyle{\rm Spec}(\mathcal{Z}_{0}(U_{\epsilon}))$ (whence of
$\displaystyle\mathcal{G}\cdot\chi$) belongs to the domains $\displaystyle
D(\phi_{s})$ and $\displaystyle D(\psi_{s})$ for $\displaystyle s$ small
enough. Since the one-parameter groups
$\displaystyle(\phi_{s})_{s\in\mathbb{C}}$ and
$\displaystyle(\psi_{s})_{s\in\mathbb{C}}$ are obtained by integrating the
derivations $\displaystyle{\mathcal{E}}$ and $\displaystyle{\mathcal{F}}$, it
follows from Lemma 7.11 that $\displaystyle\mathcal{G}\cdot\chi$ contains a
neighborhood in $\displaystyle(\bar{\psi}\circ\bar{\sigma}^{-1})(\Gamma\cap
G^{0})$ of each of its points. By the same reason any limit point of
$\displaystyle\mathcal{G}\cdot\chi$ in $\displaystyle{\rm
Spec}(\mathcal{Z}_{0}(U_{\epsilon}))$ has a neighborhood where the partial
$\displaystyle{\mathcal{G}}$-action is defined and which intersects
$\displaystyle\mathcal{G}\cdot\chi$. Therefore it belongs to
$\displaystyle\mathcal{G}\cdot\chi$, which shows that
$\displaystyle\mathcal{G}\cdot\chi$ is also closed in
$\displaystyle(\bar{\psi}\circ\bar{\sigma}^{-1})(\Gamma\cap G^{0})$. Because
the covering map $\displaystyle\tau$ is unramified of degree $\displaystyle
l$, the result for $\displaystyle{\rm Spec}(\mathcal{Z}(U_{\epsilon}))$
follows at once.

(c) Any $\displaystyle\chi\in{\rm Spec}(\mathcal{Z}_{0}(U_{\epsilon}))$
belongs to the domains $\displaystyle D(\phi_{s})$, $\displaystyle
D(\psi_{s})$ for $\displaystyle s$ small enough. Solving the equations
$\displaystyle\phi_{s}(\chi)=\chi$ and $\displaystyle\psi_{s}(\chi)=\chi$ by
using the formulas in Proposition 7.14 imposes $\displaystyle\chi=(0,0,\pm
1)$. This gives the fixed point set of $\displaystyle{\rm
Spec}(\mathcal{Z}_{0}(U_{\epsilon}))$; the result for $\displaystyle{\rm
Spec}(\mathcal{Z}(U_{\epsilon}))$ follows immediately. As for the first claim,
note that for any conjugacy class $\displaystyle\Gamma$ of maximal dimension
the set $\displaystyle\Gamma\cap G^{0}$ contains a diagonal matrix, and that
the union of such sets forms a Zariski open and dense subset of $\displaystyle
G$. Hence (b) above implies that a central invariant element is completely
determined by its value at points $\displaystyle(0,0,z_{\chi},\Omega_{\chi})$.
Therefore it is non zero if and only if it belongs to
$\displaystyle\mathbb{C}[\Omega]$.

(d) The groups $\displaystyle\mathcal{G}$ and $\displaystyle\mathcal{G}_{DCK}$
are obtained by integrating the derivations $\displaystyle{\mathcal{E}}$,
$\displaystyle{\mathcal{F}}$, and $\displaystyle D_{e}$, $\displaystyle D_{f}$
respectively, and the Lie algebras generated by these two pairs of derivations
have the same span at every point of $\displaystyle{\rm
Spec}(\mathcal{Z}_{0}(U_{\epsilon}))$ or $\displaystyle{\rm
Spec}(\mathcal{Z}(U_{\epsilon}))$. Then the conclusion follows from Theorem
7.9 (f). $\displaystyle\Box$

######  Remark 7.16.

A difficulty with the group $\displaystyle\mathcal{G}$ is that its elements
act only on subsets of $\displaystyle Spec(\mathcal{Z}(U_{\epsilon}))$: in the
formulas (129), the parameter $\displaystyle t\in\mathbb{C}$ must be such that
the lower right entries of the computed matrices are non zero. However it has
various merits as compared to the group $\displaystyle\mathcal{G}_{DCK}$ of
Theorem 7.9:

(i) $\displaystyle\mathcal{G}$ is finite dimensional, associated to the Lie
algebra $\displaystyle sl(2)$ (by Lemma 7.11), whereas
$\displaystyle\mathcal{G}_{DCK}$ is infinite-dimensional.

(ii) The partial action of $\displaystyle\mathcal{G}$ on $\displaystyle{\rm
Spec}(\mathcal{Z}(U_{\epsilon}))$ is by birational transformations (by
Proposition 7.14), whereas $\displaystyle\mathcal{G}_{DCK}$ is a subgroup of
$\displaystyle{\rm Aut}(\hat{U}_{\epsilon})$, acting on $\displaystyle{\rm
Spec}(\mathcal{Z}(U_{\epsilon}))$ by holomorphic (entire) transformations.

(iii) $\displaystyle\mathcal{G}$ can be generalized straightforwardly to
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$, whereas
$\displaystyle\mathcal{G}_{DCK}$ does not. This is our main motivation for
developing this construction. We do it in the next section.

######  Remark 7.17.

We think useful to have in mind the following description of
$\displaystyle{\rm Spec}(\mathcal{Z}(U_{\epsilon}))$ (see [29]). Denote by
$\displaystyle G/\\!/G$ the affine variety with coordinate ring
$\displaystyle\mathcal{O}(G)^{G}$, the ring of regular functions on
$\displaystyle G$ invariant under the coadjoint action of $\displaystyle G$.
We have an isomorphism

(132) $G/\\!/G\cong\mathbb{C}^{*}/(t\sim t^{-1}).$

In fact, denote by $\displaystyle T$ the torus of $\displaystyle G$ formed by
the diagonal matrices, and by $\displaystyle\mathcal{O}(T)$ its coordinate
ring. Then $\displaystyle T\cong\mathbb{C}^{*}$ and
$\displaystyle\mathcal{O}(T)\cong\mathbb{C}[t,t^{-1}]$, where $\displaystyle
t$ is the coordinate function of the upper left entry of elements of
$\displaystyle T$. The Weyl group $\displaystyle W$ of $\displaystyle G$ acts
on $\displaystyle T$ by inversion, and

(133) $\mathcal{O}(G)^{G}\cong\mathcal{O}(T)^{W}=\mathbb{C}[t+t^{-1}].$

Consider the maps

(134) $p\colon G\longrightarrow G/\\!/G\ ,\ p_{k}:G/\\!/G\longrightarrow
G/\\!/G,$

where $\displaystyle p\colon G\rightarrow G/\\!/G$ is the quotient map, and
$\displaystyle p_{k}$ is induced by the $\displaystyle k$-th power map
$\displaystyle g\mapsto g^{k}$, $\displaystyle g\in G$, $\displaystyle
k\in\mathbb{Z}$. Note here that on coordinate functions we have $\displaystyle
p_{k}(t+t^{-1})=t^{k}+t^{-k}$, so $\displaystyle p_{k}$ is just realized by
the $\displaystyle k$-th Chebyshev polynomial $\displaystyle T_{k}$. Consider
the fibered product of $\displaystyle p$ and $\displaystyle p_{l}$, that is,
the affine variety

(135) $G\times_{G/\\!/G}G/\\!/G=\\{(g,[t])\in G\times G/\\!/G\ |\
p(g)=p_{l}([t])\\}.$

Set

(136) $\tilde{G}=G\times_{G/\\!/G}G/\\!/G,\
\tilde{G}^{0}=G^{0}\times_{G/\\!/G}G/\\!/G.$

Then Spec$\displaystyle(\mathcal{Z}(U_{\epsilon}))$ is isomorphic to
$\displaystyle\tilde{G}^{0}$. In fact, by Theorem 7.8 and the defining
relation (82) we know that $\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ is
isomorphic to $\displaystyle\mathcal{O}(G^{0})$, and that
$\displaystyle\mathcal{Z}(U_{\epsilon})=\mathcal{Z}_{0}(U_{\epsilon})\otimes_{\mathcal{Z}_{0}(U_{\epsilon})\cap\mathbb{C}[\Omega]}\mathbb{C}[\Omega]$.
By the quantum Harish-Chandra homomorphism, see [29] for details,
$\displaystyle\mathbb{C}[\Omega]\cong\mathbb{C}[K+K^{-1}]$ and
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})\cap\mathbb{C}[\Omega]\cong\mathbb{C}[K^{l}+K^{-l}]$.
Let us identify $\displaystyle\mathbb{C}[K,K^{-1}]$ with
$\displaystyle\mathcal{O}(T)$. Then
$\displaystyle\mathbb{C}[\Omega]\cong\mathcal{O}(T)^{W}$ and
$\displaystyle\mathbb{C}[K^{l}+K^{-l}]\cong\mathcal{O}(T/\mu_{l})^{W}$, where
$\displaystyle\mu_{l}$ is the subgroup of $\displaystyle T$ corresponding to
the $\displaystyle l$th-roots of unity under the isomorphism $\displaystyle
T\cong\mathbb{C}^{*}$. Hence

$\displaystyle\mathcal{Z}(U_{\epsilon})\cong\mathcal{O}(G^{0})\otimes_{\mathcal{O}(T/\mu_{l})^{W}}\mathcal{O}(T)^{W}.$

The isomorphism of Spec$\displaystyle(\mathcal{Z}(U_{\epsilon}))$ with
$\displaystyle G^{0}\times_{G/\\!/G}G/\\!/G$ follows by duality.

####  7.2.2. Extension to $\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$

The results of the previous section extend naturally to
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$. First we consider the
generalization of Theorem 7.8.

The bracket $\displaystyle\\{\ ,\ \\}_{FR}$ on $\displaystyle G$ has been
extended to $\displaystyle G^{n}$ by Fock-Rosly [49]. From its very
definition, it is readily checked to be defined by

(137) $\displaystyle\displaystyle\\{\stackrel{{\scriptstyle
1}}{{L}}\\!{}^{(i)},\stackrel{{\scriptstyle 2}}{{L}}\\!{}^{(i)}\\}_{FR}$
$\displaystyle\displaystyle=\mathfrak{r}\stackrel{{\scriptstyle
1}}{{L}}\\!{}^{(i)}\\!\\!\stackrel{{\scriptstyle
2}}{{L}}\\!{}^{(i)}-\stackrel{{\scriptstyle
1}}{{L}}\\!{}^{(i)}\\!\\!\stackrel{{\scriptstyle
2}}{{L}}\\!{}^{(i)}\mathfrak{r}^{\prime}+\stackrel{{\scriptstyle
2}}{{L}}\\!{}^{(i)}\mathfrak{r}^{\prime}\\!\stackrel{{\scriptstyle
1}}{{L}}\\!{}^{(i)}-\stackrel{{\scriptstyle
1}}{{L}}\\!{}^{(i)}\mathfrak{r}\\!\stackrel{{\scriptstyle 2}}{{L}}\\!{}^{(i)}$
(138) $\displaystyle\displaystyle\\{\stackrel{{\scriptstyle
1}}{{L}}\\!{}^{(i)},\stackrel{{\scriptstyle 2}}{{L}}\\!{}^{(j)}\\}_{FR}$
$\displaystyle\displaystyle=\mathfrak{r}\stackrel{{\scriptstyle
1}}{{L}}\\!{}^{(i)}\\!\\!\stackrel{{\scriptstyle
2}}{{L}}\\!{}^{(j)}+\stackrel{{\scriptstyle
1}}{{L}}\\!{}^{(i)}\\!\\!\stackrel{{\scriptstyle
2}}{{L}}\\!{}^{(j)}\mathfrak{r}-\stackrel{{\scriptstyle
2}}{{L}}\\!{}^{(j)}\mathfrak{r}\\!\stackrel{{\scriptstyle
1}}{{L}}\\!{}^{(i)}-\stackrel{{\scriptstyle
1}}{{L}}\\!{}^{(i)}\mathfrak{r}\\!\stackrel{{\scriptstyle 2}}{{L}}\\!{}^{(j)}$

where $\displaystyle L^{(i)}=\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(i)}$ (see (95)), $\displaystyle i,j\in\\{1,\ldots,n\\}$, and
$\displaystyle i<j$.

The formula (111) extends naturally to define derivations $\displaystyle
D_{a}\colon{}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}\rightarrow{}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}$
preserving $\displaystyle\mathcal{Z}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})$,
for every $\displaystyle
a\in\mathcal{Z}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})$. Hence we can define
a Poisson bracket $\displaystyle\\{\ ,\ \\}_{\tt QCA}$ on
$\displaystyle\mathcal{Z}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})$ (keeping
voluntarily the same notation as on $\displaystyle\mathcal{Z}(U_{\epsilon})$)
by

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

for every $\displaystyle
a,b\in\mathcal{Z}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}).$ As
$\displaystyle\Phi_{n}$ is an isomorphism of algebras, both derivations and
Poisson bracket can be defined and computed by means of their pushforwards on
$\displaystyle U_{\epsilon}^{\otimes n}$. Indeed, for any $\displaystyle
a\in\mathcal{Z}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})$,
$\displaystyle\Phi_{n}(a)\in\mathcal{Z}(U_{\epsilon}^{\otimes n})$, so the
derivation $\displaystyle D_{\Phi_{n}(a)}\colon U_{\epsilon}^{\otimes
n}\rightarrow U_{\epsilon}^{\otimes n}$ is defined and we can put
$\displaystyle D_{a}=\Phi_{n}^{-1}D_{\Phi_{n}(a)}\Phi_{n}$. For every
$\displaystyle a\in\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})$, by the
inclusion
$\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}\subset{}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}$
it yields a derivation $\displaystyle
D_{a}\colon{\mathcal{L}}_{0,n}^{\epsilon}\rightarrow{\mathcal{L}}_{0,n}^{\epsilon}$
preserving $\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})$.

Denote by $\displaystyle\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})$
(resp. $\displaystyle\mathcal{Z}_{0}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})$)
the subalgebras of $\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})$
(resp. $\displaystyle\mathcal{Z}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})$)
generated by $\displaystyle T_{l}(\omega^{(i)})$, $\displaystyle b^{(i)l}$,
$\displaystyle c^{(i)l}$, $\displaystyle d^{(i)l}$, (resp. $\displaystyle
T_{l}(\omega^{(i)})$, $\displaystyle b^{(i)l}$, $\displaystyle c^{(i)l}$,
$\displaystyle d^{(i)l}$ and $\displaystyle\delta^{(i)\pm l})$ for
$\displaystyle i=1,\ldots,n$. Recall the Frobenius morphism $\displaystyle
Fr:({\mathcal{L}}_{0,1}^{1})^{\otimes
n}\rightarrow\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})$ (see (95)), and
the identification $\displaystyle{\mathcal{L}}_{0,1}^{1}=\mathcal{O}(G)$.

######  Theorem 7.18.

The map

$\displaystyle\Phi:=(\bar{\sigma}^{-1*}\circ\bar{\psi}^{*}{})^{\otimes
n}\circ\Phi_{1}^{\otimes
n}\colon(\mathcal{Z}_{0}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}),\\{\
,\\}_{\tt QCA})\rightarrow(\mathcal{O}(G^{0})^{\otimes n},\\{\ ,\ \\}_{FR})$

is an isomorphism of Poisson algebras, and $\displaystyle\Phi^{-1}$ restricted
to $\displaystyle\mathcal{O}(G)^{\otimes n}\subset\mathcal{O}(G^{0})^{\otimes
n}$ coincides with the Frobenius homomorphism. Hence $\displaystyle
Fr\colon(\mathcal{O}(G)^{\otimes n},\\{\ ,\
\\}_{FR})\rightarrow(\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon}),\\{\
,\\}_{\tt QCA})$ is an isomorphism of Poisson algebras.

Proof. The map
$\displaystyle\bar{\sigma}^{-1*}\circ\bar{\psi}^{*}{}\colon(\mathcal{Z}_{0}(U_{\epsilon}),\\{\
,\ \\}_{\tt QCA})\rightarrow(\mathcal{O}(G^{0}),\\{\ ,\ \\}_{FR})$ is an
isomorphism of Poisson algebras by the results recalled in the previous
section. Since
$\displaystyle\Phi_{1}\colon{}_{loc}{\mathcal{L}}_{0,1}^{A}\rightarrow
U_{A}^{\prime}$ is an algebra isomorphism by Lemma 5.7, when specializing at
$\displaystyle q=\epsilon$ it yields a Poisson isomorphism between
$\displaystyle(\mathcal{Z}_{0}({}_{loc}{\mathcal{L}}_{0,1}^{\epsilon}),\\{\
,\\}_{\tt QCA})$ and $\displaystyle(\mathcal{Z}_{0}(U_{\epsilon}),\\{\ ,\
\\}_{\tt QCA})$. This proves the case $\displaystyle n=1$.

When $\displaystyle n\geq 2$, the map is well-defined because
$\displaystyle\mathcal{Z}_{0}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})={}_{loc}\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})$
(the localization by the powers $\displaystyle\delta^{(i)lk}$, $\displaystyle
k\in\mathbb{Z}$),
$\displaystyle\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})=\mathcal{Z}_{0}({\mathcal{L}}_{0,1}^{\epsilon})^{\otimes
n}$ by Proposition 7.3, and $\displaystyle\Phi_{1}^{\otimes n}$ extends from
$\displaystyle\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})$ to
$\displaystyle{}_{loc}\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})$ for
obvious reasons (the algebra being commutative). On another hand, since
$\displaystyle\Phi_{n}\colon{}_{loc}{\mathcal{L}}_{0,n}^{A}\rightarrow
U_{A}^{\otimes n}$ is an equivariant algebra isomorphism and
$\displaystyle\Phi_{n}(\omega^{(i)})=\Omega^{(i)}$, when specializing at
$\displaystyle q=\epsilon$ it yields a Poisson isomorphism between
$\displaystyle(\mathcal{Z}_{0}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}),\\{\
,\\}_{\tt QCA})$ and $\displaystyle(\mathcal{Z}_{0}(U_{\epsilon}),\\{\ ,\
\\}_{\tt QCA})^{\otimes n}$, ie. the algebra
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})^{\otimes n}$ with the product
Poisson structure. Hence it is enough to prove that

$\displaystyle\Phi\circ\Phi_{n}^{-1}\colon(\mathcal{Z}_{0}(U_{\epsilon}),\\{\
,\ \\}_{\tt QCA})^{\otimes n}\rightarrow(\mathcal{O}(G^{0})^{\otimes n},\\{\
,\ \\}_{FR})$

is an isomorphism of Poisson algebras. This can be checked on generators,
which is most easily done by using the inverse map. Put
$\displaystyle\Pi=id_{V_{2}}\otimes(\Phi_{n}\circ(\Phi_{1}^{\otimes
n})^{-1})$. By Corollary 7.5 we have

(139)
$\Pi\left(\mathcal{M}^{(i)}\right)=\mathcal{R}^{(i)}\mathcal{M}^{(i)}\mathcal{R}^{(i)-1}.$

Moreover, (98) and (120) show that
$\displaystyle\bar{\sigma}^{-1*}\circ\bar{\psi}^{*}$ identifies the matrix
coefficients of
$\displaystyle\Phi_{1}(\mathcal{Z}_{0}({\mathcal{L}}_{0,1}^{\epsilon}))$ in
the fundamental representation $\displaystyle V_{2}$ of $\displaystyle
U_{\epsilon}$, and the matrix coefficients of the fundamental representation
of $\displaystyle G$ on $\displaystyle\mathbb{C}^{2}$. Therefore it is enough
to check that

(140) $\\{\Pi^{*}(\stackrel{{\scriptstyle
1}}{{L}}\\!{}^{(i)}),\Pi^{*}(\stackrel{{\scriptstyle
2}}{{L}}\\!{}^{(j)})\\}_{\tt QCA}^{\otimes n}=\\{\stackrel{{\scriptstyle
1}}{{L}}\\!{}^{(i)},\stackrel{{\scriptstyle 2}}{{L}}\\!{}^{(j)}\\}_{FR}$

for every $\displaystyle i,j\in\\{1,\ldots,n\\}$, where
$\displaystyle\Pi^{*}(L^{(i)})$ is the pull-back of $\displaystyle L^{(i)}$
via $\displaystyle\Pi$, ie. its expression as a function of the matrices
$\displaystyle\mathcal{M}^{(i)}$. Now we have

(141) $\mathcal{M}^{(i)}=\mathcal{M}^{(i)}_{+}\mathcal{M}^{(i)-1}_{-}$

where

$\displaystyle\mathcal{M}^{(i)}_{+}=\begin{pmatrix}z^{\prime}{}^{(i)}&z^{\prime}{}^{(i)}y{}^{(i)}\\\
0&z^{\prime}{}^{(i)-1}\end{pmatrix},\
\mathcal{M}^{(i)}_{-}=\begin{pmatrix}z^{\prime}{}^{(i)-1}&0\\\
z^{\prime}{}^{(i)}x{}^{(i)}&z^{\prime}{}^{(i)}\end{pmatrix}.$

Note also that (99)-(100) imply

(142) $\mathcal{R}^{(i)}=\mathcal{M}^{(n)}_{+}\ldots\mathcal{M}^{(i+1)}_{+}.$

Using the map $\displaystyle\sigma^{-1*}\circ\psi^{*}$ to identify
$\displaystyle\mathcal{M}^{(i)}_{+}$, $\displaystyle\mathcal{M}^{(i)}_{-}$
with matrices $\displaystyle L_{+}$, $\displaystyle L_{-}$ of coordinate
functions in $\displaystyle G^{*}$, we see that they satisfy the bracket
identities (110). Since $\displaystyle G^{*}$ is a Poisson-Lie group, products
of matrices $\displaystyle L_{+}$ have the same bracket as $\displaystyle
L_{+}$. In particular this applies to $\displaystyle\mathcal{R}^{(i)}$, and we
can write

$\displaystyle\Pi^{*}(L^{(i)})=\mathcal{R}^{(i)}\\!\\!\mathcal{M}^{(i)}_{+}\\!\\!\mathcal{M}^{(i)-1}_{-}\mathcal{R}^{(i)-1}$

where all matrices in the product have known brackets. Then (140) can be
straightforwardly compared with (137)-(138), using the Leibniz rule for
simplifications, and the fact that $\displaystyle r+r^{\prime}$ is ad-
invariant. We leave the verifications to the reader.

The equality of $\displaystyle\Phi^{-1}$ with $\displaystyle Fr$ on
$\displaystyle\mathcal{O}(G)^{\otimes n}$ follows immediatly from the fact
that $\displaystyle\bar{\sigma}^{-1*}\circ\bar{\psi}^{*}$ identifies matrix
coefficients, as discussed above. The image of $\displaystyle Fr$ is
$\displaystyle\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})$, by Proposition
7.3. This achieves the proof. $\displaystyle\Box$

Next we turn to our generalization of Theorem 7.15. First we define a partial
action of the group $\displaystyle\mathcal{G}$ on $\displaystyle{\rm
Spec}({\mathcal{Z}}_{0}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}))$ and
$\displaystyle{\rm
Spec}({\mathcal{Z}}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}))$ by generalizing
the method we used in the $\displaystyle n=1$ case.

Recall the derivations $\displaystyle
D_{a}\colon{}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}\rightarrow{}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}$,
defined for every $\displaystyle
a\in\mathcal{Z}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})$. Denote by
$\displaystyle\hat{x}^{(i)},\hat{y}^{(i)},\hat{z}^{(i)}\in\mathcal{Z}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})$
the inverse images by $\displaystyle\Phi_{n}$ of the elements $\displaystyle
x^{(i)},y^{(i)},z^{(i)}$, and let
$\displaystyle{\mathcal{E}}^{(i)},{\mathcal{F}}^{(i)},{\mathcal{H}}^{(i)}$ be
the derivations of $\displaystyle{}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}$
defined by

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

Note that because of the relations (101) and the definition of
$\displaystyle\delta^{(i)}$ we can obtain simple formulas for
$\displaystyle\hat{z}^{(i)}$ and $\displaystyle\hat{x}^{(i)},$ namely:

(144) $\hat{z}^{(i)\pm 1}=\delta^{(i)\mp l}\
,\hat{x}^{(i)}=-c^{(i)l}\prod_{k=i+1}^{n}\delta^{(k)-l}.$

Set

(145) ${\mathcal{E}}^{\Delta}=\sum_{i=1}^{n}{\mathcal{E}}^{(i)}\
,{\mathcal{F}}^{\Delta}=\sum_{i=1}^{n}{\mathcal{F}}^{(i)}\
,{\mathcal{H}}^{\Delta}=\sum_{i=1}^{n}{\mathcal{H}}^{(i)}.$

######  Proposition 7.19.

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

$[\mathcal{H}^{\Delta},\mathcal{E}^{\Delta}]=2\mathcal{E}^{\Delta},[\mathcal{H}^{\Delta},\mathcal{F}^{\Delta}]=-2\mathcal{F}^{\Delta},[\mathcal{E}^{\Delta},\mathcal{F}^{\Delta}]=\mathcal{H}^{\Delta}.$

(2) The power series $\displaystyle\exp(t{\mathcal{E}}^{\Delta})$,
$\displaystyle\exp(t{\mathcal{F}}^{\Delta})$ define morphisms of algebras from
$\displaystyle{}_{loc}{{\mathcal{L}}}_{0,n}^{\epsilon}$ to
$\displaystyle{}_{loc}{{\mathcal{L}}}_{0,n}^{\epsilon}[[t]]$, sending
$\displaystyle{\mathcal{Z}}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})$ to
$\displaystyle{\mathcal{Z}}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})[[t]]$ and
fixing the elements $\displaystyle\omega^{(i)}$. Moreover

$\displaystyle\begin{array}[]{c}\exp(t\Phi_{n}({\mathcal{E}}^{\Delta}))({\mathcal{M}}^{(i)})=\exp(-tX){\mathcal{M}}^{(i)}\exp(tX)\\\
\exp(t\Phi_{n}({\mathcal{F}}^{\Delta}))({\mathcal{M}}^{(i)})=\exp(-tY){\mathcal{M}}^{(i)}\exp(tY).\end{array}$

As a result $\displaystyle\exp(t{\mathcal{E}}^{\Delta})$ and
$\displaystyle\exp(t{\mathcal{F}}^{\Delta})$ are sending
$\displaystyle{\mathcal{Z}}_{0}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})$ to
$\displaystyle{\mathcal{Z}}_{0}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})[[t]]$.

Proof. (1) Straighforward from (125) and from the local Poisson-commutativity,
i.e the property that $\displaystyle\\{a,b\\}=0$ when $\displaystyle
a\in\\{x^{(i)},y^{(i)},z^{(i)}\\}$, $\displaystyle
b\in\\{x^{(j)},y^{(j)},z^{(j)}\\}$, $\displaystyle i\not=j$.

(2) The first claim is straightforward, the second is a direct application of
(129) and the local Poisson-commutativity. $\displaystyle\Box$

As in (130) let $\displaystyle s\in{\mathbb{C}}$, and denote by $\displaystyle
D(\phi_{s}^{(i)})$ the set of points $\displaystyle\chi\in{\rm
Spec}({\mathcal{Z}}_{0}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}))$ such that
for all $\displaystyle
u\in{\mathcal{Z}}_{0}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})$ the series
$\displaystyle(\exp(s{\mathcal{E}}^{(i)})(u))(\chi)$ is normally convergent in
a small neighborhood of $\displaystyle s$. Thus $\displaystyle
D(\phi_{s}^{(i)})=\\{\chi\in{\rm
Spec}({\mathcal{Z}}_{0}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})),|s\hat{x}^{(i)}_{\chi}\hat{z}^{(i)}_{\chi}|<1\\}$.
Define an action of the element $\displaystyle\phi_{s}^{(i)}$ on
$\displaystyle D(\phi_{s}^{(i)})$ by

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

Similarly, put $\displaystyle D(\psi_{s}^{(i)})=\\{\chi\in
Spec({\mathcal{Z}}_{0}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})),|s\hat{y}^{(i)}_{\chi}\hat{z}^{(i)}_{\chi}|<1\\}$,
and define an action of $\displaystyle\psi_{s}^{(i)}$ on $\displaystyle
D(\psi_{s}^{(i)})$ by

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

Denote by $\displaystyle{\mathcal{G}}^{(i)}$ the group generated by
$\displaystyle\phi_{s}^{(i)},\psi_{s}^{(i)},s\in{\mathbb{C}}$. It is
isomorphic to $\displaystyle{\mathcal{G}}$. Denote by
$\displaystyle{\mathcal{G}}^{tot}$ the direct product of the groups
$\displaystyle{\mathcal{G}}^{(i)}$, and by
$\displaystyle{\mathcal{G}}^{\Delta}$ the subgroup of
$\displaystyle{\mathcal{G}}^{tot}$ generated by the diagonal elements
$\displaystyle\phi_{s}^{\Delta}=(\phi_{s}^{(1)},...,\phi_{s}^{(n)})$ (resp.
$\displaystyle\psi_{s}^{\Delta}=(\psi_{s}^{(1)},\ldots,\psi_{s}^{(n)})$).
These elements act on $\displaystyle
D(\phi_{s}^{\Delta})=\cap_{i=1}^{n}D(\phi_{s}^{(i)}),$ (resp $\displaystyle
D(\psi_{s}^{\Delta})=\cap_{i=1}^{n}D(\psi_{s}^{(i)}))$, by acting dually with
the series $\displaystyle\exp(s{\mathcal{E}}^{\Delta})$ (resp.
$\displaystyle\exp(s{\mathcal{F}}^{\Delta})$).

As in the case $\displaystyle n=1$, the results of Exel [37, 38] imply that
(146) and (147) define partial action of the group
$\displaystyle{\mathcal{G}}^{tot}\cong{\mathcal{G}}^{n}$ and
$\displaystyle{\mathcal{G}}^{\Delta}\cong{\mathcal{G}}$ on $\displaystyle{\rm
Spec}({\mathcal{Z}}_{0}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon})).$ We call the
first one the total partial quantum coadjoint action and the second one the
diagonal partial quantum coadjoint action.

Next we need the following result of Fock-Rosly. Recall that we denote by
$\displaystyle X_{G}(\Sigma)$ the variety of $\displaystyle G$-characters of
the sphere $\displaystyle\Sigma$ with $\displaystyle n+1$ punctures, and by
$\displaystyle\\{\ ,\ \\}_{Gold}$ the Goldman Poisson bracket on
$\displaystyle X_{G}(\Sigma)$. Denote by $\displaystyle G^{n}/\\!/G$ the
algebraic quotient of $\displaystyle G^{n}$ by the adjoint action of
$\displaystyle G$. It is the affine variety with coordinate ring
$\displaystyle\mathcal{O}(G^{n})^{G}$, the ring of regular functions on
$\displaystyle G^{n}$ invariant under the coadjoint action of $\displaystyle
G$. The points of $\displaystyle X_{G}(\Sigma)$ are in one-to-one
correspondence with the trace equivalence classes of representations
$\displaystyle\pi_{1}(\Sigma)\rightarrow G$. Therefore, choosing a basepoint
and generators of the fundamental group of $\displaystyle\Sigma$ affords an
isomorphism of algebraic sets

$\displaystyle\mathfrak{c}\colon X_{G}(\Sigma)\rightarrow G^{n}/\\!/G.$

######  Theorem 7.20.

([49]) The adjoint action of the Poisson-Lie group $\displaystyle(G,\\{\ ,\
\\})$ on the Poisson manifold $\displaystyle(G^{n},\\{\ ,\ \\}_{FR})$ is a
Poisson map. Hence $\displaystyle\\{\ ,\ \\}_{FR}$ defines a Poisson bracket
on $\displaystyle G^{n}/\\!/G$. Moreover the map $\displaystyle\mathfrak{c}$
is a Poisson isomorphism: $\displaystyle\mathfrak{c}_{*}\\{\ ,\
\\}_{Gold}=\\{\ ,\ \\}_{FR}$.

The theorems 7.18 and 7.20 relate
$\displaystyle\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})$ with
$\displaystyle\mathcal{O}(G^{n})$ and $\displaystyle\mathcal{O}(G^{n})^{G}$
with $\displaystyle\mathcal{O}(X_{G}(\Sigma))$. We need to “lift” these
results to the whole center
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})$ and corresponding
rings of regular functions.

At first, recall the isomorphism
$\displaystyle\mathcal{O}(\tilde{G}^{0})\cong\mathcal{Z}(U_{\epsilon})$ of
Remark 7.17. It maps $\displaystyle\mathcal{O}(T)^{W}$ to
$\displaystyle\mathbb{C}[\Omega]$, the $\displaystyle\mathcal{G}$-invariant
subalgebra of $\displaystyle\mathcal{Z}(U_{\epsilon})$. Composing it with
$\displaystyle\Phi_{1}^{-1}$, we get an isomorphism of
$\displaystyle\mathcal{O}(\tilde{G}^{0})$ with
$\displaystyle\mathcal{Z}({}_{loc}{\mathcal{L}}_{0,1}^{\epsilon})$, mapping
the subalgebras $\displaystyle\mathcal{O}(\tilde{G})$ and
$\displaystyle\mathcal{O}(G)$ to
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,1}^{\epsilon})$ and
$\displaystyle\mathcal{Z}_{0}({\mathcal{L}}_{0,1}^{\epsilon})$ respectively.
It follows straightforwardly from the arguments of Theorem 7.18 that
$\displaystyle Fr$ extends to an isomorphism

$\displaystyle\widetilde{Fr}\colon(\mathcal{O}(\tilde{G})^{\otimes n},\\{\ ,\
\\}_{FR})\rightarrow(\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon}),\\{\
,\\}_{\tt QCA})$

mapping the $\displaystyle n$ copies of $\displaystyle\mathcal{O}(T)^{W}$
associated to the factors of $\displaystyle\mathcal{O}(\tilde{G}^{n})$ to
$\displaystyle\mathbb{C}[\omega^{(1)}],\ldots,\mathbb{C}[\omega^{(n)}]$. Note
in particular that $\displaystyle\\{\ ,\ \\}_{FR}$ extends trivially from
$\displaystyle\mathcal{O}(G)^{\otimes n}$ to
$\displaystyle\mathcal{O}(\tilde{G})^{\otimes n}$. Also, the restriction map
(keeping voluntarily the notation of (105))

(148) $\tau\colon{\rm
Spec}(\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon}))\longrightarrow{\rm
Spec}(\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})),$

which by Proposition 7.3 is a regular map of degree $\displaystyle l^{n}$,
corresponds under $\displaystyle\widetilde{Fr}$ to the projection map
$\displaystyle\tilde{G}^{n}\rightarrow G^{n}$.

Now, let us identify as above $\displaystyle G^{n}$ with the space of
representation $\displaystyle R_{G}(\Sigma)=Hom(\pi_{1}(\Sigma),G)$ (fixing a
basepoint and generators of $\displaystyle\pi_{1}(\Sigma)$). The conjugation
action extends trivially from $\displaystyle G^{n}$ to
$\displaystyle\tilde{G}^{n}$. Then the projection map
$\displaystyle\tilde{G}^{n}\rightarrow G^{n}$ provides an identification of
$\displaystyle\tilde{G}^{n}$ with a branched covering space
$\displaystyle\tilde{R}_{G}^{\prime}(\Sigma)$ of $\displaystyle
R_{G}(\Sigma)$, endowed with the conjugation action of $\displaystyle G$. The
points of $\displaystyle\tilde{R}_{G}^{\prime}(\Sigma)$ are given by
representations $\displaystyle\rho\colon\pi_{1}(\Sigma)\rightarrow G$ endowed
with a choice of solution $\displaystyle x\in\mathbb{C}$ of the equation
$\displaystyle T_{l}(x)=Tr(\rho(\gamma_{i}))$, $\displaystyle 1\leq i\leq n$,
where $\displaystyle\gamma_{1},\ldots,\gamma_{n}$ are the $\displaystyle n$
chosen generators of $\displaystyle\pi_{1}(\Sigma)$. Taking algebraic
quotients yields a branched covering map
$\displaystyle\tilde{X}_{G}^{\prime}(\Sigma)\rightarrow X_{G}(\Sigma)$ of the
same degree $\displaystyle l^{n}$, and one can lift Theorem 7.20 to an
isomorphism (with, again, $\displaystyle\\{\ ,\ \\}_{Gold}$ trivially lifted
from $\displaystyle X_{G}(\Sigma)$):

$\displaystyle\tilde{\mathfrak{c}}\colon(\tilde{X}_{G}^{\prime}(\Sigma),\\{\
,\ \\}_{Gold})\rightarrow(\tilde{G}^{n}/\\!/G,\\{\ ,\ \\}_{FR}).$

We can now state and prove our generalization of Theorem 7.15. When
considering the (partial) action of $\displaystyle\mathcal{G}$ on
Spec$\displaystyle(\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon}))$ it will
always be meant to be the diagonal action, by means of
$\displaystyle\mathcal{G}^{\Delta}$.

######  Corollary 7.21.

(1) The dual diffeomorphism $\displaystyle Fr^{*}\colon({\rm
Spec}(\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})),\\{\ ,\\}_{\tt
QCA})\rightarrow(G^{n},\\{\ ,\ \\}_{FR})$ maps the action of
$\displaystyle{\mathcal{G}}$ on the tangent spaces of $\displaystyle(0,0,\pm
1)^{n}$ onto the coadjoint action of $\displaystyle G$ on $\displaystyle
sl(2,\mathbb{C})^{*n}$.

(2) For any conjugacy class $\displaystyle\Gamma$ in $\displaystyle G^{n}$,
$\displaystyle(Fr^{*})^{-1}(\Gamma)$ is a (non empty)
$\displaystyle{\mathcal{G}}$-orbit in
Spec$\displaystyle(\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon}))$, and
$\displaystyle\tau^{-1}$ of this orbit has $\displaystyle l^{n}$ connected
components, all of whose are $\displaystyle{\mathcal{G}}$-orbits in
Spec$\displaystyle(\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon}))$.

(3) The map $\displaystyle\widetilde{Fr}\circ\tilde{\mathfrak{c}}^{-1*}$ takes
values in
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}}$, and
therefore affords an isomorphism of Poisson algebras

$\displaystyle\widetilde{Fr}\circ\tilde{\mathfrak{c}}^{-1*}\colon(\mathcal{O}(\tilde{X}_{G}^{\prime}(\Sigma)),\\{\
,\
\\}_{Gold})\rightarrow(\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}},\\{\
,\ \\}_{\tt QCA}).$

(4) The orbits of the group $\displaystyle{\mathcal{G}}^{tot}$ in
Spec$\displaystyle(\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon}))$ are the
symplectic leaves of $\displaystyle\\{\ ,\ \\}_{\tt QCA}$. These project onto
the symplectic leaves of $\displaystyle\\{\ ,\ \\}_{\tt QCA}$ in
Spec$\displaystyle(\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}})$.

Proof. (1) Using as usual the isomorphism
$\displaystyle\bar{\sigma}^{-1*}\circ\bar{\psi}^{*}$ to identify
$\displaystyle\mathcal{Z}_{0}(U_{\epsilon})$ with
$\displaystyle\mathcal{O}(G^{0})$, it follows easily from (139), (141) and
(142) that the automorphism
$\displaystyle\Phi_{n}\circ(\Phi_{1}^{-1})^{\otimes n}$ of
$\displaystyle\mathcal{O}(G^{0})^{\otimes n}$ is equivariant with respect to
the coadjoint action of $\displaystyle G$. Since
$\displaystyle\Phi_{n}^{-1*}((0,0,\pm 1)^{n})=(\pm I,\ldots,\pm I)$, the
formulas in Proposition 7.19 (2) show that $\displaystyle d\Phi_{n}^{-1*}$
maps the action of $\displaystyle{\mathcal{E}}^{\Delta}$,
$\displaystyle{\mathcal{F}}^{\Delta}$ on the tangent spaces of
$\displaystyle(0,0,\pm 1)^{n}$ to the coadjoint action of $\displaystyle X$,
$\displaystyle Y$ on $\displaystyle sl(2,\mathbb{C})^{*n}$. Post-composing
$\displaystyle\Phi_{n}^{-1*}$ with $\displaystyle(\Phi_{1}^{-1*})^{\otimes
n}\circ\Phi_{n}^{*}$, Proposition 7.19 (1) proves that
$\displaystyle\Phi^{-1*}$ maps the action of $\displaystyle{\mathcal{G}}$ on
the tangent spaces of $\displaystyle(0,0,\pm 1)^{n}\in{\rm
Spec}(\mathcal{Z}_{0}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}))$ onto the
coadjoint action of $\displaystyle G$ on $\displaystyle
sl(2,\mathbb{C})^{*n}\cong T_{(\pm I,\ldots,\pm I)}^{*}(G^{0})^{n}$. The
result follows, $\displaystyle Fr^{*}$ being an extension of
$\displaystyle\Phi^{-1*}$.

(2) This is an integrated version of (1) above. It follows from the arguments
used to prove Theorem 7.15 (c), by replacing (129) with the formulas in
Proposition 7.19 (2), and Lemma 7.11 with Proposition 7.19 (1).

(3) It is enough to prove that $\displaystyle Fr\circ\mathfrak{c}^{-1*}$
establishes a Poisson isomorphism between
$\displaystyle\mathcal{O}(X_{G}(\Sigma))$ and
$\displaystyle\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}}$.
By Theorem 7.18, $\displaystyle
Fr=\Phi^{-1}_{|\mathcal{O}(G^{n})}\colon(\mathcal{O}(G)^{\otimes n},\\{\ ,\
\\}_{FR})\rightarrow(\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon}),\\{\
,\\}_{\tt QCA})$ is an isomorphism of Poisson algebras. By (2) above
$\displaystyle Fr$ maps invariants functions to invariant functions. That the
bracket $\displaystyle\\{\ ,\ \\}_{\tt QCA}$ is well-defined on
$\displaystyle\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}}$
is an immediate consequence of its definition and the structure of module
algebra of $\displaystyle{\mathcal{L}}_{0,n}^{\epsilon}$. Then the conclusion
follows from the last claim of Theorem 7.20.

(4) The first claim follows from Theorem 7.15 (d), and the facts that
$\displaystyle\Phi_{n}$ is a Poisson isomorphism from
$\displaystyle(\mathcal{Z}_{0}({}_{loc}{\mathcal{L}}_{0,n}^{\epsilon}),\\{\
,\\}_{\tt QCA})$ to $\displaystyle(\mathcal{Z}_{0}(U_{\epsilon}),\\{\ ,\
\\}_{\tt QCA})^{\otimes n}$ (see the proof of Theorem 7.18), and that
$\displaystyle\Phi_{n}^{-1*}$ maps the partial action of
$\displaystyle{\mathcal{G}}^{tot}$ on
Spec$\displaystyle(\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon}))$ to the
partial action of $\displaystyle{\mathcal{G}}^{n}$ (the $\displaystyle n$-fold
direct product) on $\displaystyle{\rm
Spec}(\mathcal{Z}_{0}(U_{\epsilon}))^{n}$ (by the definition of
$\displaystyle{\mathcal{G}}^{tot}$). The second claim follows from the first
one in Theorem 7.20. $\displaystyle\Box$

##  8\. Topological formulation

###  8.1. The Wilson loop functor

Recall that we denote by $\displaystyle\mathcal{C}_{A}$ the semisimple
category of finite dimensional $\displaystyle U_{A}^{res}$-modules of type
$\displaystyle 1$, and that
$\displaystyle\mathcal{C}_{A}\otimes{\mathbb{Z}}[q^{1/D},q^{-1/D}]$ is a
ribbon category. Recall also the following notions (see [74]). Denote by
$\displaystyle Rib_{\mathcal{C}_{A}}$ the category whose morphisms are the
isotopy classes rel$\displaystyle(\partial)$ of oriented ribbon graphs in
$\displaystyle[0,1]^{3}$ colored over $\displaystyle\mathcal{C}_{A}$ (ie. with
each component labelled by an object of $\displaystyle\mathcal{C}_{A}$), with
boundary segments (if any) in
$\displaystyle]0,1[\times\\{1/2\\}\times\\{i\\}$, where $\displaystyle
i\in\\{0,1\\}$. The objects of $\displaystyle Rib_{\mathcal{C}_{A}}$ are the
tuples
$\displaystyle((V_{1},\varepsilon_{1}),\ldots,(V_{k},\varepsilon_{k}))$, where
$\displaystyle\varepsilon_{1},\ldots,\varepsilon_{k}=\pm$ and $\displaystyle
V_{1},\ldots,V_{k}$ are objects of $\displaystyle\mathcal{C}_{A}$. The source
and target object of a morphism of $\displaystyle Rib_{\mathcal{C}_{A}}$ thus
correspond to tuples of segments in
$\displaystyle]0,1[\times\\{1/2\\}\times\\{i\\}$, $\displaystyle
i\in\\{0,1\\}$, endowed with normal co-orientations specifying the associated
signs $\displaystyle\pm$. We denote by $\displaystyle\mathbb{RT}$ the
corresponding Reshetikhin-Turaev functor,

$\displaystyle\mathbb{RT}\colon
Rib_{\mathcal{C}_{A}}\to\mathcal{C}_{A}\otimes{\mathbb{Z}}[q^{1/D},q^{-1/D}].$

Now, fix points $\displaystyle p_{1}<\cdots<p_{n}$ in $\displaystyle]0,1/2[$,
and define $\displaystyle Rib_{n,\mathcal{C}_{A}}$ as the category with the
same objects as $\displaystyle Rib_{\mathcal{C}_{A}}$ but morphisms the
isotopy classes rel$\displaystyle(\partial)$ of oriented ribbon graphs in
$\displaystyle[0,1]^{3}\setminus(\\{p_{1},\ldots,p_{n}\\}\times\\{1/2\\}\times[0,1])$,
colored over $\displaystyle\mathcal{C}_{A}$ and with boundary segments (if
any) in $\displaystyle]1/2,1[\times\\{1/2\\}\times\\{i\\}$, $\displaystyle
i\in\\{0,1\\}$. The following picture shows an example.

$\displaystyle p_{1}$$\displaystyle p_{3}$$\displaystyle p_{2}$

Figure 8.1: a morphism of $\displaystyle Rib_{n,\mathcal{C}_{A}}$.

The composition of morphisms of $\displaystyle Rib_{n,\mathcal{C}_{A}}$ is
defined as for $\displaystyle Rib_{\mathcal{C}_{A}}$. That is, given morphisms
$\displaystyle T_{1}$, $\displaystyle T_{2}$ with a same pattern of co-
oriented boundary segments on the bottom of $\displaystyle T_{1}$ and top of
$\displaystyle T_{2}$, $\displaystyle T_{1}\circ T_{2}$ is obtained by placing
$\displaystyle T_{1}$ atop $\displaystyle T_{2}$, gluing the corresponding
boundary segments, and deforming the result by isotopy into
$\displaystyle[0,1]^{3}\setminus(\\{p_{1},\ldots,p_{n}\\}\times\\{1/2\\}\times[0,1])$.
Identifying $\displaystyle[0,1]^{3}$ with the “right half” cube
$\displaystyle[1/2,1]\times[0,1]^{2}$ in
$\displaystyle[0,1]^{3}\setminus(\\{p_{1},\ldots,p_{n}\\}\times\\{1/2\\}\times[0,1])$
yields an obvious faithful functor

$\displaystyle\iota\colon Rib_{\mathcal{C}_{A}}\rightarrow
Rib_{n,\mathcal{C}_{A}}.$

The morphisms of $\displaystyle Rib_{n,\mathcal{C}_{A}}$ are obtained by
composing morphisms of $\displaystyle Rib_{\mathcal{C}_{A}}$ and elementary
morphisms as shown in the following picture, for $\displaystyle a=1,\ldots,n$.

$\displaystyle 1$$\displaystyle a$$\displaystyle n$$\displaystyle
V_{1}\otimes\cdots\otimes V_{k}$

Figure 8.2: a generating morphism of $\displaystyle Rib_{n,\mathcal{C}_{A}}$
not in $\displaystyle Rib_{\mathcal{C}_{A}}$.

Recall that we denote by $\displaystyle\Sigma$ the sphere with $\displaystyle
n+1$ open disks removed. Denote by
$\displaystyle\mathcal{L}_{\mathcal{C}_{A}}(\Sigma)$ the $\displaystyle
A$-module freely generated by the empty set and the isotopy classes of closed
oriented ribbon graphs in $\displaystyle\Sigma\times[0,1]$ colored over
$\displaystyle\mathcal{C}_{A}$. The stacking product $\displaystyle
L.L^{\prime}$ of elements $\displaystyle
L,L^{\prime}\in\mathcal{L}_{\mathcal{C}_{A}}(\Sigma)$ is defined as the
isotopy class of the disjoint union of representatives of $\displaystyle L$
and $\displaystyle L^{\prime}$, pushed in $\displaystyle\Sigma\times[0,1/2]$
and $\displaystyle\Sigma\times[1/2,1]$ respectively. The stacking product
makes $\displaystyle\mathcal{L}_{\mathcal{C}_{A}}(\Sigma)$ an $\displaystyle
A$-algebra. Since $\displaystyle\Sigma$ is diffeomorphic to the closure of
$\displaystyle[0,1]^{2}\setminus(\\{p_{1},\ldots,p_{n}\\}\times\\{1/2\\})$,
$\displaystyle\mathcal{L}_{\mathcal{C}_{A}}(\Sigma)$ can be identified with
the $\displaystyle A$-algebra generated by the morphisms
$\displaystyle\emptyset\rightarrow\emptyset$ of $\displaystyle
Rib_{n,\mathcal{C}_{A}}$, the stacking product being the $\displaystyle
A$-linear extension of the composition of morphisms.

Define a new category $\displaystyle\mathcal{C}_{A}({\mathcal{L}}_{0,n}^{A})$
with objects the couples $\displaystyle(n,V)$, where $\displaystyle V$ is an
object of $\displaystyle\mathcal{C}_{A}\otimes{\mathbb{Z}}[q^{1/D},q^{-1/D}]$,
and with spaces of morphisms

$\displaystyle
Hom((n,V),(n,W))=\left(Hom_{U_{A}^{res}}(V,W)\otimes{\mathbb{Z}}[q^{1/D},q^{-1/D}]\right)\otimes{\mathcal{L}}_{0,n}^{A}$

where the composition is the tensor product of the product in
$\displaystyle{\mathcal{L}}_{0,n}^{A}$ and the composition of $\displaystyle
U_{A}^{res}$-homomorphisms of modules.

The following result follows from the arguments in Section 6.1–6.2 of Faitg’s
PhD thesis [40], to which we refer for full details. These arguments
generalize and simplify those of [18]. In these works only the Wilson loop map
$\displaystyle W$ was considered, but the extension to
$\displaystyle\mathbb{W}$ is straightforward.

######  Theorem 8.1.

There is unique functor
$\displaystyle{\mathbb{W}}:Rib_{n,\mathcal{C}_{A}}\rightarrow\mathcal{C}_{A}({\mathcal{L}}_{0,n}^{A})$
such that on objects we have
$\displaystyle{\mathbb{W}}((V_{1},\varepsilon_{1}),\ldots,(V_{k},\varepsilon_{k}))=(n,V_{1}^{\varepsilon_{1}}\otimes\ldots\otimes
V_{k}^{\varepsilon_{k}})$, where $\displaystyle V_{j}^{+}=V_{j}$ and
$\displaystyle V_{j}^{-}=V_{j}^{*}$ (the dual of $\displaystyle V_{j}$), and
on morphisms:

  * •

$\displaystyle{\mathbb{W}}(\iota(T))=\mathbb{RT}(T)\otimes 1$ for every
morphism $\displaystyle T$ of $\displaystyle Rib_{\mathcal{C}_{A}}$;

  * •

$\displaystyle{\mathbb{W}}$ gives the value
$\displaystyle\stackrel{{\scriptstyle V}}{{M}}{}^{\\!\\!(a)}$ to the morphism
shown in Figure 8.2, in the case where there is a single ribbon colored by
$\displaystyle V:=V_{1}$.

Moreover the morphism of algebras $\displaystyle
W\colon\mathcal{L}_{\mathcal{C}_{A}}(\Sigma)\rightarrow{\mathcal{L}}_{0,n}^{A}\otimes{\mathbb{Z}}[q^{1/D},q^{-1/D}]$
obtained by restricting $\displaystyle\mathbb{W}$ to
$\displaystyle\mathcal{L}_{\mathcal{C}_{A}}(\Sigma)$ takes values in the
invariant subalgebra
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}\otimes{\mathbb{Z}}[q^{1/D},q^{-1/D}]$.

We call $\displaystyle{\mathbb{W}}$ the Wilson loop functor, and
$\displaystyle W$ the Wilson loop map. By construction, for every element
$\displaystyle\hat{T}\in\mathcal{L}_{\mathcal{C}_{A}}(\Sigma)$, and any
morphism $\displaystyle T$ of $\displaystyle Rib_{n,\mathcal{C}_{A}}$ obtained
from $\displaystyle\hat{T}$ by cutting open ribbons with colors $\displaystyle
V_{1},\ldots,V_{k}$, setting $\displaystyle V=V_{1}\otimes\ldots\otimes V_{k}$
we have

(149) $W(\hat{T})=qTr_{V}\left({\mathbb{W}}(T)\right).$

Here is an alternative way of defining $\displaystyle{\mathbb{W}}$, purely in
terms of $\displaystyle\mathbb{RT}$. Define a category
$\displaystyle\mathcal{C}_{A}(\tilde{U}_{A}^{\otimes n})$ with same objects
$\displaystyle(n,V)$ as
$\displaystyle\mathcal{C}_{A}({\mathcal{L}}_{0,n}^{A})$ but with spaces of
morphisms

$\displaystyle
Hom((n,V),(n,W))=\left(Hom_{U_{A}^{res}}(V,W)\otimes{\mathbb{Z}}[q^{1/D},q^{-1/D}]\right)\otimes\tilde{U}_{A}^{\otimes
n}$

where the composition is the tensor product of the product in
$\displaystyle\tilde{U}_{A}^{\otimes n}$ and the composition of $\displaystyle
Hom_{U_{A}^{res}}(V,W)$. The Alekseev map being a morphism of algebra, it
defines a functor

$\displaystyle\Phi_{n}:\mathcal{C}_{A}({\mathcal{L}}_{0,n}^{A})\rightarrow{\mathcal{C}}_{A}(\tilde{U}_{A}^{\otimes
n})$

by setting $\displaystyle\Phi_{n}(f\otimes a)=f\otimes\Phi_{n}(a)$ for every
$\displaystyle a\in{\mathcal{L}}_{0,n}^{A}$, $\displaystyle f\in
Hom_{U_{A}^{res}}(V,W)$. To any morphism $\displaystyle T$ of $\displaystyle
Rib_{n,\mathcal{C}_{A}}$ we can associate a colored oriented ribbon graphs in
$\displaystyle[0,1]^{3}$,

$\displaystyle
T^{\sharp}=(\cup_{i=1}^{n}[p_{i}-\delta,p_{i}+\delta]\times\\{1/2\\}\times[0,1])_{\tilde{U}_{A}}\cup
T,$

where $\displaystyle\delta>0$ is small and the cores of the ribbons
$\displaystyle[p_{i}-\delta,p_{i}+\delta]\times\\{1/2\\}\times[0,1]$ are
oriented from $\displaystyle\\{p_{i}\\}\times\\{1/2\\}\times\\{1\\}$ to
$\displaystyle\\{p_{i}\\}\times\\{1/2\\}\times\\{0\\}$ and colored by the
regular representation of $\displaystyle\tilde{U}_{A}$. Denote by
$\displaystyle\overline{Rib}_{\mathcal{C}_{A}}$ the category with morphisms
given by the ribbon graphs $\displaystyle T^{\sharp}$; clearly the map
$\displaystyle T\mapsto T^{\sharp}$ yields a functor

$\displaystyle\sharp\colon
Rib_{n,\mathcal{C}_{A}}\rightarrow\overline{Rib}_{n,\mathcal{C}_{A}}.$

Associating $\displaystyle\sigma\circ(id\otimes\pi_{V})(R^{\pm})\colon
End(V)\otimes\tilde{U}_{A}\rightarrow\tilde{U}_{A}\otimes End(V)$ to crossings
of index $\displaystyle\pm 1$ colored over $\displaystyle\mathcal{C}_{A}$ and
$\displaystyle\tilde{U}_{A}$, in the same way as
$\displaystyle\sigma\circ(\pi_{W}\otimes\pi_{V})(R^{\pm 1})$ is associated by
$\displaystyle\mathbb{RT}$ to crossings colored by objects $\displaystyle V,W$
of $\displaystyle\mathcal{C}_{A}$, extends $\displaystyle\mathbb{RT}$ to a
functor

$\displaystyle\mathbb{RT}\colon\overline{Rib}_{n,\mathcal{C}_{A}}\rightarrow\mathcal{C}_{A}(\tilde{U}_{A}^{\otimes
n}).$

The next fact is a direct consequence of the definition of
$\displaystyle\Phi_{n}$, and its representation by Figure 6.1. Note that
$\displaystyle\Phi_{n}$ being injective, it defines
$\displaystyle{\mathbb{W}}$ uniquely from $\displaystyle\mathbb{RT}$.

######  Proposition 8.2.

We have a commutative diagram of functors:

$\displaystyle\textstyle{Rib_{n,\mathcal{C}_{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$♯$\displaystyle\scriptstyle{{\mathbb{W}}}$$\displaystyle\textstyle{\overline{Rib}_{n,\mathcal{C}_{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{\mathbb{RT}}$$\displaystyle\textstyle{\mathcal{C}_{A}({\mathcal{L}}_{0,n}^{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{\Phi_{n}}$$\displaystyle\textstyle{{\mathcal{C}}_{A}(\tilde{U}_{A}^{\otimes
n}).}$

###  8.2. The Wilson loop isomorphism

Define a one-coupon multicurve as an element of
$\displaystyle\mathcal{L}_{\mathcal{C}_{A}}(\Sigma)$ that can be represented
by a (oriented, $\displaystyle\mathcal{C}_{A}$-colored) ribbon graph embedded
in $\displaystyle\Sigma\times\\{0\\}$ and having a single coupon. An example
of graph in $\displaystyle\Sigma\times[0,1]$ representing a one-coupon
multicurve is shown in Figure 8.3, where $\displaystyle\Sigma$ is the four-
holed sphere (ie. $\displaystyle n=3$).

$\displaystyle a_{[\lambda]}$

Figure 8.3: a one-coupon multicurve in $\displaystyle\Sigma_{0,4}\times[0,1]$.

Recall the peculiar expression (71) of
$\displaystyle\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}$. Let
$\displaystyle a_{[\lambda]}\in End_{U_{A}^{res}}(V_{[\lambda]})$,
$\displaystyle[\lambda]\in P_{+}^{3}$, and denote by $\displaystyle
L(a_{[\lambda]})$ the one-coupon multicurve in Figure 8.3. It is clear that

(150)
$W(L(a_{[\lambda]}))=qTr_{V_{[\lambda]}}\left(a_{[\lambda]}\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}\right).$

This generalizes immediately to any $\displaystyle n$. Using Proposition 6.20
we deduce:

######  Lemma 8.3.

The Wilson loop map $\displaystyle
W\colon\mathcal{L}_{\mathcal{C}_{A}}(\Sigma)\rightarrow({\mathcal{L}}_{0,n}^{A})^{U_{A}}\otimes_{A}{\mathbb{Z}}[q^{1/D},q^{-1/D}]$
is surjective.

In the sequel we assume that $\displaystyle\mathfrak{g}=sl(2)$.

Set $\displaystyle\zeta:=iq^{1/2}$, and denote by
$\displaystyle\mathcal{L}_{\zeta}(\Sigma)\subset\mathcal{L}_{\mathcal{C}_{A}}(\Sigma)\otimes{\mathbb{Z}}[\zeta,\zeta^{-1}]$
the subalgebra freely generated as a
$\displaystyle\mathbb{Z}[\zeta,\zeta^{-1}]$-module by the empty set and the
isotopy classes of oriented ribbon links in $\displaystyle\Sigma\times[0,1]$
colored by the fundamental representation $\displaystyle V_{2}$. Recall that
the Kauffman bracket skein algebra $\displaystyle K_{\zeta}(\Sigma)$ is the
$\displaystyle\mathbb{Z}[\zeta,\zeta^{-1}]$-algebra obtained from
$\displaystyle\mathcal{L}_{\zeta}(\Sigma)$ by forgetting the link
orientations, and taking the quotient by the ideal generated by the relations

(151) $\displaystyle\displaystyle L$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\zeta L_{+}+\zeta^{-1}L_{-}$ (152)
$\displaystyle\displaystyle L\sqcup\bigcirc$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-(\zeta^{2}+\zeta^{-2})L$

where in the second identity $\displaystyle\bigcirc$ is the trivial ribbon
link in a ball disjoint from $\displaystyle L$, and in the first identity
$\displaystyle L,L_{+},L_{-}\in\mathcal{L}_{\zeta}(\Sigma)$ are identical up
to isotopy except in a ball in which they look like (the strands representing
flat ribbons on a meridional projection disk):

$\displaystyle L$$\displaystyle L_{+}$$\displaystyle L_{-}$

Figure 8.4: skein related ribbon links.

The following result is essentially an integral version (ie. over the ring
$\displaystyle A$) of a combination of Theorem 10 of [23] and Theorem 1 of
[24]. For completeness we give a proof, by using the notions we have
introduced.

######  Theorem 8.4.

The linear map defined by $\displaystyle\mathcal{W}(L)=i^{lk(L)}W(L)$, where
$\displaystyle L\in\mathcal{L}_{\zeta}(\Sigma)$ and $\displaystyle lk(L)$ is
the linking number of $\displaystyle L$, descends to an isomorphism of
algebras (where $\displaystyle\zeta:=iq^{1/2}$):

$\displaystyle\mathcal{W}\colon
K_{\zeta}(\Sigma)\rightarrow({\mathcal{L}}_{0,n}^{A})^{U_{A}}\otimes_{A}\mathbb{Z}[\zeta,\zeta^{-1}].$

Proof. The facts that the values of $\displaystyle\mathcal{W}$ do not depend
on the link orientations, and that $\displaystyle\mathcal{W}$ maps to
$\displaystyle 0$ the ideal of $\displaystyle\mathcal{L}_{\zeta}(\Sigma)$
generated by the skein relations (151)-(152), follow from Proposition 8.2,
injectivity of $\displaystyle\Phi_{n}$, and the properties of
$\displaystyle\mathbb{RT}$ and the $\displaystyle R$-matrix $\displaystyle
R_{V_{2},V_{2}}$ proved eg. in [59], Lemma 3.18, Theorem 4.3 and Corollary
4.13. We stress that the skein relations require the choice of variable
$\displaystyle\zeta:=iq^{1/2}$, and the normalisation by $\displaystyle
i^{lk(L)}$. Therefore $\displaystyle\mathcal{W}$ descends to a module map
defined on $\displaystyle K_{\zeta}(\Sigma)$. Since the stacking product is
induced from $\displaystyle\mathcal{L}_{\zeta}(\Sigma)$,
$\displaystyle\mathcal{W}$ is a morphism of algebras.

To show that $\displaystyle\mathcal{W}$ is an isomorphism we use the following
basis of $\displaystyle K_{\zeta}(\Sigma)$ (see eg. [57, 63]). Let
$\displaystyle\Gamma$ be a trivalent ribbon graph onto which
$\displaystyle\Sigma$ retracts by deformation. Recall that an admissible
coloring of $\displaystyle\Gamma$ is an assignment of a nonnegative integer to
each edge, called the color of the edge, so that the colors adjacent to each
vertex form admissible triples. A triple of colors $\displaystyle(a,b,c)$ is
admissible if $\displaystyle a\leq b+c$, $\displaystyle b\leq a+c$,
$\displaystyle c\leq a+b$ and $\displaystyle a+b+c$ is even. There is a skein
$\displaystyle\Gamma_{\gamma}$ in $\displaystyle K_{\zeta}(\Sigma)$
corresponding to every admissible coloring $\displaystyle\gamma$ of
$\displaystyle\Gamma$, obtained by replacing an edge with color $\displaystyle
m$ by the $\displaystyle m$-th Jones–Wenzl idempotent and vertices with
Kauffman triads (these being defined in eg. [63], page 136 and Fig. 14.7).
Because $\displaystyle\Gamma$ is a spine of $\displaystyle\Sigma$, the set of
skeins $\displaystyle\Gamma_{\gamma}$ forms a basis of $\displaystyle
K_{\zeta}(\Sigma)$.

By an isotopy of
$\displaystyle\Sigma\times[0,1]\cong[0,1]^{3}\setminus(\\{p_{1},\ldots,p_{n}\\}\times\\{1/2\\}\times[0,1])$
we can deform $\displaystyle\Gamma$ (keeping the same notation) so that a
neighborhood of the vertices lies inside a coupon embedded in
$\displaystyle]1/2,1[\times\\{1/2\\}\times[0,1]$, and the $\displaystyle n$
ribbons forming the portion $\displaystyle\Gamma^{\prime}$ of
$\displaystyle\Gamma$ outside the coupon are attached to its left side,
encircling $\displaystyle\\{p_{i}\\}\times\\{1/2\\}\times[0,1]$, for
$\displaystyle i=1,\ldots,n$. Let $\displaystyle\gamma$ be an admissible
coloring of $\displaystyle\Gamma$. Put
$\displaystyle\Gamma^{\prime\prime}=\Gamma\setminus\Gamma^{\prime}$, and
denote by $\displaystyle\Gamma^{\prime}_{\gamma}$,
$\displaystyle\Gamma^{\prime\prime}_{\gamma}$ the graphs
$\displaystyle\Gamma^{\prime}$, $\displaystyle\Gamma^{\prime\prime}$ with
edges colored by $\displaystyle\gamma$. Denote by $\displaystyle
e_{1},\ldots,e_{n}$ the edges of $\displaystyle\Gamma^{\prime}$ ordered by
increasing height, by $\displaystyle\gamma_{i}$ the color of $\displaystyle
e_{i}$ plus $\displaystyle 1$, and fix some orientation of the edges of
$\displaystyle\Gamma$ so that each $\displaystyle e_{i}$ is oriented from top
to bottom. In order to fit with the standard framework for computations with
$\displaystyle\mathbb{RT}$, let us rotate the cube $\displaystyle[0,1]^{3}$
clockwise by an angle of $\displaystyle\pi/2$ around the axis
$\displaystyle\\{1/2\\}\times[0,1]\times\\{1/2\\}$. Then, using Theorem 8.1
and the fact that the $\displaystyle m$-th Jones–Wenzl idempotent projects
$\displaystyle V_{2}^{\otimes m}$ onto a subspace isomorphic to $\displaystyle
V_{m+1}$, we see that

$\displaystyle\mathbb{W}(\Gamma^{\prime}_{\gamma})=\stackrel{{\scriptstyle
V_{\gamma_{1}}}}{{M}}{}^{\\!\\!(1)}\otimes\ldots\otimes\stackrel{{\scriptstyle
V_{\gamma_{n}}}}{{M}}{}^{\\!\\!(n)},$

which is an element of
$\displaystyle\otimes_{i=1}^{n}\left(V_{\gamma_{i}}\otimes
V_{\gamma_{i}}^{*}\right)\otimes{\mathcal{L}}_{0,n}^{A}$ by identifying
$\displaystyle End(V)$ with $\displaystyle V\otimes V^{*}$, and

$\displaystyle\mathbb{W}(\Gamma^{\prime\prime}_{\gamma})\in
Hom_{U_{A}^{res}}\left(\otimes_{i=1}^{n}\left(V_{\gamma_{i}}\otimes
V_{\gamma_{i}}^{*}\right),1\right)\otimes_{A}\mathbb{Z}[\zeta,\zeta^{-1}].$

By definitions we have
$\displaystyle\mathcal{W}(\Gamma_{\gamma})=\mathbb{W}(\Gamma^{\prime\prime}_{\gamma})(\mathbb{W}(\Gamma^{\prime}_{\gamma}))$,
and the map
$\displaystyle\gamma\mapsto\mathbb{W}(\Gamma^{\prime\prime}_{\gamma})$ maps
the admissible colorings of $\displaystyle\Gamma$ to a basis of the space of
invariant elements of
$\displaystyle\otimes_{i=n}^{1}End(V_{\gamma_{i}})^{*}\otimes_{A}\mathbb{Z}[\zeta,\zeta^{-1}]$
for the action $\displaystyle(ad^{r})^{\otimes n}$. Therefore the set
$\displaystyle\\{\mathcal{W}(\Gamma_{\gamma}),\gamma\ {\rm admissible}\\}$ is
a basis of the space of invariant elements of
$\displaystyle({\mathcal{L}}_{0,1}^{A})^{\otimes
n}\otimes_{A}\mathbb{Z}[\zeta,\zeta^{-1}]$ for the action
$\displaystyle(coad^{r})^{\otimes n}$. By applying to it the linear
isomorphism $\displaystyle\Phi_{n}^{-1}\circ\psi\circ\Phi_{1}^{\otimes
n}\colon({\mathcal{L}}_{0,1}^{A})^{\otimes
n}\rightarrow{\mathcal{L}}_{0,n}^{A}$, with $\displaystyle\psi$ the
intertwiner of $\displaystyle(ad^{r})^{\otimes n}$ and $\displaystyle
ad^{r}_{n}$ defined in the the proof of Proposition 6.5, we get a basis of
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}\otimes_{A}\mathbb{Z}[\zeta,\zeta^{-1}]$.
This concludes the proof. $\displaystyle\Box$

######  Remark 8.5.

Above we can replace $\displaystyle\mathbb{W}(\Gamma^{\prime}_{\gamma})$ with
$\displaystyle\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}$, where
$\displaystyle[\lambda]=(\gamma_{1},\ldots,\gamma_{n})$, by “parsing” the ends
of $\displaystyle e_{1},\ldots,e_{n}$ as do the strands just above the coupon
$\displaystyle a_{[\lambda]}$ in Figure 8.3. Correspondingly
$\displaystyle\mathbb{W}(\Gamma^{\prime\prime}_{\gamma})$ becomes an element
of $\displaystyle
Hom_{U_{A}^{res}}\left(1,(\otimes_{i=1}^{n}V_{\gamma_{i}})\otimes(\otimes_{i=1}^{n}V_{\gamma_{i}})^{*}\right)\otimes_{A}\mathbb{Z}[\zeta,\zeta^{-1}]$.
With these choices, the pairing of
$\displaystyle\mathbb{W}(\Gamma^{\prime}_{\gamma})$ and
$\displaystyle\mathbb{W}(\Gamma^{\prime\prime}_{\gamma})$ is equivariant with
respect to the actions $\displaystyle coad^{r}_{n}$ and $\displaystyle
ad^{r}_{n}$, and it puts in duality the product of
$\displaystyle{\mathcal{L}}_{0,n}^{A}$ and the comultiplication map
constructed in [23]. $\displaystyle\Box$

###  8.3. The threading map and $\displaystyle{\mathcal{G}}$-invariant central
elements

A multicurve on $\displaystyle\Sigma$ is a union of disjoint simple non
trivial (ie. not bounding a disk) closed curves considered up to isotopy. Any
multicurve $\displaystyle\gamma$ is a stacking product
$\displaystyle\textstyle\prod_{i=1}^{k}\gamma_{i}^{c_{i}}$, where
$\displaystyle\gamma_{1},\ldots,\gamma_{k}$ are disjoint (hence commuting)
non-isotopic simple non trivial closed curves on $\displaystyle\Sigma$, and
$\displaystyle\gamma_{i}^{c_{i}}$, $\displaystyle c_{i}\in\mathbb{N}$,
consists of $\displaystyle c_{i}$ parallel copies of
$\displaystyle\gamma_{i}$.

Denote by $\displaystyle\mathcal{S}(\Sigma)$ the set of multicurves on
$\displaystyle\Sigma$. Recall the normalized Chebyshev polynomials
$\displaystyle T_{k}$, $\displaystyle k\in\mathbb{N}$, defined in (83). For
$\displaystyle\gamma\in\mathcal{S}(\Sigma)$,
$\displaystyle\textstyle\gamma=\prod_{i=1}^{k}\gamma_{i}^{c_{i}}$, set

$\displaystyle T(\gamma):=\textstyle\prod_{i=1}^{k}T_{c_{i}}(\gamma_{i}).$

It is standard that $\displaystyle\mathcal{S}(\Sigma)$ is a
$\displaystyle\mathbb{Z}[\zeta,\zeta^{-1}]$-basis of $\displaystyle
K_{\zeta}(\Sigma)$ (see eg. [70], Theorem 7) and that
$\displaystyle\\{T_{k}\\}_{k}$ is a basis of $\displaystyle\mathbb{Z}[X]$.
Therefore $\displaystyle\\{T(\gamma),\gamma\in\mathcal{S}(\Sigma)\\}$ is a
$\displaystyle\mathbb{Z}[\zeta,\zeta^{-1}]$-basis of $\displaystyle
K_{\zeta}(\Sigma)$. It is called the Chebyshev basis (see [47] and the
references therein).

Theorem 8.4 implies that
$\displaystyle\\{\mathcal{W}(T(\gamma)),\gamma\in\mathcal{S}(\Sigma)\\}$ is a
$\displaystyle\mathbb{Z}[\zeta,\zeta^{-1}]$-basis of
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}\otimes_{A}\mathbb{Z}[\zeta,\zeta^{-1}]$.
Note that it is a peculiar instance of the basis obtained in Proposition 6.20,
by taking powers of $\displaystyle V_{2}$ in place of the colors
$\displaystyle\lambda_{j}\in P_{+}$ (as explained in the proof of Theorem 8.6
below), and an $\displaystyle A$-basis of the Temperley-Lieb algebra for the
coupons.

Now let as usual $\displaystyle l\geq 3$ be an odd integer. Define
$\displaystyle\mathcal{S}_{l}(\Sigma)\subset\mathcal{S}(\Sigma)$ as the set of
multicurves of the form
$\displaystyle\textstyle\gamma_{\partial}\prod_{i=1}^{k}\gamma_{i}^{c_{i}}$,
where $\displaystyle\gamma_{\partial}$ is peripheral, ie. a monomial in the
skein classes of the boundary components of $\displaystyle\Sigma$,
$\displaystyle\gamma_{i}$ is a non peripheral curve, and $\displaystyle l$
divides $\displaystyle c_{i}$ for every $\displaystyle i\in\\{1,\ldots,k\\}$.

Let $\displaystyle\epsilon$ be a primitive root of unity of odd order
$\displaystyle l$. Recall
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}}$ and
the derivations $\displaystyle
D_{a}:{\mathcal{L}}_{0,n}^{\epsilon}\rightarrow{\mathcal{L}}_{0,n}^{\epsilon}$,
$\displaystyle a\in\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})$, defined in
Section 7.2.2.

######  Theorem 8.6.

(1) The set
$\displaystyle\\{\mathcal{W}(T(\gamma)),\gamma\in\mathcal{S}_{l}(\Sigma)\\}$
is a $\displaystyle\mathbb{C}$-basis of the algebra generated by
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}}$ and
$\displaystyle\eta$. In particular, it is a central subalgebra of
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}_{\epsilon}$.

(2) The derivations $\displaystyle D_{a}$, $\displaystyle
a\in\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}}$, act on
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}_{\epsilon}$.

Proof. (1) First note that for every
$\displaystyle\gamma\in\mathcal{S}(\Sigma)$,
$\displaystyle\textstyle\gamma=\gamma_{\partial}\prod_{i=1}^{k}\gamma_{i}^{c_{i}}$
with $\displaystyle\gamma_{\partial}$ peripheral, we have

(153)
$\mathcal{W}(T(\gamma))=\mathcal{W}(T(\gamma_{\partial}))\prod_{i=1}^{k}T_{c_{i}}(\mathcal{W}(\gamma_{i})).$

Denote by $\displaystyle\partial_{1},\ldots,\partial_{n+1}$ the boundary
components of $\displaystyle\Sigma$, ordered so that the diffeomorphism
identifying $\displaystyle\Sigma$ with the closure of
$\displaystyle[0,1]^{2}\setminus(\\{p_{1},\ldots,p_{n}\\}\times\\{1/2\\})$
maps $\displaystyle\partial_{n+1}$ to $\displaystyle\partial([0,1]^{2})$, and
$\displaystyle\partial_{i}$ to a small loop encircling
$\displaystyle(p_{i},1/2)$, $\displaystyle 1\leq i\leq n$. By (149) we have

(154) $\mathcal{W}(\partial_{n+1})=qTr\left(\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(1)}\ldots\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(n)}\right)=\eta,\
\mathcal{W}(\partial_{i})=qTr\left(\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(i)}\right)=\omega^{(i)},\ 1\leq i\leq n.$

Therefore the elements $\displaystyle\mathcal{W}(T(\gamma_{\partial}))$ form a
basis of $\displaystyle\mathbb{C}[\omega^{(1)},\ldots,\omega^{(n)},\eta]$,
which is a central subalgebra of
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}_{\epsilon}$ (see Theorem
6.24). We can give formulas of the other terms as follows. Deform a ribbon
neighborhood of $\displaystyle\gamma$ in $\displaystyle\Sigma\times[0,1]$ so
that for every $\displaystyle i\in\\{1,\ldots,k\\}$, $\displaystyle\gamma_{i}$
is represented by a one-coupon multicurve like in Figure 8.3, the coupon being
filled with $\displaystyle r$ simple arcs (possibly pairwise intersecting),
for some $\displaystyle r\in\mathbb{N}$. Since $\displaystyle\gamma_{i}$ is
colored by $\displaystyle V_{2}$, we see that (150) takes the form

(155) $\mathcal{W}(\gamma_{i})=qTr_{V_{2}^{\otimes
r}}\left(a(\gamma_{i})\mathbb{M}(\gamma_{i})\right)$

where $\displaystyle a(\gamma_{i})\in End_{U_{A}^{res}}(V_{2}^{\otimes r})$,
and $\displaystyle\mathbb{M}(\gamma_{i})$ is defined as
$\displaystyle\stackrel{{\scriptstyle[\lambda]}}{{\mathbb{M}}}$, replacing the
modules $\displaystyle V_{\lambda_{1}},\ldots,V_{\lambda_{n}}$ associated to
the sequence $\displaystyle[\lambda]=(\lambda_{1},\ldots,\lambda_{n})$ by
$\displaystyle V_{2}$ or the trivial module $\displaystyle V_{1}$ (this latter
case happens when the module labels a strand which has to be removed to get
the one-coupon multicurve representing $\displaystyle\gamma_{i}$). In order to
get simultaneously a simple expression of all the elements
$\displaystyle\mathcal{W}(\gamma_{1}),\ldots,\mathcal{W}(\gamma_{k})$, note
that each $\displaystyle\gamma_{i}$, being a simple closed curve, separates
$\displaystyle\Sigma$ in two disks with punctures. We can choose the above
diffeomorphism of $\displaystyle\Sigma$ with the closure of
$\displaystyle[0,1]^{2}\setminus(\\{p_{1},\ldots,p_{n}\\}\times\\{1/2\\})$ so
that the ordering of $\displaystyle p_{1},\ldots,p_{n}$ makes
$\displaystyle\gamma_{i}$ bounding a punctured disk $\displaystyle
D_{i}\subset[0,1]^{2}$ with successive punctures $\displaystyle
p_{i_{1}},p_{i_{1}+1},\ldots,p_{i_{1}+n_{i}-1}$. Then, for every
$\displaystyle i\in\\{1,\ldots,k\\}$ we have

(156) $\mathcal{W}(\gamma_{i})=qTr_{V_{2}}\left(\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(i_{1})}\ldots\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(i_{1}+n_{i}-1)}\right).$

The algebras
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}\otimes_{A}\mathbb{Z}[\zeta,\zeta^{-1}]$,
and therefore the expressions of $\displaystyle\mathcal{W}(\gamma_{i})$,
associated to different presentations of $\displaystyle\Sigma$ as above are
related by isomorphisms induced by the mapping class group of
$\displaystyle\Sigma$.

Now assume that $\displaystyle\gamma\in\mathcal{S}_{l}(\Sigma)$, so
$\displaystyle l$ divides $\displaystyle c_{i}$, $\displaystyle 1\leq i\leq
k$, in (153). Put $\displaystyle c^{\prime}_{i}:=c_{i}/l$. Then

$\displaystyle\displaystyle T_{c_{i}}\left(\mathcal{W}(\gamma_{i})\right)$
$\displaystyle\displaystyle=T_{c_{i}^{\prime}}\left(T_{l}\left(\mathcal{W}(\gamma_{i})\right)\right)$
(157)
$\displaystyle\displaystyle=T_{c_{i}^{\prime}}\left(Tr\left(Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(i_{1})}\ldots Fr\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(i_{1}+n_{i}-1)}\right)\right)$

where we use the standard identity $\displaystyle
T_{c_{i}}=T_{c_{i}^{\prime}}\circ T_{l}$ in the first equality, and
Proposition 7.7 in the second. By the First Fundamental Theorem of classical
invariant theory for $\displaystyle SL_{2}$ (see eg. [61]) and the Cayley-
Hamilton identity, the set of trace functions

$\displaystyle
t_{i_{1},\ldots,i_{1}+n_{i}-1}:(\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(1)},\ldots,\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(n)})\mapsto Tr\left(\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(i_{1})}\ldots\underline{\stackrel{{\scriptstyle
V_{2}}}{{M}}}{}^{(i_{1}+n_{i}-1)}\right)$

for all possible tuples $\displaystyle(i_{1},\ldots,i_{1}+n_{i}-1)$ are
generating functions of $\displaystyle\mathcal{O}(G^{n})^{G}$. By (157) they
are sent by $\displaystyle Fr$ to $\displaystyle
T_{l}\left(\mathcal{W}(\gamma_{i})\right)$ for some $\displaystyle\gamma_{i}$.
By Corollary 7.21 (3), $\displaystyle\widetilde{Fr}$ is an isomorphism from
$\displaystyle\mathcal{O}(\tilde{G}^{n})^{G}$ to
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}}$,
mapping $\displaystyle\mathcal{O}(G^{n})^{G}$ to
$\displaystyle\mathcal{Z}_{0}({\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}}$,
and the $\displaystyle n$ copies of $\displaystyle\mathcal{O}(T)^{W}$
associated to the factors of $\displaystyle\mathcal{O}(\tilde{G}^{n})$ to
$\displaystyle\mathbb{C}[\omega^{(1)}],\ldots,\mathbb{C}[\omega^{(n)}]\subset({\mathcal{L}}_{0,n}^{A})^{U_{A}}_{\epsilon}$.
Therefore
$\displaystyle\mathcal{Z}_{(}{\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}}$ is
spanned over $\displaystyle\mathbb{C}$ by the elements
$\displaystyle\mathcal{W}(T(\gamma))$ with
$\displaystyle\gamma\in\mathcal{S}_{l}(\Sigma)$ and
$\displaystyle\gamma_{\partial}$ a monomial in
$\displaystyle\partial_{1},\ldots,\partial_{n}$. In particular
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}}$ is a
(central) subalgebra of
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}_{\epsilon}$.

(2) It is enough to show that the derivations $\displaystyle D_{a}$,
$\displaystyle a\in\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}}$,
restrict to endomorphisms of
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}_{\epsilon}$. This follows from
the inclusion
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}}\subset({\mathcal{L}}_{0,n}^{A})^{U_{A}}_{\epsilon}$,
since $\displaystyle[\tilde{a},u]\in({\mathcal{L}}_{0,n}^{A})^{U_{A}}$ for
every $\displaystyle\tilde{a},u\in({\mathcal{L}}_{0,n}^{A})^{U_{A}}$
($\displaystyle{\mathcal{L}}_{0,n}^{A}$ being an $\displaystyle U_{A}$-module
algebra), which by the definition of $\displaystyle D_{a}$ implies that
$\displaystyle
D_{a}(({\mathcal{L}}_{0,n}^{A})^{U_{A}}_{\epsilon})\subset({\mathcal{L}}_{0,n}^{A})^{U_{A}}_{\epsilon}$
when $\displaystyle
a\in\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}}$.
$\displaystyle\Box$

######  Remark 8.7.

(Threading $\displaystyle T_{l}(V_{2})$) For every $\displaystyle 1\leq i\leq
n$, $\displaystyle 0\leq k\leq n-i$ we have

$\displaystyle T_{l}\left(qTr_{V_{2}}\left(\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(i)}\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(i+1)}\ldots\stackrel{{\scriptstyle
V_{2}}}{{M}}{}^{\\!\\!(i+k)}\right)\right)=qTr_{T_{l}(V_{2})}\left(\stackrel{{\scriptstyle
T_{l}(V_{2})}}{{M}}{}^{\\!\\!(i)}\stackrel{{\scriptstyle
T_{l}(V_{2})}}{{M}}{}^{\\!\\!(i+1)}\ldots\stackrel{{\scriptstyle
T_{l}(V_{2})}}{{M}}{}^{\\!\\!(i+k)}\right)$

where $\displaystyle T_{l}(V_{2})$ is the virtual representation in the
Grothendieck ring of $\displaystyle U_{A}^{res}$-modules, obtained by plugging
$\displaystyle V_{2}$ in the $\displaystyle l$-th Chebyshev polynomial
$\displaystyle T_{l}$. Indeed

$\displaystyle T_{l}\left(qTr\left(\stackrel{{\scriptstyle
V_{2}}}{{M}}\right)\right)=qTr\left(\\!\stackrel{{\scriptstyle\
T_{l}(V_{2})}}{{M}}\right)$

because (using eg. the second picture in Figure 6.2)

$\displaystyle qTr\left(\stackrel{{\scriptstyle\ V_{2}^{\otimes k}}}{{M\
}}\right)=\left(qTr\left(\\!\stackrel{{\scriptstyle
V_{2}}}{{M}}\right)\right)^{k}.$

This and the computations in the first half of the proof of Proposition 7.7
prove our claim.

######  Remark 8.8.

By a result of [46], the set
$\displaystyle\\{T(\gamma),\gamma\in\mathcal{S}_{l}(\Sigma)\\}$ is a
$\displaystyle\mathbb{C}$-basis of the center of $\displaystyle
K_{\epsilon^{\prime}}(\Sigma)$. Therefore the proof of (1) shows that
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}}$ and
$\displaystyle\eta$ generate
$\displaystyle\mathcal{Z}(({\mathcal{L}}_{0,n}^{A})^{U_{A}}_{\epsilon})$.

######  Remark 8.9.

As in the proof of (1), denote by
$\displaystyle\partial_{1},\ldots,\partial_{n+1}$ the boundary components of
$\displaystyle\Sigma$, and fix the generators of
$\displaystyle\pi_{1}(\Sigma)$ to be
$\displaystyle\partial_{1},\ldots,\partial_{n}$ (for some choice of
basepoints). Consider the fiber product (with the notations of Remark 7.17)

$\displaystyle\tilde{G}^{n}\times_{G/\\!/G}G/\\!/G=\left\\{((\tilde{g}_{1},\ldots,\tilde{g}_{n}),[t])\in\tilde{G}^{n}\times
G/\\!/G\ |\ p(g_{1}\ldots g_{n})=p_{l}([t])\right\\}.$

As before Corollary 7.21, one can identify
$\displaystyle\tilde{G}^{n}\times_{G/\\!/G}G/\\!/G$ with an $\displaystyle
l$-fold branched covering space $\displaystyle\tilde{R}_{G}(\Sigma)$ of
$\displaystyle\tilde{R}_{G}^{\prime}(\Sigma)$, whose points are given by
representations $\displaystyle\rho\colon\pi_{1}(\Sigma)\rightarrow G$ endowed
with a choice of solution $\displaystyle x\in\mathbb{C}$ of the equation
$\displaystyle T_{l}(x)=Tr(\rho(\partial_{i}))$, for every $\displaystyle
i=1,\ldots,n$ and $\displaystyle i=n+1$. Taking algebraic quotients yields a
branched covering map $\displaystyle\tilde{X}_{G}(\Sigma)\rightarrow
X_{G}(\Sigma)$ of degree $\displaystyle l^{n+1}$, and one can lift
$\displaystyle\widetilde{Fr}\circ\mathfrak{c}^{*-1}$ to an isomorphism from
$\displaystyle\mathcal{O}(\tilde{X}_{G}(\Sigma))$ onto the algebra generated
by $\displaystyle\eta$ and
$\displaystyle\mathcal{Z}({\mathcal{L}}_{0,n}^{\epsilon})^{\mathcal{G}}$.

##  9\. Applications to skein algebras

In this Section we use Theorem 8.4 to reformulate some of our results on
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}_{\epsilon}$, in the case of
$\displaystyle\mathfrak{g}=sl(2)$.

In [70] it was proved that $\displaystyle K_{\zeta}(\Sigma)$ does not have non
trivial zero divisors, and its center was computed by topological means. On
the contrary the proof we give below of these two facts is purely algebraic,
based on properties of $\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}$ proved
in Section 6, that hold true for any complex finite dimensional simple Lie
algebra $\displaystyle\mathfrak{g}$.

######  Corollary 9.1.

The skein algebra $\displaystyle K_{\zeta}(\Sigma)$ does not have non trivial
zero divisors, and its center is the polynomial algebra over
$\displaystyle\mathbb{Z}[\zeta,\zeta^{-1}]$ generated by the classes
$\displaystyle\partial_{1},\ldots,\partial_{n+1}$ of the boundary components
of $\displaystyle\Sigma$.

Proof. The claims are direct consequences of Theorem 8.4, the formulas (154),
and Proposition 6.9 and Theorem 6.24 in the case of
$\displaystyle\mathfrak{g}=sl(2)$ (the basis of
$\displaystyle{\mathcal{L}}_{0,n}^{U_{q}}$ described there being indeed a
basis of its $\displaystyle A$-sublattice
$\displaystyle({\mathcal{L}}_{0,n}^{A})^{U_{A}}$). $\displaystyle\Box$

Next we deduce from the results of Section 7 some properties of the center of
the specializations of $\displaystyle K_{\zeta}(\Sigma)$ at roots of unity of
order $\displaystyle 4l$, $\displaystyle l\geq 3$ odd. Let
$\displaystyle\epsilon^{\prime}$ be such that
$\displaystyle(\epsilon^{\prime})^{2}=-\epsilon$. Define

$\displaystyle
K_{\epsilon^{\prime}}(\Sigma):=K_{\zeta}(\Sigma)\otimes_{\mathbb{Z}[\zeta,\zeta^{-1}]}\mathbb{C}_{\epsilon^{\prime}}$

where $\displaystyle\mathbb{C}_{\epsilon^{\prime}}=\mathbb{C}$ as a vector
space, and as an $\displaystyle A^{\prime}$-module, the action of
$\displaystyle\zeta$ on $\displaystyle\mathbb{C}_{\epsilon^{\prime}}$ is by
multiplication by $\displaystyle\epsilon^{\prime}$. Note that
$\displaystyle\epsilon^{\prime}$ is a primitive root of unity of order
$\displaystyle 4l$, and that $\displaystyle\mathcal{W}$ yields an isomorphism
of algebras

$\displaystyle\mathcal{W}\colon
K_{\epsilon^{\prime}}(\Sigma)\rightarrow({\mathcal{L}}_{0,n}^{A})^{U_{A}}_{\epsilon}.$

Since $\displaystyle K_{\zeta}(\Sigma)$ is a free
$\displaystyle\mathbb{Z}[\zeta,\zeta^{-1}]$-module, the sets
$\displaystyle\mathcal{S}(\Sigma)$ and
$\displaystyle\\{T(\gamma),\gamma\in\mathcal{S}(\Sigma)\\}$ are
$\displaystyle\mathbb{C}$-basis of $\displaystyle
K_{\epsilon^{\prime}}(\Sigma)$.

Denote by $\displaystyle\mathcal{Z}(K_{\epsilon^{\prime}}(\Sigma))$ the center
of $\displaystyle K_{\epsilon^{\prime}}(\Sigma)$. Theorem 8.4 and Theorem 8.6
(1) imply immediately:

######  Corollary 9.2.

The set $\displaystyle\\{T(\gamma),\gamma\in\mathcal{S}_{l}(\Sigma)\\}$ is a
$\displaystyle\mathbb{C}$-basis of a central subalgebra
$\displaystyle\mathcal{Z}^{\prime}(K_{\epsilon^{\prime}}(\Sigma))$ of
$\displaystyle\mathcal{Z}(K_{\epsilon^{\prime}}(\Sigma))$, which is generated
by $\displaystyle\eta$ and the image of the embedding

$\displaystyle
Ch_{\mathcal{W}}:=\mathcal{W}^{-1}\circ\widetilde{Fr}\circ\tilde{\mathfrak{c}}^{*-1}\colon\mathcal{O}(\tilde{X}_{G}^{\prime}(\Sigma))\rightarrow\mathcal{Z}(K_{\epsilon^{\prime}}(\Sigma)).$

Therefore $\displaystyle\mathcal{Z}^{\prime}(K_{\epsilon^{\prime}}(\Sigma))$
is endowed with a natural Poisson bracket, the image of $\displaystyle\\{\ ,\
\\}_{Gold}$, which extends to an action by derivations of
$\displaystyle\mathcal{Z}^{\prime}(K_{\epsilon^{\prime}}(\Sigma))$ on
$\displaystyle K_{\epsilon^{\prime}}(\Sigma)$.

In fact
$\displaystyle\mathcal{Z}^{\prime}(K_{\epsilon^{\prime}}(\Sigma))=\mathcal{Z}(K_{\epsilon^{\prime}}(\Sigma))$
(see [46]). One can check that $\displaystyle Ch_{\mathcal{W}}$ is a version
of the threading map $\displaystyle
Ch\colon\mathcal{K}_{\epsilon^{{}^{\prime}l^{2}}}(\Sigma)\rightarrow\mathcal{Z}_{\epsilon^{\prime}}(\Sigma)$
of Bonahon-Wong (see [19], and also [47]). The point is that it affords an
explicit realization of $\displaystyle Ch$ in classical invariant theory terms
(via the formulas (153), (154) and (157)), and pulls the geometric tools of
Section 7 onto $\displaystyle K_{\epsilon^{\prime}}(\Sigma)$. A more symmetric
statement is obtained by extending $\displaystyle Ch_{\mathcal{W}}$ to the
ring $\displaystyle\mathcal{O}(\tilde{X}_{G}(\Sigma))$ of Remark 8.9, so that
$\displaystyle\partial_{n+1}$ belongs to its image.

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