Classification of semi-weight representations of reduced stated skein algebras]Classification of semi-weight representations of reduced stated skein algebras

We classify the finite dimensional semi-weight representations of the  reduced stated skein algebras at odd roots of unity of connected essential marked surfaces which either have a boundary component with at least two boundary edges or which do not have any unmarked boundary component.  We deduce computations of the PI-degrees and Azumaya loci of unreduced stated skein algebras of essential marked surfaces having at most one boundary puncture per boundary component and of the  of lattice gauge field theory.

§ INTRODUCTION

§.§ Background on reduced stated skein algebras and their representations

Let $\Sigma$ be an oriented compact surface and $\mathcal{A}$ a finite set of disjoint arcs embedded in $\partial \Sigma$ named boundary edges. The pair $\mathbf{\Sigma}:=(\Sigma, \mathcal{A})$ will be called a marked surface.
For $A\in \mathbb{C}^*$ a root of unity of odd order $N$, the reduced stated skein algebra $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ was introduced in [33] as the quotient of the stated skein algebra by the kernel of Bonahon-Wong quantum trace. In particular, when $\mathcal{A}=\emptyset$, $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ is the usual Kauffman-bracket skein algebra.
Its representations appear in quantum hyperbolic geometry and are conjectured to form the building blocks of some $\SL_2$-HQFT (see [10, 11, 13, 12, 14]). Let $Z_{\mathbf{\Sigma}}$ denote the center of  $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$. By [61, 66] (after the original work in [28]), there exists a Frobenius morphism

$$ Fr_{\mathbf{\Sigma}} : \overline{\mathcal{S}}_{+1}(\mathbf{\Sigma}) \hookrightarrow Z_{\mathbf{\Sigma}}$$

which embeds the commutative algebra $\overline{\mathcal{S}}_{+1}(\mathbf{\Sigma}) $ at $A=+1$ into $Z_{\mathbf{\Sigma}}$. Let $Z_{\mathbf{\Sigma}}^0 \subset Z_{\mathbf{\Sigma}}$ denote the image of the Frobenius morphism.
A representation $r: \overline{\mathcal{S}}_A(\mathbf{\Sigma})\to \End(V)$ is a weight representation if $V$ is semi-simple as a module over $Z_{\mathbf{\Sigma}}$ and is called a semi-weight representation if $V$ is semi-simple as a $Z_{\mathbf{\Sigma}}^0$ module. The purpose of this paper is to make progresses towards the following problem.

Classify all finite dimensional weight and semi-weight representations of  $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$.

We will solve Problem <ref> in the case where $\mathbf{\Sigma}=(\Sigma, \mathcal{A})$ is a connected marked surface which either has a boundary component with at least two boundary edges or which has at least one boundary edge and no unmarked boundary component.

The center $Z_{\mathbf{\Sigma}}$ was computed in [58] and is described as follows.
Partition the set of boundary components of $\Sigma$ into two subsets $\pi_0(\partial \Sigma)= \mathring{\mathcal{P}} \bigsqcup \Gamma^{\partial}$ where
$\mathring{\mathcal{P}}$ is the subset of boundary components  which do not intersect $\mathcal{A}$ and $\Gamma^{\partial}$ the boundary components which contain some boundary edges. For each $p\in \mathring{\mathcal{P}}$, there is a central element $\gamma_p \in Z_{\mathbf{\Sigma}}$ and for each $\partial \in \Gamma^{\partial}$ there is an invertible central element $\alpha_{\partial} \in Z_{\mathbf{\Sigma}}$ such that $Z_{\mathbf{\Sigma}}$ is generated by the image of the Frobenius morphism together with the elements $\gamma_p$ and $\alpha_{\partial}^{\pm 1}$. Let $\widehat{X}(\mathbf{\Sigma}):=\Specm(Z_{\mathbf{\Sigma}})$,  $X(\mathbf{\Sigma}):=\Specm(Z_{\mathbf{\Sigma}}^0)$,  and $\pi: \widehat{X}(\mathbf{\Sigma})\to X(\mathbf{\Sigma})$ the map defined by the inclusion $Z_{\mathbf{\Sigma}}^0 \subset Z_{\mathbf{\Sigma}}$. We also denote by $\widehat{\mathcal{X}}(\mathbf{\Sigma})\subset\Spec(Z_{\mathbf{\Sigma}})$ the set of prime ideals $I\subset Z_{\mathbf{\Sigma}}$ such that $I\cap Z^0_{\mathbf{\Sigma}}$ is maximal in $Z^0_{\mathbf{\Sigma}}$ so we still have a projection $\widehat{\pi}: \widehat{\mathcal{X}}(\mathbf{\Sigma}) \to X(\mathbf{\Sigma})$ sending $I$ to $I\cap Z^0_{\mathbf{\Sigma}}$ extending $\pi$.
For a point $\widehat{x} \in \widehat{X}(\mathbf{\Sigma})$ (i.e. a maximal ideal in $Z_{\mathbf{\Sigma}}$), we denote by $\chi_{\widehat{x}} : Z_{\mathbf{\Sigma}} \to \mathbb{C}$ the corresponding character with kernel $\widehat{x}$ and use similar notations for $X(\mathbf{\Sigma})$. The variety  $\widehat{X}(\mathbf{\Sigma})$ is then described as

\begin{multline*}  \widehat{X}(\mathbf{\Sigma}) := \{ \widehat{x}= (x, h_p, h_{\partial})_{p,\partial}, x\in X(\mathbf{\Sigma}),
h_p\in \mathbb{C} \mbox{ is s.t. } T_N(h_p)=\chi_x(\gamma_p) \mbox{ for }p\in \mathring{P},
\\ h_{\partial} \in \mathbb{C}^* \mbox{is s.t. } h_{\partial}^N = \chi_x(\alpha_{\partial}) \mbox{ for }\partial \in \Gamma^{\partial} \}.\end{multline*}

Here $T_N(X)$ denotes the $N$-th Chebyshev polynomial of the first kind. The morphism $\pi$ is just the projection $\pi(x, h_p, h_{\partial})= x$. It was also proved in [58] that $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ is finitely generated over $Z_{\mathbf{\Sigma}}$. Moreover, if $Q(Z_{\mathbf{\Sigma}})$ denotes the fraction field of $Z_{\mathbf{\Sigma}}$, the dimension $D=D_{\mathbf{\Sigma}}:= \dim_{Q(Z_{\mathbf{\Sigma}})} \overline{\mathcal{S}}_A(\mathbf{\Sigma})\otimes_{Z_{\mathbf{\Sigma}}} Q(Z_{\mathbf{\Sigma}})$, named $\textit{PI-degree}$,  was computed explicitly in  [58].
For $\widehat{x} \in \widehat{\mathcal{X}}(\mathbf{\Sigma})$ write

$$ \overline{\mathcal{S}}_A(\mathbf{\Sigma})_{\widehat{x}}:= \quotient{ \overline{\mathcal{S}}_A(\mathbf{\Sigma})}{{\widehat{x}\overline{\mathcal{S}}_A(\mathbf{\Sigma})}}.$$

Let $r: \overline{\mathcal{S}}_A(\mathbf{\Sigma})\to \End(V)$ be an indecomposable semi-weight representation.
It classical shadow is the maximal ideal $x_r:= \ker(r)\cap Z_{\mathbf{\Sigma}}^0 \in X(\mathbf{\Sigma})$. Its full shadow is the prime ideal $\mathcal{I}_r:=\ker(r)\cap Z_{\mathbf{\Sigma}}\in \widehat{\mathcal{X}}(\mathbf{\Sigma})$. It maximal shadow is the unique maximal ideal $\widehat{x}_r \in \widehat{X}(\mathbf{\Sigma})$ containing $\mathcal{I}_r$ (see Theorem <ref> for a proof that $\mathcal{I}_r$ is prime and that $\widehat{x}_r$ is unique).
Note that
$r$ is a weight representation if and only if $\mathcal{I}_r=\widehat{x_r}\in \widehat{X}(\mathbf{\Sigma})$ is maximal.
The Azumaya locus of $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ is

$$ \mathcal{AL}(\mathbf{\Sigma}):= \{ \widehat{x} \in \widehat{X}(\mathbf{\Sigma})\mbox{ such that } \overline{\mathcal{S}}_A(\mathbf{\Sigma})_{\widehat{x}}\cong \Mat_D(\mathbb{C}) \}$$

and the fully Azumaya locus is

$$ \mathcal{FAL}(\mathbf{\Sigma}):= \{ x\in X(\mathbf{\Sigma}) \mbox{ such that } \pi^{-1}(x)\subset \mathcal{AL}(\mathbf{\Sigma})\}.$$

Since the matrix algebra $\Mat_D(\mathbb{C}) $ has a unique indecomposable representation (up to isomorphism) which is irreducible, for each $\widehat{x} \in \widehat{X}(\mathbf{\Sigma})$ there exists a unique indecomposable weight representation with classical shadow $\widehat{x}$ (up to isomorphism) named the Azumaya representation at  $\widehat{x}$.
A marked surface $\mathbf{\Sigma}=(\Sigma, \mathcal{A})$ is said essential if each of its connected component contains at least one boundary edge in $\mathcal{A}$.
In this case, it was proved  in [61] that the variety $X(\mathbf{\Sigma})$ admits a geometric interpretation as follows. For each boundary edge $a\in \mathcal{A}$, fix a point $v_a \in a$ and write $\mathbb{V}:=\{v_a, a\in \mathcal{A}\}$. Let $\Pi_1(\Sigma, \mathbb{V})$ denote the full subcategory of the fundamental groupoid $\Pi_1(\Sigma)$ generated by $\mathbb{V}$; so the set of objects of $\Pi_1(\Sigma, \mathbb{V})$ is $\mathbb{V}$ and a morphism $\alpha : v_a \to v_b$ is an homotopy class of a path starting at $v_a$ and ending at $v_b$. Let $\SL_2$ be the category with a single element whose endomorphism set is $\SL_2(\mathbb{C})$ and let $\mathcal{R}_{\SL_2}(\mathbf{\Sigma})$ denote the set of functors $\rho : \Pi_1(\Sigma, \mathbb{V})\to \SL_2$. In  [61, 33] it is proved that $X(\mathbf{\Sigma})$ embeds into $\mathcal{R}_{\SL_2}(\mathbf{\Sigma})$ (see Section <ref> for details).
For $x\in X(\mathbf{\Sigma})$, let $\rho_x$ be the associated functor.
For $p\in \mathring{\mathcal{P}}$, let $\gamma_p \in \Pi_1(\Sigma, \mathbb{V})$ be the homotopy class of a peripheral curve encircling $p$ once. Then $x$ is called central at  $p$ if $\rho_x(\gamma_p)=\pm \mathds{1}_2$. This property is clearly independent on the choices of $a$ and $\gamma_p$.

It was proved in [39] for unmarked surfaces and in [58] for marked surfaces that $\mathcal{AL}(\mathbf{\Sigma})$ is open dense so a generic indecomposable weight representation is irreducible and determined by its full shadow. However the computation of the Azumaya locus remains a challenging problem. For closed (and thus unmarked) surfaces, the Azumaya loci was computed in [53] using the original results in [45].
For open unmarked surfaces, many partial results were found in [40, 53, 78].
For $\mathbf{\Sigma}=(\Sigma_{g,1}, \{a\})$ a genus $g$ surface with a single boundary component and a single boundary edge, it was proved in [59] that $ \overline{\mathcal{S}}_{A}(\mathbf{\Sigma})$ is Azumaya, i.e. that $\mathcal{AL}(\mathbf{\Sigma})=\widehat{X}(\mathbf{\Sigma})$. For $\mathbf{\Sigma}=\mathbb{T}:=(\mathbb{D}^2, \{a,b,c\})$ a disc with three boundary edges (usually called the triangle), it was showed in [33] that $ \overline{\mathcal{S}}_{A}(\mathbb{T})$ is isomorphic to a quantum torus, thus it is Azumaya as well.

§.§ Main results

In order to state our main theorem, let us introduce some notations. Write $\Delta_+(\widehat{\mathfrak{sl}}_2):= \{ (n,m) \in \mathbb{N}^2 | (n-m)^2 \leq 1\}$ and decompose it as
$\Delta_+(\widehat{\mathfrak{sl}}_2)= \{(0,0), (0,1), (1,0)\} \bigsqcup \Delta^{\Re}_{++}(\widehat{\mathfrak{sl}}_2)\bigsqcup \Delta^{Im}_+(\widehat{\mathfrak{sl}}_2)$ where

$$  \Delta^{\Re}_{++}(\widehat{\mathfrak{sl}}_2)= \{ (k, k+1), k\geq1\} \cup \{(k+1, k), k\geq 1\}, \quad \Delta^{Im}_+(\widehat{\mathfrak{sl}}_2)=\{ (k,k), k\geq 1\}.$$

Further write

$$ \Delta:= \{S, P\} \sqcup   \Delta^{\Re}_+(\widehat{\mathfrak{sl}}_2) \sqcup  \left(\Delta^{Im}_+(\widehat{\mathfrak{sl}}_2)\times \mathbb{CP}^1\right), $$

where $S,P$ are some formal parameters and write $\Delta \sqcup \overline{\Delta}:= \Delta \times \{0,1\}$ (two copies of $\Delta$ where we denote an element $(\alpha, 0)$ simply by $\alpha$ and an element $(\alpha, 1)$ by $\overline{\alpha}$). Let $\widehat{x}=(x,h_p,h_{\partial})_{p, \partial} \in \widehat{X}(\mathbf{\Sigma})$ and decompose the set of inner punctures as $\mathring{\mathcal{P}}= \mathring{\mathcal{P}}_0 \sqcup \mathring{\mathcal{P}}_1 \sqcup \mathring{\mathcal{P}}_2$ where

\begin{align*}
&  \mathring{\mathcal{P}}_2:= \{p\in \mathring{\mathcal{P}}, \mbox{ such that } x \mbox{ is central at }p \mbox{ and }h_p \neq \pm 2\} \\
&  \mathring{\mathcal{P}}_1:= \{p\in \mathring{\mathcal{P}}\setminus \mathring{\mathcal{P}}_2, \mbox{ such that } \chi_x(\gamma_p) = \pm 2 \mbox{ and }h_p \neq \pm 2\}.
\end{align*}

A map $\sigma : \mathring{\mathcal{P}}\to \Delta \bigsqcup \overline{\Delta} $ is called a coloring compatible with $\widehat{x}$ if $(1)$ for $p\in \mathring{\mathcal{P}}_0$, then $\sigma(p)=S$ and $(2)$ if $p\in \mathring{\mathcal{P}}_1$, then $\sigma(p)\in \{S,P\}$. By convention, if $\mathring{\mathcal{P}}=\emptyset$, we consider that every $\widehat{x}$ has a unique coloring. We denote by $\col(\widehat{x})$ the set of colorings compatible with $\widehat{x}$.
We write
$ m=m(\widehat{x}):= | \mathring{\mathcal{P}}_2|$.
Let $\widehat{X}^{reg}(\mathbf{\Sigma})$ denote the regular part (smooth points) of $\widehat{X}(\mathbf{\Sigma})$. Let  $\overline{\mathcal{C}}$ be the category of semi-weight modules. Let $\Indecomp(\mathbf{\Sigma})$ be the set of isomorphism classes of indecomposable semi-weight $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ modules and let $\Indecomp(\mathbf{\Sigma}, \widehat{x})\subset \Indecomp(\mathbf{\Sigma})$ the subset of these modules with maximal shadow $\widehat{x}$.
The main result of this paper is the following

Let $\mathbf{\Sigma}=(\Sigma, \mathcal{A})$ be a connected essential marked surface  which either has a boundary component with at least two boundary edges or which does not have any inner puncture.
Let  $\widehat{x}=(x, h_p, h_{\partial}) \in \widehat{X}(\mathbf{\Sigma})$.

* We have $\mathcal{AL}(\mathbf{\Sigma})= \widehat{X}^{reg}(\mathbf{\Sigma})$, i.e. the Azumaya locus of $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ is equal to the regular locus of $\widehat{X}(\mathbf{\Sigma})$. Moreover
$\widehat{x}\in \mathcal{AL}(\mathbf{\Sigma})$ if and only if $m(\widehat{x})=0$.

* The fully Azumaya locus $\mathcal{FAL}(\mathbf{\Sigma})$ of  $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ is the locus of representations which are not central at any inner puncture. For $x\in \mathcal{FAL}(\mathbf{\Sigma})$ the indecomposable semi-weight representations with classical shadow $x$ are classified (up to isomorphism) by their full shadows, so their set  is in $1:1$ correspondence with the fiber $\widehat{\pi}^{-1}(x) \subset \widehat{\mathcal{X}}(\mathbf{\Sigma})$.

* The indecomposable semi-weight representations with maximal shadow  $\widehat{x}$ are in $1:1$ correspondence with the set of colorings compatible with $\widehat{x}$ so one has

$$ \Indecomp(\mathbf{\Sigma})= \bigsqcup_{\widehat{x}\in \widehat{X}(\mathbf{\Sigma})} \Indecomp(\mathbf{\Sigma}, \widehat{x}) \stackrel{1:1}{\longleftrightarrow} \bigsqcup_{\widehat{x}\in \widehat{X}(\mathbf{\Sigma})} \col(\widehat{x}).$$

* The indecomposable weight representations with full shadow  $\widehat{x}$ correspond to the colorings taking values in $\{S,\overline{S}, ((1,1), 1), \overline{((1,1), 1)}\}$: there are thus $4^m$ such representations.

* The irreducible representations with full shadow  $\widehat{x}$ correspond to the colorings taking values in $\{S, \overline{S}\}$: there are thus $2^m$ such representations.

* The indecomposable semi-weight representations with maximal shadow $\widehat{x}$ which are projective objects in $\overline{\mathcal{C}}$ correspond to the colorings sending the elements of $\mathring{\mathcal{P}}_1$ to $P$ and the elements of $\mathring{\mathcal{P}}_2$ to an element in $\{P, \overline{P}\}$: there are thus $2^m$ such representations.

We will construct explicitly the  indecomposable semi weight representations of Theorem <ref>, so it solves Problem <ref> for these marked surfaces.

The proof of Theorem <ref> uses a preliminary result which is interesting on its own.  For $\mathbf{\Sigma}_1, \mathbf{\Sigma}_2$ two  marked surfaces and $a_i$ a boundary edges of $\mathbf{\Sigma}_i$ ($i=1,2$), we denote by $\mathbf{\Sigma}_1\cup_{a_1\# a_2} \mathbf{\Sigma}_2$ the marked surface obtained by gluing $a_1$ and $a_2$ together. There exists an injective algebra morphism, named the splitting morphism

$$ \theta_{a_1 \# a_2}: \overline{\mathcal{S}}_A(\mathbf{\Sigma}_1\cup_{a_1 \# a_2} \mathbf{\Sigma}_2) \hookrightarrow \overline{\mathcal{S}}_A(\mathbf{\Sigma}_1)\otimes \overline{\mathcal{S}}_A(\mathbf{\Sigma}_2).$$

If $V_1$ and $V_2$ are modules over  $\overline{\mathcal{S}}_A(\mathbf{\Sigma}_1)$ and $ \overline{\mathcal{S}}_A(\mathbf{\Sigma}_2)$, then by precomposing with $ \theta_{a_1 \# a_2}$,
$V_1\otimes V_2$ becomes a module over $\overline{\mathcal{S}}_A(\mathbf{\Sigma}_1\cup_{a_1 \# a_2} \mathbf{\Sigma}_2)$.

Let  $\mathbf{\Sigma}_1$ and $\mathbf{\Sigma}_2$ be marked surfaces and for $i=1,2$ let $a_i$ be a boundary edge of $\mathbf{\Sigma}_i$ such that the connected component of $\partial \Sigma_i$ which contains $a_i$ also contains at least another boundary edge.
Write $\mathbf{\Sigma}:=\mathbf{\Sigma}_1 \cup_{a_1\# a_2} \mathbf{\Sigma}_2$.
Then  any indecomposable (semi) weight $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$-module is isomorphic to a module $V_1\otimes V_2$ with $V_i$ a (semi) weight indecomposable $ \overline{\mathcal{S}}_A(\mathbf{\Sigma}_i)$ module. Conversely, any such $ \overline{\mathcal{S}}_A(\mathbf{\Sigma})$ module $V_1\otimes V_2$ is (semi) weight indecomposable.

The proof of Theorem <ref> goes as follows. A 2P marked surface is a connected marked surface which has at least two boundary edges and no unmarked boundary component.
When $\mathbf{\Sigma}$ is 2P marked surfaces, it follows from the work of Müller [70] and Lê-Yu [68] that $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ is isomorphic to a quantum cluster algebra. The Azumaya loci of quantum cluster algebras have been studied in [74] and we will deduce from this study that $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ is Azumaya.
For $\mathbf{\Sigma}=\mathbb{D}_1:=(\Sigma_{0,2}, \{a,b\})$ an annulus with two boundary arcs in the same boundary component (also called punctured bigon), it was proved in [55] that $ \overline{\mathcal{S}}_{A}(\mathbb{D}_1) \cong \Dq$ is isomorphic to the Drinfel'd double of the quantum Borel algebra whose representation theory is very similar to the one of $U_q\mathfrak{sl}_2$.
The simple representations of $U_q\mathfrak{sl}_2$ were classified in [35, 7], the finite dimensional indecomposable representations of the small quantum group $\mathfrak{u}_q\mathfrak{sl}_2$ were classified in [76],  and many more indecomposable representations were found in [8]. However, at the authors knowledge, the indecomposable $U_q\mathfrak{sl}_2$ modules have never been classified. We will provide a classification of all indecomposable semi-weight $\Dq$ modules and prove that
Theorem <ref>  holds for $\mathbb{D}_1$. If  $\mathbf{\Sigma}$ is a connected marked surface which has a boundary component with at least two boundary edges, it can be obtained by gluing several copies of $\mathbb{D}_1$ to a 2P marked surface so Theorem <ref> will follow from Theorem <ref>. When $\mathbf{\Sigma}$ is an essential marked surface without inner puncture, it is either a 2P marked surface or is has a single boundary arc in which case its reduced stated skein algebra is Azumaya by [59].

The techniques of the present paper can also be used to study the representations of the unreduced stated skein algebras. Let $\mathbf{\Sigma}$ be an essential marked surface and let $\mathbf{\Sigma}^*$ be the same marked surface where each boundary edge has been replaced by a pair of boundary edges inside the same boundary component. Lê and Yu constructed in [68] an embedding $j: \mathcal{S}_A(\mathbf{\Sigma}) \hookrightarrow \overline{\mathcal{S}}_A(\mathbf{\Sigma}^*)$ of the unreduced stated skein algebra of $\mathbf{\Sigma}$ into the reduced stated skein algebra of $\mathbf{\Sigma}^*$. Therefore every module over $\overline{\mathcal{S}}_A(\mathbf{\Sigma}^*)$ is also a module over $\mathcal{S}_A(\mathbf{\Sigma})$ so Theorem <ref> provides a large family  of representations for the unreduced stated skein algebras as well. We will deduce the following:

Let $\mathbf{\Sigma}=(\Sigma_{g,n}, \mathcal{A})$ be a connected essential marked surface of genus $g$ with $n$ boundary components such each boundary component contains at most one boundary edge. So it has $k:= n-|\mathcal{A}|$ inner punctures $p_1, \ldots, p_k$ and $|\mathcal{A}|$ boundary punctures $p_{\partial_1}, \ldots, p_{\partial_{|\mathcal{A}|}}$.

* The center $\underline{Z}_{\mathbf{\Sigma}}$ of $\mathcal{S}_A(\mathbf{\Sigma})$ is generated by the image $\underline{Z}^0_{\mathbf{\Sigma}}$ of the Frobenius morphism together with the peripheral curves $\gamma_{p_1}, \ldots, \gamma_{p_k}$. So the closed points of $\widehat{\underline{X}}(\mathbf{\Sigma}):= \Specm(\underline{Z}_{\mathbf{\Sigma}})$ are in $1:1$ correspondence with the set of elements $\widehat{\rho}=(\rho, h_{p_1}, \ldots, h_{p_k})$ with $\rho: \Pi_1(\Sigma, \mathbb{V})\to \SL_2$ and $h_{p_i}\in \mathbb{C}$ is such that $T_N(h_{p_i})=-\tr(\rho(\gamma_{p_i}))$.

* The PI-degree $\underline{D}_{\mathbf{\Sigma}}$ of  $\mathcal{S}_A(\mathbf{\Sigma})$ is equal to $N^{3g-3+n+2|\mathcal{A}|}(= D_{\mathbf{\Sigma}^*})$.

* For $\rho: \Pi_1(\Sigma, \mathbb{V})\to \SL_2$, write $\mu(\rho):= (\rho(\alpha(p_{\partial_1})), \ldots, \rho(\alpha(p_{\partial_{|\mathcal{A}|}}))) \in (\SL_2)^{|\mathcal{A}|}$.
Let  $\widehat{\rho}=(\rho, h_{p_i}) \in \widehat{\underline{X}}(\mathbf{\Sigma})$.

(i) If $\mu(\rho)\in (\SL_2^0)^{|\mathcal{A}|}$ and for each $1\leq i \leq k$, either $\tr(\rho(\gamma_{p_i}))\neq \pm 2$ or $\tr(\rho(\gamma_{p_i}))=\pm 2$ and $h_{p_i}=\mp 2$, then $\widehat{\rho}$ belongs to the Azumaya locus of $\mathcal{S}_A(\mathbf{\Sigma})$.

(ii) Suppose that either $\mu(\rho)\in (\SL_2^1)^{|\mathcal{A}|}$ or that $\mu(\rho)\in (\SL_2^0)^{|\mathcal{A}|}$ and there exists $i$ such that $\tr(\rho(\gamma_{p_i}))=\pm 2$ and $h_{p_i}\neq \mp 2$, then $\widehat{\rho}$ does not belong to the Azumaya locus of $\mathcal{S}_A(\mathbf{\Sigma})$.

In this theorem, we have considered the simple Bruhat decomposition $\SL_2= \SL_2^0 \sqcup \SL_2^1$ where $\SL_2^0$ is the subset of matrices $\begin{pmatrix} a & b \\ c & d\end{pmatrix}$ such that $a\neq 0$ and $\SL_2^1$ the set of such matrices with $a=0$. Soon after the prepublication of the present paper, Yu computed in [77] the center and PI-degrees of every stated skein algebras, thus generalizing the first and second items of Theorem <ref>.
In the particular case where $|\mathcal{A}|=1$, i.e. when $\mathbf{\Sigma}=(\Sigma_{g,n+1}, \{a\})$ is a genus $g$ surface with $n+1$ boundary components and a single boundary arc, the algebra $\mathcal{L}_{g,n}:= \mathcal{S}_A(\mathbf{\Sigma})$ is called the  in lattice gauge field theory and first  appeared in the work of Buffenoir-Roche and Alekseev-Grosse-Schomerus ([1, 2, 9, 23, 24]). For these algebras, Theorem <ref> specializes to

* The center of $\mathcal{L}_{g,n}$ is generated by the image of the Frobenius morphism and the peripheral curves $\gamma_{p_1}, \ldots, \gamma_{p_n}$.

* The PI-degree of  $\mathcal{L}_{g,n}$ is $N^{3g+n}$.

* The Azumaya locus of $\mathcal{L}_{g,n}$ is the locus of elements $\widehat{\rho}=(\rho, h_{p_i})$ such that $\mu(\rho)\in \SL_2^0$ and $\tr(\rho(\gamma_{p_i}))=\pm 2 \Rightarrow h_{p_i}=\mp 2$.

When $n=0$, Corollary <ref> was proved in <cit.>. When $g=0$, the first point of Corollary <ref> was proved in <cit.>, while the second point was proved in <cit.>. S.Baseilhac and P.Roche have informed the authors that an alternative proof of the whole Corollary <ref> and a generalization for arbitrary gauge group (the case considered here is $\SL_2$) will appear in [18].

Reduced stated skein algebras have been recently generalized for $\SL_n$ gauge group in [67, 69] where similar relations with quantum cluster algebras are proved so it seems natural to expect that the techniques of the present paper should extend to higher rank cases. However, it is proved in [42, 43] that the small quantum group $\mathfrak{u}_q\mathfrak{g}$ is wild for simple Lie algebras $\mathfrak{g}$ of rank at least $3$: the problem of classifying its indecomposable representations is unsolvable. Therefore there is little hope that the problem of classifying the semi-weight indecomposable representations of $\SL_n$ reduced stated skein algebras for $n\geq 3$ could be solvable.

Let us conclude this introduction by explaining why we choose to classify the semi-weight representations. The long-term objective is to construct a $\SL_2$ HQFT whose vector spaces associated to (decorated) surfaces would be indecomposable representations of the reduced stated skein algebras; such HQFT are expected to extend the construction in [15] and to refine the ones in [12] and  the links invariants of these conjectural HQFT were built in [55] using such representations. Since we want to decorate our surfaces by some flat connections thought as points in $X(\mathbf{\Sigma})$, it is natural to consider at least the weight representations. Moreover, in non semi-simple TQFTs [15] some spaces are semi-weight modules for the skein algebras which are not weight modules; by analyzing the construction carefully we see that this necessity follows from the observation in [8] that the category of semi-weight representations is the smallest  full subcategory of $U_q\mathfrak{sl}_2-\Mod$ which contains the simple modules and is stable by $\otimes$. So it seems natural to consider the class of semi-weight representations.

§.§ Plan of the paper

In Section <ref>, we review the definition and main properties of the reduced stated skein algebras. Section <ref> is devoted to the study of the schemes $X(\mathbf{\Sigma})$ and $\widehat{X}(\mathbf{\Sigma})$.
We then study the representation theory of $ \overline{\mathcal{S}}_{A}(\mathbb{D}_1) \cong \Dq$ in Section <ref>. We then study the case of 2P marked surfaces  in Section <ref> using quantum cluster algebras. Theorem <ref> is proved in Section <ref> and  Theorem <ref> is proved in Section <ref>. In Section <ref>, we prove Theorem <ref> and Corollary <ref>. In Appendix <ref> we summarize the terminologies and notations used in the paper.

Acknowledgments. The authors thank S.Baseilhac, D.Calaque,  F.Costantino, M.Faitg,  T.Lê and P.Roche  for valuable conversations. The skein interpretation of Alekseev's morphism which appears in Section <ref> was suggested to the authors by T.Lê which we warmly thank. J.K. acknowledges support from  the European Research Council (ERC DerSympApp) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 768679).
H.K. was partially supported by JSPS KAKENHI Grant Numbers JP22K20342, JP23K12976.

§ REDUCED STATED SKEIN ALGEBRAS

§.§ Definition of reduced stated skein algebras

* A marked surface is a pair $\mathbf{\Sigma}=(\Sigma, \mathcal{A})$ where $\Sigma$ is a compact oriented surface and $\mathcal{A}$ is a finite set of pairwise disjoint closed intervals embedded in $\partial \Sigma$ named boundary edges.
The orientation of $\Sigma$ induces an orientation of $\partial \Sigma$ and thus an orientation of each boundary edge.
We impose that each boundary edge $a\subset \partial \Sigma$ is equipped with a parametrization $\varphi_a: [0,1] \hookrightarrow \partial \Sigma$ such that $\varphi_a([0,1])=a$ and such that $\varphi_a$ is an oriented embedding (where $[0,1]$ is oriented from $0$ to $1$). A marked surface  $\mathbf{\Sigma}=(\Sigma, \mathcal{A})$ is called unmarked if $\mathcal{A}=\emptyset$ and is said essential if each connected component of $\Sigma$ contains at least one boundary edge.

* A connected component of $\partial \Sigma \setminus \mathcal{A}$ is called a puncture and, pictorially, we will represent them as a punctures $\bullet$. There are two kinds of punctures: the ones homeomorphic to a circle, which correspond to unmarked boundary components are called inner punctures and their set is denoted by $\mathring{\mathcal{P}}$; the ones homeomorphic to an open interval, which lies between two boundary edges of the same boundary component, are called boundary punctures and their set is denoted by $\mathcal{P}^{\partial}$. We denote by $\Gamma^{\partial}$ the set of boundary components which intersect $\mathcal{A}$ non trivially.

* If $\mathbf{\Sigma}_1=(\Sigma_1, \mathcal{A}_1)$ and $\mathbf{\Sigma}_2=(\Sigma_2, \mathcal{A}_2)$ are two marked surfaces and $a_1, a_2$ are boundary edges of $\mathbf{\Sigma}_1$ and $\mathbf{\Sigma}_2$ respectively, we denote by $\mathbf{\Sigma}_1\cup_{a_1\# a_2} \mathbf{\Sigma}_2:= (\Sigma_1\cup_{a_1\# a_2}\Sigma_2, (\mathcal{A}_1 \cup \mathcal{A}_2)\setminus \{a_1, a_2\})  $ the marked surface obtained by gluing $a_1$ to $a_2$ where

$$ \Sigma_1\cup_{a_1\# a_2}\Sigma_2:= \quotient{ \Sigma_1 \bigsqcup \Sigma_2}{ (\varphi_{a_1}(1-t)=\varphi_{a_2}(t), t\in [0,1])}.$$

Let $\mathbf{\Sigma}=(\Sigma, \mathcal{A})$ be a marked surface.

* A tangle in $ \mathbf{\Sigma} \times (0,1)$   is a  compact framed, properly embedded $1$-dimensional manifold $T\subset \Sigma\times (0,1)$ such that  $\partial T \subset \mathcal{A}\times (0,1)$ and
for every point of $\partial T$ the framing is parallel to the $(0,1)$ factor and points to the direction of $1$.  Here, by framing, we refer to a section of the unitary normal bundle of $T$. The height of $(v,h)\in \Sigma\times (0,1)$ is $h$.  If $a\in \mathcal{A}$ is a boundary edge, we impose that no two points in $\partial_aT:= \partial T \cap a\times(0,1)$  have the same heights, hence the set $\partial_aT$ is totally ordered by the heights. Two tangles are isotopic if they are isotopic through the class of tangles that preserve the boundary height orders. By convention, the empty set is a tangle only isotopic to itself.

* Let $\pi : \Sigma\times (0,1)\to \Sigma$ be the projection with $\pi(v,h)=v$. A tangle $T$ is in a standard position if for each of its points, the framing is parallel to the $(0,1)$ factor and points in the direction of $1$ and is such that $\pi_{\big| T} : T\rightarrow \Sigma$ is an immersion with at most transversal double points in the interior of $\Sigma$. Every tangle is isotopic to a tangle in a standard position. We call diagram  the image $D=\pi(T)$ of a tangle in a standard position, together with the over/undercrossing information at each double point. An isotopy class of diagram $D$ together with a total order of $\partial_a D:=\partial D\cap a$ for each boundary arc $a$, define uniquely an isotopy class of tangle. When choosing an orientation $\mathfrak{o}(a)$ of a boundary edge $a$ and a diagram $D$, the set $\partial_aD$ receives a natural order by setting that the points are increasing when going in the direction of $\mathfrak{o}(a)$. Pictorially, we will depict tangles by drawing a diagram and an orientation (an arrow) for each boundary edge, as in Figure <ref>. When a boundary edge $a$ is oriented we assume that $\partial_a D$ is ordered according to the orientation.

* A connected open diagram without double point is called an arc. A closed connected diagram without double point is called a loop.

* A state of a tangle is a map $s:\partial T \rightarrow \{-, +\}$. A pair $(T,s)$ is called a stated tangle. We define a stated diagram $(D,s)$ in a similar manner.

On the left: a stated tangle. On the right: its associated diagram. The arrows represent the height orders.

([27, 65])
Let $k$ be a commutative unital ring and $A^{1/2}\in k^{\times}$ and invertible element.
The stated skein algebra  $\mathcal{S}_A(\mathbf{\Sigma})$ is the  free $k$-module generated by isotopy classes of stated tangles in $\mathbf{\Sigma}\times (0, 1)$ modulo the following relations (<ref>) and (<ref>),

\begin{equation}\label{eq: skein 1}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\draw[line width=1.2,-] (-0.4,-0.52) -- (.4,.53);
\draw[line width=1.2,-] (0.4,-0.52) -- (0.1,-0.12);
\draw[line width=1.2,-] (-0.1,0.12) -- (-.4,.53);
\end{tikzpicture}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\draw[line width=1.2] (-0.4,-0.52) ..controls +(.3,.5).. (-.4,.53);
\draw[line width=1.2] (0.4,-0.52) ..controls +(-.3,.5).. (.4,.53);
\end{tikzpicture}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,rotate=90]
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\draw[line width=1.2] (-0.4,-0.52) ..controls +(.3,.5).. (-.4,.53);
\draw[line width=1.2] (0.4,-0.52) ..controls +(-.3,.5).. (.4,.53);
\end{tikzpicture}
\hspace{.5cm}
\text{ and }\hspace{.5cm}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,rotate=90]
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\draw[line width=1.2,black] (0,0)  circle (.4)   ;
\end{tikzpicture}
= -(A^2+A^{-2})
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,rotate=90]
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\end{tikzpicture}
\end{equation}

\begin{equation}\label{eq: skein 2}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.7,-.75)  rectangle (.4,.75)   ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (0,-.3);
\draw[line width=1.2] (0.4,0.3) to (0,.3);
\draw[line width=1.1] (0,0) ++(90:.3) arc (90:270:.3);
\draw (0.65,0.3) node {\scriptsize{$+$}};
\draw (0.65,-0.3) node {\scriptsize{$+$}};
\end{tikzpicture}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.7,-.75)  rectangle (.4,.75)   ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (0,-.3);
\draw[line width=1.2] (0.4,0.3) to (0,.3);
\draw[line width=1.1] (0,0) ++(90:.3) arc (90:270:.3);
\draw (0.65,0.3) node {\scriptsize{$-$}};
\draw (0.65,-0.3) node {\scriptsize{$-$}};
\end{tikzpicture}
\hspace{.2cm}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.7,-.75)  rectangle (.4,.75)   ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (0,-.3);
\draw[line width=1.2] (0.4,0.3) to (0,.3);
\draw[line width=1.1] (0,0) ++(90:.3) arc (90:270:.3);
\draw (0.65,0.3) node {\scriptsize{$+$}};
\draw (0.65,-0.3) node {\scriptsize{$-$}};
\end{tikzpicture}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.7,-.75)  rectangle (.4,.75)   ;
\draw[-] (0.4,-0.75) to (.4,.75);
\end{tikzpicture}
\hspace{.1cm} \text{ and }
\hspace{.1cm}
\heightexch{->}{-}{+}
- A^{5/2}
\heightexch{->}{+}{-}
\heightcurve.
\end{equation}

The product of two classes of stated tangles $[T_1,s_1]$ and $[T_2,s_2]$ is defined by  isotoping $T_1$ and $T_2$  in $\Sigma \times (1/2, 1) $ and $\Sigma \times (0, 1/2)$ respectively and then setting $[T_1,s_1]\cdot [T_2,s_2]=[T_1\cup T_2, s_1\cup s_2]$. Figure <ref> illustrates this product.

For an unmarked surface, $\mathcal{S}_A(\mathbf{\Sigma})$ coincides with the classical  Kauffman-bracket skein algebra.

An illustration of the product in stated skein algebras.

* Let $\mathbf{\Sigma}$ a marked surface and $p\in \mathcal{P}^{\partial}$ a boundary puncture between two consecutive boundary edges $a$ and $b$ on the same boundary component $\partial\in \Gamma^{\partial}$.
The orientation of $\Sigma$ induces an orientation of $\partial$ so a cyclic ordering of the elements of $\partial \cap \mathcal{A}$ we suppose that $b$ is followed by $a$ in this ordering.  We denote by $\alpha(p)$ the arc with one endpoint $v_a\in a$ and one endpoint $v_b \in b$ such that $\alpha(p)$ can be isotoped inside $\partial$. $\alpha(p)$ is called a corner arc.
Let $\alpha(p)_{ij}\in \mathcal{S}_A(\mathbf{\Sigma})$ be the class of the stated arc $(\alpha(p), s)$ where $s(v_a)=i$ and $s(v_b)=j$.

* We call bad arc associated to $p$ the element $\alpha(p)_{-+}\in \mathcal{S}_A(\mathbf{\Sigma})$ (see Figure <ref>). The reduced stated skein algebra $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ is the quotient of $\mathcal{S}_A(\mathbf{\Sigma})$ by the ideal generated by all bad arcs.

A bad arc.

The quantum trace is an algebra morphism $\Tr_q : \mathcal{S}_A(\mathbf{\Sigma}) \to \mathcal{Z}_A(\mathbf{\Sigma}, \Delta)$ between the stated skein algebra and the balanced Chekhov-Fock algebra (a variation of the quantum Teichmüller space) which is a quantum torus associated to an ideal triangulation $\Delta$ of $\mathbf{\Sigma}$ (see [27, 65]). Its kernel is precisely the ideal generated by bad arcs (<cit.>), hence it defines an embedding $\Tr_q: \overline{\mathcal{S}}_A(\mathbf{\Sigma}) \hookrightarrow  \mathcal{Z}_A(\mathbf{\Sigma}, \Delta)$. The spaces which appear in quantum hyperbolic geometry are representations of $\mathcal{Z}_A(\mathbf{\Sigma}, \Delta)$, therefore of $ \overline{\mathcal{S}}_A(\mathbf{\Sigma}) $ as well. This is the motivation for the introduction of reduced stated skein algebras and for the study of their representations.

The main interest in extending skein algebras to marked surfaces is the existence of splitting morphisms which we now describe.
Let $a$, $b$ be two distinct boundary edges of $\mathbf{\Sigma}$, denote by $\pi : \Sigma\rightarrow \Sigma_{a\#b}$ the projection and $c:=\pi(a)=\pi(b)$. Let $(T_0, s_0)$ be a stated framed tangle of $\Sigma_{a\#b}\times (0,1)$ transverse to $c\times (0,1)$ and such that the heights of the points of $T_0 \cap c\times (0,1)$ are pairwise distinct and the framing of the points of $T_0 \cap c\times (0,1)$ is vertical towards $1$. Let $T\subset \Sigma \times (0,1)$ be the framed tangle obtained by cutting $T_0$ along $c$.
Any two states $s_a : \partial_a T \rightarrow \{-,+\}$ and $s_b : \partial_b T \rightarrow \{-,+\}$ give rise to a state $(s_a, s, s_b)$ on $T$.
Both the sets $\partial_a T$ and $\partial_b T$ are in canonical bijection with the set $T_0\cap c$ by the map $\pi$. Hence the two sets of states $s_a$ and $s_b$ are both in canonical bijection with the set $\mathrm{St}(c):=\{ s: c\cap T_0 \rightarrow \{-,+\} \}$.

The splitting morphism $\theta_{a\#b}: \mathcal{S}_{A}(\mathbf{\Sigma}_{a\#b}) \rightarrow \mathcal{S}_{A}(\mathbf{\Sigma})$ is the linear map given, for any $(T_0, s_0)$ as above, by:

$$ \theta_{a\#b} \left( [T_0,s_0] \right) := \sum_{s \in \mathrm{St}(c)} [T, (s, s_0 , s) ].$$

An illustration of the splitting morphism $\theta_{a\#b}$.

(<cit.>, <cit.>)
The linear map $\theta_{a\#b}: \mathcal{S}_A(\mathbf{\Sigma}_{a\#b}) \hookrightarrow \mathcal{S}_A(\mathbf{\Sigma})$ is an injective morphism of algebras. It passes to the quotient to define an injective morphism (still denoted by the same letter)
$\theta_{a\#b}: \overline{\mathcal{S}}_A(\mathbf{\Sigma}_{a\#b}) \hookrightarrow \overline{\mathcal{S}}_A(\mathbf{\Sigma})$.
Moreover the gluing operation is coassociative in the sense that if $a,b,c,d$ are four distinct boundary arcs, then we have $\theta_{a\#b} \circ \theta_{c\#d} = \theta_{c\#d} \circ \theta_{a\#b}$.

§.§ Bases

We now define two bases for the reduced stated skein algebras. Let us first make a preliminary remark: if $T$ is a tangle in standard position and $D$ its planar diagram projection, we cannot recover the isotopy class $T$ from the isotopy class of  $D$ since we lost the information about the height orders of the sets $\partial_aT$, $a\in \mathcal{A}$ in the projection.

Let $(D,s)$ be a stated diagram in $\mathbf{\Sigma}$. Recall that each boundary edge $a\in \mathcal{A}$ receives an orientation from the orientation of $\Sigma$ and that for $v,w \in a$ we write $v<_{\mathfrak{o}^+} w$ if $a$ is oriented from $v$ to $w$. Let $(T,s)$ be a stated tangle such that $(1)$ $(T,s)$ is in standard position (in the sense of Definition <ref>) and its planar projection is $(D,s)$ and $(2)$ for every $a\in \mathcal{A}$, if $v, w \in \partial_a D$ are such that $v<_{\mathfrak{o}^+}w$ then $h(v)<h(w)$. Conditions $(1)$ and $(2)$ completely determine the isotopy class of $(T,s)$ and we will write $[D,s]:= [T,s]\in \overline{\mathcal{S}}_A(\mathbf{\Sigma})$.

* A closed component of a diagram $D$ is trivial if it bounds an embedded disc in $\Sigma$. An open component of $D$ is trivial if it can be isotoped, relatively to its boundary, inside some boundary arc. A diagram is simple if it has neither double point nor trivial component. By convention, the empty set is a simple diagram. Let $\mathfrak{o}^+$ denote the orientation of the boundary edges of $\mathbf{\Sigma}$ induced by the orientation of $\Sigma$ as in Definition <ref>. For each boundary edge $a$ we write $<_{\mathfrak{o}^+}$ the induced total order on $\partial_a D$. A state $s: \partial D \rightarrow \{ - , + \}$ is $\mathfrak{o}^+-$increasing if for any boundary edge $a$ and any two points $x,y \in \partial_a D$, then $x<_{\mathfrak{o}^+} y$ implies $s(x)\leq s(y)$, with the convention $- < +$.

* We denote by $\mathcal{B}^{L}\subset \overline{\mathcal{S}}_{A}(\mathbf{\Sigma})$ the set of classes $[D,s]$ of stated diagrams such that $D$ is simple,  $s$ is $\mathfrak{o}^+$-increasing and such that $(D,s)$ does not contain bad arc component.

(<cit.>) $\mathcal{B}^{L}$ is a basis of $\overline{\mathcal{S}}_{A}(\mathbf{\Sigma})$.

(<cit.>) Let $p \in \mathcal{P}^{\partial}$. Then $\alpha(p)_{--}$ is the inverse of $\alpha(p)_{++}$ in $ \overline{\mathcal{S}}_{A}(\mathbf{\Sigma})$ (i.e. $\alpha(p)_{++}\alpha(p)_{--}=1$).

Let $\mathcal{B}^M \subset  \overline{\mathcal{S}}_{A}(\mathbf{\Sigma})$ be the set of classes of stated diagrams $[D,s]$ such that:

* $D$ is simple,

* if $\alpha \subset D$ is an open connected component of $D$ which is not a corner arc, then both endpoints of $\alpha$ have state $+$,

* for each $p\in \mathcal{P}^{\partial}$, there exists a sign $\varepsilon_p\in \{-,+\}$ such that every component of $D$ parallel to $\alpha(p)$ has both endpoints with state $\varepsilon_p$.

$\mathcal{B}^M$ is a basis of $\overline{\mathcal{S}}_{A}(\mathbf{\Sigma})$.

Let $(D,s)$ be a stated diagram which contains a bad arc $\alpha_{bad}=\alpha(p)_{-+}\subset (D,s)$. Further suppose that every stated arc $\alpha(p)_{ij} \subset (D,s)$ parallel to $\alpha(p)$ and distinct from $\alpha_{bad}$ has states $i=j=-$. Then $[D,s]=0$ in $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$.

The proof is illustrated in Figure <ref> and goes as follows. Let $a$ be the boundary edge adjacent to $p$ containing the endpoint of $\alpha_{bad}$ with state $-$. Let $\alpha_1, \ldots, \alpha_k$ be the subarcs of $D$ which are parallel to $\alpha(p)$ so the underlying arc of $\alpha_{bad}$ is $\alpha_n$ for $1\leq n \leq k$. We suppose that $\alpha_1$ is the arc closest to $p$ in the sense that there exists another arc $\beta\subset \partial \Sigma$ such that $\alpha_1\cup \beta$ bounds a disc which does not intersect any $\alpha_i$ for $i\neq 1$. Let $s'$ be the state of $D$ which is equal to $s$ for every arcs distinct from the $\alpha_i$ and such that $\alpha_1$ has states $(-,+)$ (so is a bad arc) and all $\alpha_i$ for $i\neq 1$ have states $(-,-)$. By applying the skein relation

\heightexch{->}{-}{+}
= A^{2}
\heightexch{->}{+}{-}
+ A^{-1/2}
\heightcurve.
repeatedly and noting that the class of the stated diagram $\heightcurve$ vanishes due to the skein relation
$ \begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.7,-.75)  rectangle (.4,.75)   ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (0,-.3);
\draw[line width=1.2] (0.4,0.3) to (0,.3);
\draw[line width=1.1] (0,0) ++(90:.3) arc (90:270:.3);
\draw (0.65,0.3) node {\scriptsize{$-$}};
\draw (0.65,-0.3) node {\scriptsize{$-$}};
\end{tikzpicture}
=0$, we see that $[D,s]=A^{2n}[D,s']$.
Let $(T,s')$ be the stated tangle associated to $(D,s)$ using Convention <ref> so that $[D,s]=[T,s]$ by definition. Let $v\subset \partial \alpha_1$ be the endpoint with state $s'(v)=-$ and let  $T'$ be the tangle obtained from $T$ by isotoping $v$ in the $[0,1]$ direction such that $h(v)<h(v')$ for all $v'\in \partial_aT'$. Using the following height exchange relations, proved in <cit.>,

$$\heightexch{<-}{+}{-}=A \heightexch{->}{+}{-}, \quad \mbox{ and }\quad
\heightexch{<-}{-}{-}=A^{-1} \heightexch{->}{-}{-}$$

we see that there exists $m\in \mathbb{Z}$ such that $[T,s']=A^m [T',s']$. Let $(D_0,s_0)$ be the stated diagram obtained from $(D,s)$ by removing $\alpha_{bad}$. Then we have $[T',s']= [D_0,s_0]\cdot \alpha_{bad}$ therefore $[D,s]=A^{2n+m}[D_0,s_0]\cdot \alpha_{bad}$ is in the ideal generated by bad arcs so it vanishes in $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$. This concludes the proof.

An illustration of the proof of Lemma <ref>.

We claim that there exist a bijection $f: \mathcal{B}^M \to \mathcal{B}^L$ and a map $c: \mathcal{B}^M \to \mathbb{Z}$ such that for all $[D,s]\in \mathcal{B}^M$ we have $f([D,s])=A^{c([D,s])/2} [D,s]$. The fact that $\mathcal{B}^M$ is a basis will then follow from the fact that $\mathcal{B}^L$ is a basis. Let $[D,s] \in \mathcal{B}^M$. If $s$ is $\mathfrak{o}^+$ increasing, we set $f([D,s])=[D,s]$ and $c([D,s])=0$. Else, there exists $a\in \mathcal{A}$ and two consecutive endpoints $v_1 <_{\mathfrak{o}^+} v_2$ in $\partial_aD$ such that $s(v_1)=+$ and $s(v_2)=-$ (i.e. locally $[D,s]$ has the form
$\heightexch{->}{-}{+}$). By definition of $\mathcal{B}^M$, the arc in $D$ which contains $v_2$ is a corner arc $\alpha(p)_{--}$ with both endpoints with state $-$. Let $[D',s']$ be the class of stated diagram obtained from $[D,s]$ by gluing $v_1$ and $v_2$ together and pushing the resulting point in the interior of $\Sigma$, i.e. is obtained by the local replacement $\heightexch{->}{-}{+} \mapsto \heightcurve$.
Using the skein relation (<ref>), one has:

\heightexch{->}{-}{+}
= A^{2}
\heightexch{->}{+}{-}
+ A^{-1/2}
\heightcurve = A^{-1/2}
\heightcurve, $$

where the second equality comes from the fact that the stated diagram corresponding to $\heightexch{->}{+}{-}$ contains the bad arc $\alpha(p)_{-+}$ and thus vanishes in  $\overline{\mathcal{S}}_{A}(\mathbf{\Sigma})$ by Lemma <ref>. Therefore $[D,s]=A^{-1/2} [D',s']$. If $[D',s']$ is $\mathfrak{o}$-increasing, we set $f([D,s]):= [D',s']$ and $c([D,s])=1$. Else, $[D',s']$ contains a pair of consecutive points of the form $\heightexch{->}{-}{+}$ and we can repeat the process of replacing
$\heightexch{->}{-}{+}$ by $\heightcurve$. Since this process decreases the number of endpoints in $\partial D$, after a finite number of steps, say $n$, we obtain an element $f([D,s])\in \mathcal{B}^L$ such that $f([D,s])= A^{n/2} [D,s]$ and we set $c([D,s])=n$. It remains to prove that $f$ is a bijection; let us define an inverse map $g: \mathcal{B}^L \to \mathcal{B}^M$.
Let $[D,s]\in \mathcal{B}^L$.  If $[D,s]\in \mathcal{B}^M$, we set $g([D,s]):=[D,s]$. Else,
since $s$ is $\mathfrak{o}^+$ increasing, $(D,s)$ cannot have two sub stated arcs isotopic to $\alpha(p)_{++}$ and $\alpha(p)_{--}$ respectively for some $p\in \mathcal{P}^{\partial}$. Therefore there exists $a\in \mathcal{A}$, an arc $\alpha \subset D$
and $v\in \partial_aD \cap \partial \alpha$ such that $s(v)=-$ and such that $(\alpha, s_{| \partial \alpha})$ is not isotopic to $\alpha(p)_{--}$ for any $p \in \mathcal{P}^{\partial}$.
Choose a such $v$ which is minimal for the order $<_{\mathfrak{o}^+}$ in $a$. Let $p, p'\in \mathcal{P}^{\partial}$ be the punctures adjacent to $a$
such that the $\mathfrak{o}^+$ orientation of $a$ points from $p$ to $p'$. Let $b\in \mathcal{A}$ be the boundary edge adjacent to $p$ right before  $a$ (possibly $a=b$ if the connected component of $\partial \Sigma$ containing $a$ does not contain any other boundary edge, i.e. if $p=p'$).
If $w\in \partial_a D$ is such that $w<_{\mathfrak{o}^+}v$ and $\beta \subset D$ is the arc with endpoint $w$, then $(\beta, s_{| \partial \beta})$ is isotopic to $\alpha(p)_{--}$. Let $[D',s']$ be the element obtained from $[D,s]$ by slightly pushing $v\in a$ to $v' \in b$ above such arcs $\beta$ and by setting $s'(v')=+$ and by adding a copy of $\alpha(p)_{--}$ as illustrated in Figure <ref>. Using the same skein relation (<ref>) than previously, one finds that $[D',s']=A^{-1/2}[D,s]$. If $[D',s']\in \mathcal{B}^M$ we set $g([D,s]):=[D',s']$. Else, we can find another point $w'\in \partial D'$ and repeat the process. Since the operation $[D,s]\mapsto [D',s']$ decreases the number of endpoints of $D$ with state $-$, after a finite number of steps we obtain an element $g([D,s])\in \mathcal{B}^M$. Clearly $f$ and $g$ are inverse to each other so $f$ is a bijection and the proof is complete.

An illustration of the operation $[D,s]\mapsto [D',s']$.

Let  $(T,s)$ and $(T',s')$ be two stated tangles in $\mathbf{\Sigma}\times (0,1)$ in standard position (in the sense of Definition <ref>)  whose projection on $\mathbf{\Sigma}\times \{1/2\}$ are both equal to the same element $(D,s)$ such that either $[D,s]\in \mathcal{B}^L$ or $[D,s]\in \mathcal{B}^M$.  Then there exists $n\in \mathbb{Z}$ such that $[T,s]=A^{n/2}[T',s']$.

As we proved during the proof of Proposition <ref>, for each $[D,s]\in \mathcal{B}^M$ there exists $[D',s']\in \mathcal{B}^L$ such that $[D,s]=A^{n/2}[D',s']$ for some $n\in \mathbb{Z}$. So we can assume that the common projection $(D,s)$ of $(T,s)$ and $(T',s')$ is such that $[D,s] \in \mathcal{B}^L$.
We say that $(T,s)$ and $(T',s')$ differ by an height exchange move if there exists $a\in \mathcal{A}$ and $v_1, v_2$ two  points in $\partial_a T$ with $h(v_1)<h(v_2)$ which are consecutive, in the sense that there is no $w\in \partial_aT$ with
$h(v_1)<_{\mathfrak{o}^+} h(w)<_{\mathfrak{o}^+} h(v_2)$, such that $(T',s')$ is obtained from $(T,s)$ by exchanging the heights of $v_1$ and $v_2$ (this means that $(T,s)\mapsto (T',s')$ is obtained by a local relation $\heightexch{->}{\varepsilon}{\varepsilon'} \mapsto \heightexch{<-}{\varepsilon}{\varepsilon'}$). Any two stated tangles in standard position with the same projection are related by a finite sequence of height exchanges so it suffices to prove the lemma in this case. Exchanging $(T,s)$ and $(T',s')$ if necessary, we can suppose without loss of generality than $v_1 <_{\mathfrak{o}^+} v_2$.
Since $[D,s]\in \mathcal{B}^L$, $s$ is $\mathfrak{o}^+$ increasing so  $( s(v_1), s(v_2)) \in \{ (+, +), (-, -), (-,+)\}$. By <cit.>, one has the following skein relations

$$ \heightexch{->}{+}{+}= A \heightexch{<-}{+}{+}, \quad  \heightexch{->}{-}{-}= A \heightexch{<-}{-}{-}, \quad  \heightexch{->}{+}{-}= A^{-1} \heightexch{<-}{+}{-}.$$

So $[T,s]=A^{\pm 1/2}[T',s']$ and the lemma is proved.

§.§ Center, Azumaya locus and shadows

From now on and in all the rest of the paper with the exception of Subsections <ref> and <ref>, we suppose that $k=\mathbb{C}$ and that $A^{1/2}$ is a root of unity of odd order $N\geq 3$ which is fixed once and for all. We denote by $Z_{\mathbf{\Sigma}}$ the center of $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$.

In this subsection, we describe $Z_{\mathbf{\Sigma}}$ and define the Azumaya locus $\mathcal{AL}(\mathbf{\Sigma})$. Stated skein algebras and their reduced versions belong to the following specific class of algebras.

A $\mathbb{C}$ associative, unital algebra $\mathscr{A}$ is said almost Azumaya if $(i)$ $\mathscr{A}$ is affine (finitely generated as an algebra), $(ii)$ $\mathscr{A}$ is prime and $(iii)$ $\mathscr{A}$ is finitely generated as a module over its center $Z$.

Let $Q(Z)$ denote the fraction field of $Z$ obtained by localizing by every non zero element. By a theorem of Posner-Formaneck (see <cit.>), $\mathscr{A}\otimes_Z Q(Z)$ is a central simple algebra and there exists a finite extension $F$ of $Q(Z)$ such that $\mathscr{A} \otimes_Z F\cong \Mat_D(F)$ is a matrix algebra of size $D$. The integer $D$ is called the PI-degree of $\mathscr{A}$ and is characterized by the formula $\dim_{Q(Z)}(\mathscr{A}\otimes_ZQ(Z)) = D^2$. Let $\mathcal{X}:= \Specm(Z)$ and for $x\in X$ corresponding to a maximal ideal $\mathfrak{m}_x$, consider the finite dimensional algebra $\mathscr{A}_x:= \quotient{\mathscr{A}}{\mathfrak{m}_x \mathscr{A}}$.

Let $\rho: \mathscr{A}\to \End(V)$ be a representation.

* $\rho$ is a  central representation  if
it sends central elements  to  scalar operators. In this case, it induces a character over the center $Z$, so a  point in $\mathcal{X}$ named its shadow.

* $\rho$ is a weight representation if $V$ is semi-simple as a $Z$-module, i.e. if $V$ is a direct sum of central representations.

Indecomposable weight representations are clearly central. A central representation  $\rho: \mathscr{A}\to \End(V)$ with shadow $x$ induces by quotient a representation $\rho : \mathscr{A}_x \to \End(V)$.

The Azumaya locus of $\mathscr{A}$ is the subset

$$ \mathcal{AL}:= \{x \in X : \mathscr{A}_x \cong \Mat_D(\mathbb{C}) \}.$$

Note that $\Mat_D(\mathbb{C})$ is semi-simple and every non trivial simple module is isomorphic to the standard module $\mathbb{C}^D$.
So if $x\in \mathcal{AL}$, there exists a unique indecomposable representation of $\mathscr{A}$ with shadow $x$ (up to isomorphism); this representation is irreducible and has dimension $D$ and is called the Azumaya representation at $x$.
The following theorem is essentially due to De Concini-Kac in their original work on quantum enveloping algebras [35] and was further generalized by several authors including [36, 29, 21, 39].
Let $\mathscr{A}$ be an almost Azumaya algebra with PI-degree $D$.

* (<cit.>) The Azumaya locus is a Zariski open dense subset.

* (<cit.>) If $\rho: \mathscr{A}\to \End(V)$ is an irreducible representation whose  shadow is not in the Azumaya locus, then $\dim(V)<D$. So the Azumaya locus admits the following alternative description:

$$\mathcal{AL}=\{ x \in \mathcal{X} | x \mbox{ is the shadow of an irreducible representation of maximal dimension }D\}.$$

Therefore, if $\rho:\mathscr{A} \to \End(V)$ is a $D$-dimensional central representation inducing $x\in \mathcal{X}$, then $\rho$ is irreducible if and only if $x\in \mathcal{AL}$.

We now explain why (reduced) stated skein algebras are almost Azumaya and describe their centers. Condition $(ii)$ follows from the
([65, 27, 75, 33]) Both $\mathcal{S}_A(\mathbf{\Sigma})$ and  $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ are domain.

More precisely, Theorem <ref> is proved in [75] for closed (thus unmarked) surfaces, in [27] for unmarked open surfaces and in [65, 33] for marked surfaces.
Consider a stated arc $\alpha_{ij}$ in some marked surface $\mathbf{\Sigma}$ and denote by $\alpha_{ij}^{(N)}$ the stated tangle obtained by taking $N$ parallel copies of $\alpha_{ij}$ pushed along the framing direction. In the case where both endpoints of $\alpha$ lye in two distinct boundary edges, one has the equality
$ \alpha_{ij}^{(N)} = (\alpha_{ij})^N$
in $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$, but when both endpoints lye in the same boundary edge, they are distinct. More precisely, suppose the two endpoints, say $v$ and $w$, of $\alpha$ lie in the same boundary edge  with $h(v)>h(w)$. Then  $\alpha_{ij}^{(N)}$ is defined by a stated tangle $(\alpha^{(N)}, s)$ where $\alpha^{(N)}=\alpha_1 \cup \ldots \cup \alpha_N$ represents $N$ copies of $\alpha$ and the endpoints $v_i, w_i$ of the copy $\alpha_i$ are chosen such that $h(v_N) > \ldots > h(v_1) > h(w_N)>\ldots >h(w_1)$.

The $N$-th Chebyshev polynomial of the first kind is the polynomial  $T_N(X) \in \mathbb{Z}[X]$ defined by the recursive formulas $T_0(X)=2$, $T_1(X)=X$ and $T_{n+2}(X)=XT_{n+1}(X) -T_n(X)$ for $n\geq 0$.

([27] for unmarked surfaces,  [61] for marked surfaces;  see also [22, 66])
There is an embedding

$$ Fr_{\mathbf{\Sigma}}: {\mathcal{S}}_{+1}(\mathbf{\Sigma}) \hookrightarrow \mathcal{Z}({\mathcal{S}}_{A}(\mathbf{\Sigma}))$$

sending the (commutative) algebra at $+1$ into the center of the stated skein algebra at $A^{1/2}$. Moreover, $Fr_{\mathbf{\Sigma}}$ is characterized by the facts that if $\gamma$ is a loop, then $Fr_{\mathbf{\Sigma}}(\gamma) = T_N(\gamma)$ and if $\alpha_{ij}$ is a stated arc, then $Fr_{\mathbf{\Sigma}}(\alpha_{ij})= \alpha_{ij}^{(N)}$. It passes to the quotient to define an embedding (still denoted by the same letter):

$$ Fr_{\mathbf{\Sigma}}: \overline{\mathcal{S}}_{+1}(\mathbf{\Sigma}) \hookrightarrow {Z}_{\mathbf{\Sigma}}.$$

$Fr_{\mathbf{\Sigma}}$ is called the Frobenius morphism.
It is proved in [61] that  ${\mathcal{S}}_A(\mathbf{\Sigma})$ is generated by the classes of loops and stated arcs so $Fr_{\mathbf{\Sigma}}$ is indeed characterized by specifying its image on these generators.
A generalization of Theorem <ref> for marked $3$-manifolds was done in [64] in the unmarked case and [22] for marked $3$-manifolds. As we shall review in Section <ref>, by <cit.> the reduced stated skein algebras are localizations of the Müller skein algebras. The existence of Frobenius morphisms for Müller skein algebras was proved in [66] from which we could derive $Fr_{\mathbf{\Sigma}}$ as well.

* For $p\in \mathring{\mathcal{P}}$ an inner puncture, we denote by $\gamma_p \in \overline{\mathcal{S}}_A(\mathbf{\Sigma})$ the class of a peripheral curve encircling $p$ once.

* For $\partial \in \Gamma^{\partial}$ a boundary component which intersects $\mathcal{A}$ non trivially, denote by $p_1, \ldots, p_n$ the boundary punctures in $\partial$ cyclically ordered by $\mathfrak{o}^+$  and define the elements in $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$:

$$ \alpha_{\partial} := \alpha(p_1)_{++} \ldots \alpha(p_n)_{++}, \quad \mbox{ and } \quad \alpha_{\partial}^{-1}:= \alpha(p_1)_{--} \ldots \alpha(p_n)_{--}.$$

The elements $\alpha_{\partial}$ and $\alpha_{\partial}^{-1}$ are inverse to each other (see [58]), hence the notation.

* A connected marked surface is small if it is either a disc with $0$ or $1$  boundary arc or an unmarked sphere.

* Let $\mathbf{\Sigma}=(\Sigma_{g,n}, \mathcal{A})$ be a connected marked surface of genus $g$ with $n$ boundary component which is not small. Set $D_{\mathbf{\Sigma}}:= N^{3g-3+n+|\mathcal{A}|}$. If $\mathbf{\Sigma}$ is small, we set $D_{\mathbf{\Sigma}}=1$. We extend it to non connected surfaces by the formula $D_{\mathbf{\Sigma}\bigsqcup \mathbf{\Sigma}'}:= D_{\mathbf{\Sigma}} D_{\mathbf{\Sigma}'}$.

Let $\mathbf{\Sigma}_1, \mathbf{\Sigma}_2$ be two marked surfaces  and let $a_1, a_2$ be some boundary edges of $\mathbf{\Sigma}_1, \mathbf{\Sigma}_2$ respectively which do not belong to a small component. Then

$$ D_{\mathbf{\Sigma}_1 \cup_{a_1 \# a_2}\mathbf{\Sigma}_2} = D_{\mathbf{\Sigma}_1} D_{\mathbf{\Sigma}_2}.$$

This follows from the facts that $\mathbf{\Sigma}_1 \cup_{a_1 \# a_2}\mathbf{\Sigma}_2$ has $| \mathcal{A}_1| + |\mathcal{A}_2| -2$ boundary edges and has $|\pi_0(\partial \Sigma_1)| + |\pi_0(\partial \Sigma_2)| -1$ boundary components.

* The elements $\gamma_p$ and $\alpha_{\partial}^{\pm 1}$ are central and $Z_{\mathbf{\Sigma}}$ is generated by the image of the Frobenius morphism together with these elements. More precisely, $Z_{\mathbf{\Sigma}}$ is isomorphic to the quotient of
$$ \overline{\mathcal{S}}_{+1}(\mathbf{\Sigma})[ \gamma_p, \alpha_{\partial}^{\pm 1}; p \in \mathring{P}, \partial \in \Gamma^{\partial} ]$$
by the relations $T_N(\gamma_p)=Fr_{\mathbf{\Sigma}}(\gamma_p)$ and $\alpha_{\partial}^N = Fr_{\mathbf{\Sigma}}(\alpha_{\partial})$.

* Both $\mathcal{S}_A(\mathbf{\Sigma})$ and $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ are finitely generated as algebras and are finitely generated over the images of the Frobenius morphisms (so over their center). Therefore they are almost Azumaya and their Azumaya loci are open dense.

* The PI-degree of $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ is $D_{\mathbf{\Sigma}}$.

* We write $X(\mathbf{\Sigma}):= \Specm( \overline{\mathcal{S}}_{+1}(\mathbf{\Sigma}) )$ and $\widehat{X}(\mathbf{\Sigma}):= \Specm(Z_{\mathbf{\Sigma}})$ and denote by $\pi: \widehat{X}(\mathbf{\Sigma}) \to {X}(\mathbf{\Sigma})$ the morphism induced by $Fr_{\mathbf{\Sigma}}$.

* We also denote by $\widehat{\mathcal{X}}(\mathbf{\Sigma})$ the set of primes  $\mathcal{I} \in \Spec(Z_{\mathbf{\Sigma}})$ such that $\mathcal{I}\cap Z_{\mathbf{\Sigma}}^0$ is maximal in $Z_{\mathbf{\Sigma}}^0$ and denote by $\widehat{\pi}: \widehat{\mathcal{X}}(\mathbf{\Sigma}) \to X(\mathbf{\Sigma})$ the projection $\mathcal{I}\mapsto \mathcal{I}\cap Z_{\mathbf{\Sigma}}^0$.

* For $\mathbf{\Sigma}_1, \mathbf{\Sigma}_2$ with boundary edges $a_1$ and $a_2$, we denote by $ p_{a_1 \# a_2} : {X}(\mathbf{\Sigma}_1)\times {X}(\mathbf{\Sigma}_2) \to {X}(\mathbf{\Sigma}_1\cup_{a_1 \# a_2} \mathbf{\Sigma}_2)$ and $\widehat{p}_{a_1 \# a_2} : \widehat{X}(\mathbf{\Sigma}_1)\times \widehat{X}(\mathbf{\Sigma}_2) \to \widehat{X}(\mathbf{\Sigma}_1\cup_{a_1 \# a_2} \mathbf{\Sigma}_2)$ the dominant maps induced by $\theta_{a_1 \# a_2}$.

Note that the Frobenius morphisms are compatible with the splitting morphisms in the sense that $Fr_{\mathbf{\Sigma}} \circ \theta_{a\# b} = \theta_{a\# b}\circ Fr_{\mathbf{\Sigma}}$.
By Theorem <ref>, as a set  $\widehat{X}(\mathbf{\Sigma})$ is described as

\begin{multline*}  \widehat{X}(\mathbf{\Sigma}) = \{ \widehat{x}= (x, h_p, h_{\partial}): x\in X(\mathbf{\Sigma}),
h_p\in \mathbb{C} \mbox{ is such that  } T_N(h_p)=\chi_x(\gamma_p) \mbox{ for }p\in \mathring{P},
\\ h_{\partial} \in \mathbb{C}^* \mbox{ is such that  } h_{\partial}^N = \chi_x(\alpha_{\partial}) \mbox{ for }\partial \in \Gamma^{\partial} \}.\end{multline*}

Let $\mathcal{AL}(\mathbf{\Sigma})\subset \widehat{X}(\mathbf{\Sigma})$ be the Azumaya locus of $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$.
The fully Azumaya locus of $ \overline{\mathcal{S}}_A(\mathbf{\Sigma})$ is the subset  $\mathcal{FAL}(\mathbf{\Sigma}) \subset {X}(\mathbf{\Sigma}) $ of elements $x\in {X}(\mathbf{\Sigma})$ such that $\pi^{-1}(x)\subset \mathcal{AL}(\mathbf{\Sigma})$.

Let $Z_{\mathbf{\Sigma}}^0\subset Z_{\mathbf{\Sigma}}$ denote the image of the Frobenius morphism (so $Z_{\mathbf{\Sigma}}^0\cong \overline{\mathcal{S}}_{+1}(\mathbf{\Sigma})$).
Recall that a representation is called semi-weight representation if the induced module over $Z_{\mathbf{\Sigma}}^0$ is semi-simple.
An indecomposable semi-weight representation $r$ induces a character over $Z_{\mathbf{\Sigma}}^0$ so defines a point in $x_r\in X(\mathbf{\Sigma})$ which we call the classical shadow of $r$.

* Let $r:\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ be an indecomposable semi-weight representation with classical shadow $x_r \in X(\mathbf{\Sigma})$. Then  $\mathcal{I}_r:=\ker(r)\cap Z_{\mathbf{\Sigma}}$ is a prime ideal of $Z_{\mathbf{\Sigma}}$, so $\mathcal{I}_r\in \widehat{\pi}^{-1}(x_r)$.

* Every prime ideal in $\widehat{\mathcal{X}}(\mathbf{\Sigma})$ is contained in a unique maximal ideal in ${\widehat{X}}(\mathbf{\Sigma})$.

* Let $\widehat{x} \in \widehat{X}(\mathbf{\Sigma})$ and decompose the set of inner punctures as $\mathring{\mathcal{P}}=\mathring{\mathcal{P}}_0 \sqcup \mathring{\mathcal{P}}_1$ where

$$ \mathring{\mathcal{P}}_1:= \left\{ p \in \mathring{\mathcal{P}}: \chi_x(\gamma_p)=\pm 2 \mbox{ and }h_p \neq \pm 2\right\}.$$

Then the set of primes contained in ${\widehat{x}}$ is in bijection with the set $\col(\widehat{x})$ of maps $c: \mathring{\mathcal{P}}\to \{S, P\}$ such that $c(p)=S$ for all $p\in \mathring{\mathcal{P}}_0$.

* If $x\in \mathcal{FAL}(\mathbf{\Sigma})$, then the map $r\mapsto \mathcal{I}_r$ induces a $1:1$ correspondance between isomorphism classes of semi-weight indecomposable representations with classical shadow $x$ and the set $\widehat{\pi}^{-1}(x)$. Therefore, for $\widehat{x}\in \pi^{-1}(x)$, the set of isomorphism classes of  semi-weight indecomposable representations such that $\mathcal{I}_r\subset \widehat{x}$ is in $1:1$ correspondance with $\col(\widehat{x})$. Under this correspondance, there is a unique simple module with $\mathcal{I}_r\subset \widehat{x}$ which corresponds to the coloring $c$ such that $c(p)=S$ for all $p\in \mathring{\mathcal{P}}$ and there is a unique projective indecomposable semi-weight module with $\mathcal{I}_r\subset \widehat{x}$ which corresponds to the coloring $c$ such that $c(p)=P$ for all $p\in \mathring{\mathcal{P}}_1$.

Let $r:\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ be an indecomposable semi-weight representation.

* The classical shadow of $r$ is the maximal ideal $x_r:= \ker(r) \cap Z^0_{\mathbf{\Sigma}} \in X(\mathbf{\Sigma})$.

* The full shadow of $r$ is the prime ideal $\mathcal{I}_r:= \ker(r)\cap Z_{\mathbf{\Sigma}} \in \widehat{\mathcal{X}}(\mathbf{\Sigma})$.

* The maximal shadow of $r$ is the (unique) maximal ideal $\widehat{x}_r \in \widehat{X}(\mathbf{\Sigma})$ containing $\mathcal{I}_r$.

So Theorem <ref> implies that semi-weight indecomposable modules with classical shadows in the fully Azumaya locus are completely classified, up to isomorphism, by their full shadows.   The proof of Theorem <ref> will occupy all the rest of the subsection.

Let $\mathscr{A}$ be an associative, unital complex algebra with center $Z$. Let $r: \mathscr{A} \to \End(V)$ be a non zero indecomposable representation. Then $\mathcal{I}_r:= \ker(r)\cap Z$ is a prime ideal in $Z$.

By contradiction, suppose that $\mathcal{I}_r$ is not prime. If $\mathcal{I}_r=Z$ then it contains the unit $1$, so $r$ is null. Else, there exist $\mathcal{J}_1, \mathcal{J}_2\subset Z$ some ideals such that $\mathcal{J}_1\mathcal{J}_2\subset \mathcal{I}_r$ and $\mathcal{J}_1\not\subset \mathcal{I}_r$, $\mathcal{J}_2 \not \subset \mathcal{I}_r$. For $i=1,2$, set

$$ V_i:= \{ v\in V : r(\mathcal{J}_i)v=0\}.$$

Clearly, $V_i$ is preserved by $\mathscr{A}$.
The condition $\mathcal{J}_i \not \subset \mathcal{I}_r$ implies that $V_i \neq V$. The condition $\mathcal{J}_1\mathcal{J}_2\subset \mathcal{I}_r$ implies that $r(\mathcal{J}_2)(V) \subset V_1$. Since $\mathcal{J}_2\not \subset \mathcal{I}_r$, there exists $v\in V$ and $x\in \mathcal{J}_2$ such that $r(x)v\neq 0$, so $V_1\neq 0$. Therefore $0\neq V_1 \subsetneq V$ is a non zero proper submodule of $V$, so $r$ is not indecomposable. This concludes the proof.

For $x\in X(\mathbf{\Sigma})$ we set

$$ \overline{\mathcal{S}}_{A}(\mathbf{\Sigma})_x:= \quotient{\overline{\mathcal{S}}_{A}(\mathbf{\Sigma})}{\mathfrak{m}_x\overline{\mathcal{S}}_{A}(\mathbf{\Sigma})} \quad \mbox{ and } Z(x):= \quotient{Z_{\mathbf{\Sigma}}}{\mathfrak{m}_x Z_{\mathbf{\Sigma}}}.$$

Note that $r$ has classical shadow $x$ if and only if it factorizes through the finite dimensional algebra $\overline{\mathcal{S}}_{A}(\mathbf{\Sigma})_x$.   The following theorem is useful to classify the indecomposable semi-weight representations whose classical shadows lye in the fully Azumaya locus.

( <cit.>) If $x\in \mathcal{FAL}(\mathbf{\Sigma})$, then
$ \overline{\mathcal{S}}_{A}(\mathbf{\Sigma})_x \cong \Mat_D(Z(x))$.

Denote by  $\mathscr{D}:= \quotient{\mathbb{C}[X]}{(X^2)}$ the dual numbers algebra.

For $x\in X(\mathbf{\Sigma})$, one has an isomorphism of algebras

$$ Z(x)\cong \left( \otimes _{p\in \mathring{\mathcal{P}}} Z(x;p) \right) \otimes \left( \otimes_{\partial \in \Gamma^{\partial}} Z(x; \partial)\right), $$

$$ Z(x;p):=
\left\{ \begin{array}{ll}
\mathbb{C}\oplus \mathscr{D} ^{\oplus \frac{N-1}{2}} &, \mbox{ if }\chi_x(\gamma_p)=\pm 2; \\
\mathbb{C}^{\oplus N} &, \mbox{ else.}
\end{array} \right.
\mbox{ and }\quad Z(x; \partial):= \mathbb{C}^{\oplus N}.$$

Therefore, if for $\widehat{x}\in \widehat{X}(\mathbf{\Sigma})$ we write $m(\widehat{x}):= | \mathring{\mathcal{P}}_1|$ in the partition $\mathring{\mathcal{P}}=\mathring{\mathcal{P}}_0 \sqcup \mathring{\mathcal{P}}_1$  of Theorem <ref>, and setting $\mathscr{D}^{\otimes 0}:= \mathbb{C}$, we have

$$ Z(x)\cong \oplus_{\widehat{x} \in \pi^{-1}(x)} Z(\widehat{x}), \quad \mbox{ where } Z(\widehat{x}):= \mathscr{D}^{\otimes m(\widehat{x})}.$$

Let us first make a preliminary remark. Let $P(X)\in\mathbb{C}[X]$ be a polynomial with decomposition $P(X)=\prod_{i=1}^d (X-\lambda_i)^{n_i}$ where $n_i \geq 1$ and the $\lambda_i$ are pairwise distinct.
Claim: We have an isomorphism of algebras:

$$ \quotient{\mathbb{C}[X]}{(P(X))} \cong \oplus_{i=1}^d \quotient{\mathbb{C}[X]}{ (X^{n_i})}.$$

Indeed, since the ideals $(X-\lambda_i)^{n_i}$ are pairwise coprime, by the chinese reminder theorem we have an isomorphism $ \quotient{\mathbb{C}[X]}{(P(X))} \cong \oplus_{i=1}^d \quotient{\mathbb{C}[X]}{ \left((X-\lambda_i)^{n_i}\right)}$ and setting $Y:=X-\lambda_i$ one has $\quotient{\mathbb{C}[X]}{ \left((X-\lambda_i)^{n_i}\right)}\cong \quotient{\mathbb{C}[Y]}{ (Y^{n_i})}$.
By Theorem <ref>, one has

\begin{multline*}
Z(x)\cong \quotient{\mathbb{C}[X_p, X_{\partial};  p\in \mathring{P}, \partial \in \Gamma^{\partial} ]}{\left( T_N(X_p)=\chi_x(\gamma_p), X_{\partial}^N = \chi_x(\alpha_{\partial})\right)} \\
\cong
\left( \otimes_{p} \quotient{\mathbb{C}[X_p]}{(T_N(X_p)-\chi_x(\alpha_p))}\right) \otimes \left(\otimes_{\partial} \quotient{\mathbb{C}[X_{\partial}]}{(X_{\partial}^N-\chi_x(\alpha_{\partial}))} \right).\end{multline*}

Let  $z\in \mathbb{C}$ and consider $\widetilde{z}$ such that $\widetilde{z}^N=z$.
Then  $X^N-z= \prod_{n\in \mathbb{Z}/N\mathbb{Z}} (X-A^n \widetilde{z})$  so $\quotient{\mathbb{C}[X]}{(X^N-z)} \cong \mathbb{C}^{\oplus N}$ (the isomorphism sends $f(X)$ to $(f(A^n \widetilde{z}))_{n\in \mathbb{Z}/N\mathbb{Z}}$).
Suppose $z\neq 0$ and set $x:= z+z^{-1}$. Then $T_N(X)-x= \prod_{n\in \mathbb{Z}/N\mathbb{Z}} (X- A^n \widetilde{z} - A^{-n}\widetilde{z}^{-1})$. If $x\neq \pm 2$ then the roots $\lambda_n:=A^n \widetilde{z} + A^{-n}\widetilde{z}^{-1}$ are pairwise distinct so $\quotient{\mathbb{C}[X]}{(T_N(x)-z)} \cong \mathbb{C}^{\oplus N}$. If $x=\pm 2$, then $T_N(x)-\pm 2= (X-\pm 2) \prod_{n=1}^{(N-1)/2}( X-\pm A^n -\pm A^{-n})^2$ so $\quotient{\mathbb{C}[X]}{(T_N(x)-z)} \cong \mathbb{C}\oplus \mathscr{D}^{\oplus \frac{N-1}{2}}$. This proves the first assertion. The second is an immediate consequence.

Let $\mathds{1}$ be the $1$-dimensional $\mathscr{D}$-module generated by the single vector $v$ such that $X\cdot v =0$. Let $W$ be the $2$-dimensional $\mathscr{D}$-module which corresponds to the left regular representation, i.e. which has basis $x,y$ such that $X\cdot x= 0$ and $X\cdot y = x$. Then every indecomposable $\mathscr{D}$-module is isomorphic to either $\mathds{1}$ or $W$ so the set of indecomposable $\mathscr{D}$ module is in $1:1$ correspondance with their annihilator (prime) ideal. Moreover we have a non split exact sequence of $\mathscr{D}$ modules

$$ 0 \to \mathds{1} \xrightarrow{i} W \xrightarrow{p} \mathds{1} \to 0,$$

where $i(v)=x$, $p(x)=0$ and $p(y)=v$. In particular $\mathds{1}$ is the only simple $\mathscr{D}$ module and $W$ is the only  projective indecomposable $\mathscr{D}$ module.

Let $r: \mathscr{D} \to \End(V)$ be an indecomposable representation. Then $r$ is determined by $\pi:= r(X)$. Up to conjugacy, we can consider that  $\pi$ is in Jordan normal form. Since $r$ is indecomposable, $\pi$ is a Jordan matrix which squares to $0$. There exists exactly two Jordan matrices which squares to $0$: the matrix $(0)$ and $\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}$. If $r(X)$ is $(0)$ then $V=\mathds{1}$; if $r(X)$ is conjugate to  $\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}$ then   $r$ is isomorphic to $W$.
The fact that the short sequence of the lemma is exact and does not split follows from a straightforward computation. This implies that $W$ is not simple and that $\mathds{1}$ is not projective. Since $W$ is free, it is projective.
This concludes the proof.

Item $(1)$ follows from Lemma <ref>. Note that the set of primes/maximal ideals $\mathcal{I}\subset Z$ containing $\widehat{x}\subset Z^0\subset Z$ is in $1:1$ correspondance with the set of primes/maximal ideals in $Z(\widehat{x})$. By Lemma <ref>, we have $Z(\widehat{x})\cong \mathscr{D}^{\otimes \mathring{\mathcal{P}}_1}$. So Item $(2)$ follows from the fact that $Z(\widehat{x})$ is local whereas Item $(3)$ follows from the fact that we have a bijection $\col(\widehat{x}) \cong \Spec(Z(\widehat{x}))$ sending $c$ to the prime ideal $\oplus_{p \in \mathring{\mathcal{P}_1}} \mathcal{I}_{c(p)} \subset\mathscr{D}^{\otimes \mathring{\mathcal{P}}_1}\cong Z(\widehat{x}) $ defined by $\mathcal{I}_{c(p)}= \left\{ \begin{array}{ll} 0 \mbox{, if }c(p)=S; \\ (X) \mbox{, if }c(p)=P. \end{array} \right.$.
Let us prove $(4)$.
Let $x\in \mathcal{FAL}(\mathbf{\Sigma})$. By definition, an indecomposable representation of $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ with classical shadow $x$ is a representation of $\overline{\mathcal{S}}_A(\mathbf{\Sigma})_x$.
By Theorem <ref>, $\overline{\mathcal{S}}_A(\mathbf{\Sigma})_x \cong \Mat_D(Z(x)) \cong \Mat_D(\mathbb{C}) \otimes Z(x)\cong \oplus_{\widehat{x}\in \pi^{-1}(x)} \Mat_D(\mathbb{C})\otimes Z(\widehat{x})$. The matrix algebra $\Mat_D(\mathbb{C})$ is semi-simple with unique simple module the standard $D$-dimensional module $V=\mathbb{C}^D$ which is projective. So the map $U \mapsto V\otimes U$ defines a $1:1$ correspondence between the indecomposable $Z(\widehat{x})$ modules and the indecomposable representations of $\overline{\mathcal{S}}_A(\mathbf{\Sigma})$ with maximal shadow $\widehat{x}$. In this correspondence $V\otimes U$ is simple (resp. projective) if and only if $U$ is simple (resp. projective). By Lemma <ref> $Z(\widehat{x})\cong \mathscr{D}^{\otimes \mathring{\mathcal{P}}_1}$ so Lemma <ref> implies that the (isomorphism classes of) indecomposable $Z(\widehat{x})$-modules are in
$1:1$ correspondance with $\Spec(Z(\widehat{x}))\cong \col(\widehat{x})$. The unique simple $Z(\widehat{x})$ module is the one dimensional module $\mathbb{C}$ which corresponds to the unique maximal ideal $0\in \Spec(Z(\widehat{x}))$ so to the coloring $c$ such that $c(p)=S$ for all $p$. The unique projective $Z(\widehat{x})$ module is the free rank one module $Z(\widehat{x})\cong W^{\otimes \mathring{\mathcal{P}}_1}$ which corresponds to the coloring sending all $p\in  \mathring{\mathcal{P}}_1$ to $P$. This concludes the proof.

§ CLASSICAL MODULI SPACES

In this subsection, we give a geometric interpretation of $X(\mathbf{\Sigma})$ following [41, 56] and study its basic properties.

§.§ Geometric interpretation of $X(\mathbf{\Sigma})$

Let $\mathbf{\Sigma}=(\Sigma, \mathcal{A})$ be an essential  marked surface.

* The fundamental groupoid $\Pi_1(\Sigma)$ is the groupoid whose objects are points in $\Sigma$ and whose morphisms $\beta: v_1 \to v_2$ are homotopy classes of path $c_{\beta}: [0,1] \to \Sigma$ such that $c_{\beta}(0)=v_1$ and $c_{\beta}(1)=v_2$. We write $v_1=s(\beta)$ (the source) and $v_2= t(\beta)$ (the target). The composition is the concatenation of paths and the unit at $v\in \Sigma$ is the class $1_v$ of the constant path. For $\beta: v_1 \to v_2$, we denote by $\beta^{-1}: v_2 \to v_1$ the class of the path $c_{\beta^{-1}}(t) = c_{\beta}(1-t)$ (so $\beta \beta^{-1}=1_{t(\beta)}$). For $a\in \mathcal{A}$ we denote by $v_a \in a$ the middle point $v_a:= \varphi_a(1/2)$ and denote by $\mathbb{V}=\{ v_a, a\in \mathcal{A}\}$ the set of such points. $\Pi_1(\Sigma, \mathbb{V})$ is the full subcategory of $\Pi_1(\Sigma)$ generated by $\mathbb{V}$. By abuse of notation, we also denote by $\Pi_1(\Sigma, \mathbb{V})$ the set of morphisms of the underlying category.

* The relative representation variety $\mathcal{R}_{\SL_2}(\mathbf{\Sigma})$ is the set of functors $\rho: \Pi_1(\Sigma, \mathbb{V}) \to \SL_2$, where $\SL_2$ is seen as a category with only one object $*$ whose set of  endomorphisms is $\SL_2(\mathbb{C})$. It admits a structure of a complex affine variety whose algebra of regular functions is

$$ \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{\Sigma})]:= \quotient{ \mathbb{C}[X_{ij}^{\beta}, i,j\in \{-,+\}, \beta \in \Pi_1(\Sigma, \mathbb{V})]}{\left( M_{\beta_1}M_{\beta_2}=M_{\beta_1 \beta_2}, \det(M_{\beta})=1 \right)}.$$

Here, for $\beta \in \Pi_1(\Sigma, \mathbb{V})$, $M_{\beta}$ represents the $2\times 2$ matrix with coefficients in the polynomial ring $ \mathbb{C}[X_{ij}^{\beta}, i,j\in \{-,+\}, \beta \in \Pi_1(\Sigma, \mathbb{V})]$ defined by
$M_{\beta}=\begin{pmatrix} X_{++}^{\beta} & X_{+-}^{\beta} \\ X_{-+}^{\beta} & X_{--}^{\beta} \end{pmatrix}$ and we quotient by the relations $\det(M_{\beta}):=X_{++}^{\beta}X_{--}^{\beta}- X_{+-}^{\beta}X_{-+}^{\beta} =1$ for all $\beta \in  \Pi_1(\Sigma, \mathbb{V})$ and by the four matrix coefficients of $ M_{\beta_1}M_{\beta_2}-M_{\beta_1 \beta_2}$ for every pair of composable paths (i.e. such that $t(\beta_2)=s(\beta_1)$).
Clearly the set of closed points of $\mathcal{R}_{\SL_2}(\mathbf{\Sigma}):= \Specm(\mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{\Sigma})])$ is in canonical bijection with the set of functors  $\rho: \Pi_1(\Sigma, \mathbb{V}) \to \SL_2$.

* A presenting graph $\Gamma$ for $\mathbf{\Sigma}$ is an embedded oriented graph $\Gamma \subset \Sigma$ whose set of vertices is $\mathbb{V}$ and such that $\Sigma$ retracts on $\Gamma$. We denote by $\mathcal{E}(\Gamma)$ the set of its oriented edges whose elements are seen as paths in $\Pi_1(\Sigma, \mathbb{V})$.

* An oriented arc $\alpha$  in $\mathbf{\Sigma}$ naturally defines a path in $\Pi_1(\Sigma, \mathbb{V})$ which we abusively also denote by $\alpha$. For $i,j \in \{-, +\}$, we denote by $\alpha_{ij}\in \mathcal{S}_A(\mathbf{\Sigma})$ the class of the arc $\alpha$ with state $i$ at its source point $s(\alpha)$ at state $j$ at its target point $t(\alpha)$. For $p\in {\mathcal{P}}^{\partial}$ we fix the canonical orientation of the corner arc $\alpha(p)$ such that $\alpha(p)_{-+}$ is a bad arc. The small Bruhat cell is the subset of $\SL_2(\mathbb{C})$ defined by

$$ \SL_2^{1}:= \{ M = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \SL_2(\mathbb{C}) \mbox{ such that }a=0 \}.$$

The reduced relative representation variety is the subvariety:

$$ \overline{\mathcal{R}}_{\SL_2}(\mathbf{\Sigma}) = \{ \rho : \Pi_1(\Sigma, \mathbb{V}) \to \SL_2 \mbox{ such that } \rho(\alpha(p))\in \SL_2^{1} \mbox{ for all }p\in {\mathcal{P}}^{\partial} \}.$$

Said differently, its algebra of regular functions is

$$ \mathcal{O}[\overline{\mathcal{R}}_{\SL_2}(\mathbf{\Sigma})] := \quotient{ \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{\Sigma})]}{ ( X_{++}^{\alpha(p)}, p\in {\mathcal{P}}^{\partial})}.$$

Let $\Arc(\mathbf{\Sigma})$ be the set of oriented arcs in $\mathbf{\Sigma}$.

(<cit.>, see also <cit.>) Let $\mathbf{\Sigma}$ be an essential marked surface. There exists a map $w: \Arc(\mathbf{\Sigma})\to \{0, 1\}$ and an isomorphism $\Psi_w: \mathcal{S}_{+1}(\mathbf{\Sigma}) \xrightarrow{\cong} \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{\Sigma})]$ characterized by the formula:

$$ \Psi_w \begin{pmatrix} \alpha_{++} & \alpha_{+-} \\ \alpha_{-+} & \alpha_{--} \end{pmatrix} = (-1)^{w(\alpha)} \begin{pmatrix} 0 & -1 \\ 1 & 0\end{pmatrix} \begin{pmatrix} X_{++}^{\alpha} & X_{+-}^{\alpha} \\ X_{-+}^{\alpha} & X_{--}^{\alpha} \end{pmatrix} = (-1)^{w(\alpha)} \begin{pmatrix} -X_{-+}^{\alpha} & -X_{--}^{\alpha} \\ X_{++}^{\alpha} & X_{+-}^{\alpha} \end{pmatrix}, \quad \mbox{ for all }\alpha \in \Arc(\mathbf{\Sigma}).$$

Let $\mathbf{\Sigma}$ be an essential marked surface. There exists a map $w: \Arc(\mathbf{\Sigma})\to \{0, 1\}$ and an isomorphism $\overline{\Psi}_w: \overline{\mathcal{S}}_{+1}(\mathbf{\Sigma}) \xrightarrow{\cong} \mathcal{O}[\overline{\mathcal{R}}_{\SL_2}(\mathbf{\Sigma})]$ characterized by the formula:

$$ \overline{\Psi}_w \begin{pmatrix} \alpha_{++} & \alpha_{+-} \\ \alpha_{-+} & \alpha_{--} \end{pmatrix} = (-1)^{w(\alpha)} \begin{pmatrix} -X_{-+}^{\alpha} & -X_{--}^{\alpha} \\ X_{++}^{\alpha} & X_{+-}^{\alpha} \end{pmatrix}, \quad \mbox{ for all }\alpha \in \Arc(\mathbf{\Sigma}).$$

Therefore $X(\mathbf{\Sigma})\cong \overline{\mathcal{R}}_{\SL_2}(\mathbf{\Sigma})$.

The isomorphism $\Psi_w$ sends the bad arc $\alpha(p)_{-+}$ to the element $(-1)^{w(\alpha(p))}X_{++}^{\alpha(p)}$ so induces $\overline{\Psi}_w$ by passing to the quotient.

§.§ Smooth loci of moduli spaces

Let $\Gamma$ be a presenting graph for $\mathbf{\Sigma}$. Since $\Sigma$ retracts to $\Gamma$, we obtain an isomorphism

$$ \varphi_{\Gamma}: \mathcal{R}_{\SL_2}(\mathbf{\Sigma}) \xrightarrow{\cong} (\SL_2(\mathbb{C}))^{\mathcal{E}(\Gamma)}, \quad \varphi_{\Gamma}: \rho \mapsto (\rho(\beta))_{\beta \in \mathcal{E}(\Gamma)}.$$

In particular $\mathcal{R}_{\SL_2}(\mathbf{\Sigma})$ is smooth.

Let $\mathbf{\Sigma}=(\Sigma_{g,n}, \mathcal{A})$ be a connected essential marked surface of genus $g$ and let us define a presenting graph $\Gamma$.
Denote by $\mathcal{A}=\{a_0, \ldots, a_{|\mathcal{A}|-1}\}$ the boundary edges, by $\pi_0(\partial \Sigma) = \{\partial_0, \ldots, \partial_{n-1}\}$ the boundary components with $a_0\subset \partial_0$ and write  $v_i:=v_{a_i}$. We order the boundary components such that for $0\leq j < n^{\partial}$ then $\partial_j\in \Gamma^{\partial}$ and for $n^{\partial}\leq j \leq n-1$ then $\partial_j \in \mathring{\mathcal{P}}$.
Let $\overline{\Sigma}$ be the surface obtained from $\Sigma$ by gluing a disc along each boundary component $\partial_i$ for $1\leq i \leq r$, and choose $\lambda_1, \mu_1, \ldots, \lambda_g, \mu_g$ some paths in $\pi_1(\Sigma, v_0)$, such that their images in $\overline{\Sigma}$ generate the free group $\pi_1(\overline{\Sigma}, v_0)$ (said differently, the $\lambda_i$ and $\mu_i$ are longitudes and meridians of $\Sigma$). For each  $p\in \mathring{\mathcal{P}}$ choose a peripheral curve $\gamma_p \in \pi_1(\Sigma, v_0)$ encircling $p$ once. Eventually, for each boundary component $\partial_j$, with $1\leq j <n^{\partial}$, containing a boundary edge $a_{k_j} \subset \partial_j$ chosen arbitrarily,  choose a path $\delta_{\partial_j} : v_0 \rightarrow v_{k_j}$. The set

$$\mathbb{G}:= \{ \lambda_i, \mu_i, \gamma_p,  \alpha(p_{\partial}), \delta_{\partial_j} | 1\leq i \leq g, p\in \mathring{\mathcal{P}}, p_{\partial} \in \mathcal{P}^{\partial},  1\leq j < n^{\partial}\}$$

is a generating set for $\Pi_1(\Sigma, \mathbb{V})$ (see Figure <ref> for an illustration).
Moreover each of its generators which is of the form $\gamma_p$ or $\alpha(p_{\partial})$ can be expressed as a composition of the other ones and their inverse, therefore a set $\mathcal{E}(\Gamma)$ obtained from $\mathbb{G}$ by removing one of the elements of the form $ \alpha(p_{\partial})$ or $\gamma_p$, forms a set of edges of a presenting graph $\Gamma$.
Note that $\mathbb{G}$ has cardinality $2g-1+|\mathcal{A}|+n$,  so one has an isomorphism $\mathcal{R}_{\SL_2}(\mathbf{\Sigma}) \cong (\SL_2(\mathbb{C}))^{2g-2+|\mathcal{A}|+n}$.

The geometric representatives of a set of generators for $\Pi_1(\Sigma, \mathbb{V})$.

Let $\mathbf{\Sigma}$ be a connected marked surface with at least two boundary arcs. Then $X(\mathbf{\Sigma})$ is a smooth variety.

For $g=\begin{pmatrix} g_{++} & g_{+-} \\ g_{-+} & g_{--} \end{pmatrix} \in \SL_2$, the Zariski tangent space $T_g\SL_2$ is:

\begin{multline*}
T_g\SL_2 = \left\{ X= \begin{pmatrix} X_{++} & X_{+-} \\ X_{-+} & X_{--} \end{pmatrix}, \mbox{ s.t. } \det(g+\varepsilon X)\equiv 1 \pmod{\varepsilon^2} \right\} \\ =  \left\{ X= \begin{pmatrix} X_{++} & X_{+-} \\ X_{-+} & X_{--} \end{pmatrix}, \mbox{ s.t. } g_{--}X_{++} + g_{++}X_{--} = g_{-+}X_{+-} + g_{+-}X_{-+} \right\}.
\end{multline*}

Similarly, if $g\in \SL_2^{1}$ (i.e. if $g_{++}=0$) then

$$ T_g\SL_2^{1}= \left\{X= \begin{pmatrix} 0& X_{+-} \\ X_{-+} & X_{--} \end{pmatrix}, \mbox{ s.t. } g_{+-}X_{-+} + g_{-+}X_{+-}=0 \right\}.$$

* Let $X=\SL_2^{1} \times \SL_2$ and consider the regular map $F: X\to \mathbb{C}$, $F(h,g)=(hg)_{--}$. Let $x=(h,g)\in X$. Then there exists $v \in T_h \SL_2^{1} \subset T_h \SL_2^{1} \oplus T_g\SL_2=T_x X$ such that $D_xF(v)\neq 0$.

* Let $Y:= \SL_2^{1} \times (\SL_2)^2$ and consider the regular map $F: Y\to \mathbb{C}$, $F(h,g,f):= (ghg^{-1}f)_{--}$. Let $y=(h,g,f)\in Y$. Then there exists $v \in T_h \SL_2^{1}\oplus T_g\SL_2 \subset T_h \SL_2^{1}\oplus T_g\SL_2\oplus T_f \SL_2=T_yY$ such that $D_yF(v)\neq 0$.

$(1)$ Let $X_1^h, X_2^h \in T_h\SL_2^{1}\subset T_xX$ be the vectors $X_1^h= \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$, $X_2^h= \begin{pmatrix} 0 & h_{+-} \\ -h_{-+} & 0 \end{pmatrix}$. Then $D_xF(X_1^h)= g_{--}$ and $D_xF(X_2^h)= -g_{+-}$. Since $\det(g)=1$, either $g_{--} \neq 0$ or $g_{+-}\neq 0$.
$(2)$ Consider the vectors $X_1^h, X_2^h, X^g \in T_yY=T_h\SL_2^{1} \oplus T_g\SL_2 \oplus T_f\SL_2$ defined by $X_1^h= \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \in T_h \SL_2^{1} \subset T_yY$, $X_2^h= \begin{pmatrix} 0 & h_{+-} \\ -h_{-+} & 0 \end{pmatrix} \in T_h \SL_2^{1} \subset T_yY$ and $X^g= \begin{pmatrix} 0 & g_{+-} \\ -g_{-+} & 0 \end{pmatrix} \in T_g \SL_2 \subset T_yY$.
One has $D_yF(X_1^h)= g_{--} (g^{-1}f)_{--}$ and $D_yF(X_2^h)= h_{+-}g_{-+}(g^{-1}f)_{--} - h_{-+} g_{--}(g^{-1}f)_{+-}$ so $D_yF(X_1^h)=D_yF(X_2^h)=0$ if and only if $g_{--}=(g^{-1}f)_{--}=0$. In this case, one finds that

$$ D_yF(X^g) = -g_{+-}(gh)_{-+}f_{--} - g_{-+}(hg^{-1}f)_{+-} + g_{-+}(gh)_{--}f_{--} = (g_{-+})^2 h_{+-}f_{--}.$$

Since $g\in \SL_2$, $g_{+-}g_{-+}=-1$ and $g_{-+}\neq 0$. Similarly, $h_{+-}\neq 0$. If $f_{--}=0$ then $f_{+-}\neq 0$ and  the equality $(g^{-1}f)_{--}=0$ would imply $g_{-+}f_{+-}=0$ which is a contradiction. Therefore $f_{--}\neq 0$ and $D_yF(X^g)\neq 0$.

Let $g$ be the genus of $\Sigma$ and $n$ the number of its boundary components. Let $\mathbb{G}$ be the generating set of Example <ref>. Let us first suppose  that $\mathring{\mathcal{P}}\neq \emptyset$. In this case, we can remove from $\mathbb{G}$ one generator $\gamma_p$ to obtain a presenting graph $\Gamma$ such that every corner arc is an edge of $\Gamma$. We obtain an isomorphism

$$ X(\mathbf{\Sigma}) \xrightarrow[\cong]{\Psi_w^*} \overline{\mathcal{R}}_{\SL_2}(\mathbf{\Sigma}) \xrightarrow[\cong]{\varphi_{\Gamma}} (\SL_2(\mathbb{C}))^{2g-2+n} \times (\SL_2^{1})^{|\mathcal{A}|} .$$

So $ X(\mathbf{\Sigma})$ is smooth and irreducible because both $\SL_2(\mathbb{C})$ and $\SL_2^{1}\cong \mathbb{C}^* \times \mathbb{C}$ are smooth and irreducible.
If $\mathring{\mathcal{P}}=\emptyset$, let $p_0 \in \mathcal{P}^{\partial}$, let $\Gamma$ be the presenting graph obtained from $\mathbb{G}$ by removing $\alpha_{p_0}$ and consider the variety

$$ \mathcal{R}' := \{ \rho : \Pi_1(\Sigma, \mathbb{V}) \to \SL_2, \mbox{ such that } \rho(\alpha(p)) \in \SL_2^{1} \mbox{ for all }p\in \mathcal{P}^{\partial} \setminus \{ p_0\} \}.$$

The morphism $\varphi_{\Gamma}$ restricts to an isomorphism $\varphi_{\Gamma}: \mathcal{R}' \cong (\SL_2(\mathbb{C}))^{2g+n-1} \times (\SL_2^{1})^{|\mathcal{A}| -1}$, so $\mathcal{R}'$ is a smooth irreducible variety.
Let $F: \mathcal{R}' \to \mathbb{C}$ be the regular function $F=X_{++}^{\alpha(p_0)}$, i.e. $F(\rho)=\rho(\alpha(p_0))_{++}$. One has $ \overline{\mathcal{R}}_{\SL_2}(\mathbf{\Sigma})= F^{-1}(0)$ so it suffices to prove that for every $\rho \in \mathcal{R}'$, one has $D_{\rho}F\neq 0$. Using $\varphi_{\Gamma}$ we identify the Zariski tangent space $T_{\rho}\mathcal{R}'$ with

$$ T_{\rho}\mathcal{R}' \cong \oplus_{p\in \mathcal{P}^{\partial} \setminus \{p_0\}} T_{\rho(\alpha(p))} \SL_2^{1} \oplus \oplus_{\beta \in \mathbb{G}\setminus\{\alpha(p)\} } T_{\rho(\beta)}\SL_2.$$

First suppose that $\mathbf{\Sigma}$ contains two boundary edges in the same boundary component $\partial$. We suppose that $p_0 \in \partial$ so one has two punctures $p_0,p_1\in \partial$ consecutive in the cyclic order of $\partial$.
By definition of $\mathbb{G}$, there exists  a path $\beta$ which is a composition of the elements of $\mathbb{G}\setminus \{\alpha(p_0), \alpha(p_1)\}$ such that $\alpha(p_0)\alpha(p_1)\beta= 1_{s(\alpha(p_0))}$. Therefore

$$ F(\rho) = \rho( \alpha(p_0))_{++}= \rho( \alpha(p_0)^{-1})_{--} = \left(\rho(\alpha(p_1)) \rho(\beta) \right)_{--}.$$

By the first item of Lemma <ref>, for every $\rho \in \mathcal{R}'$, there exists $v\in T_{\rho(\alpha(p_1))}\SL_2^{1} \subset T_{\rho}\mathcal{R}'$ such that $D_{\rho}F(v)\neq 0$.
If $\mathbf{\Sigma}$ only contains boundary components having a single boundary arc, one can find a boundary component $\partial$ whose unique puncture $p_1\in \partial$ is distinct from $p_0$. By definition of $\mathbb{G}$, there exists  a path $\beta$ which is a composition of the elements of $\mathbb{G}\setminus \{\alpha(p_0), \alpha(p_1), \delta_{\partial}\}$ such that $\alpha(p_0)^{-1}=\delta_{\partial} \alpha(p_1) \delta_{\partial}^{-1} \beta$, so

$$F(\rho)= \left( \rho(\delta_{\partial}) \rho(\alpha(p_1)) \rho(\delta_{\partial})^{-1} \rho(\beta) \right)_{--}.$$

By the second item of Lemma <ref>,  for every $\rho \in \mathcal{R}'$, there exists $v\in T_{\rho(\alpha(p_1))}\SL_2^{1}\oplus T_{\rho(\delta_{\partial})}\SL_2 \subset T_{\rho}\mathcal{R}'$ such that $D_{\rho}F(v)\neq 0$.
In every cases, we have thus proved that $D_{\rho}F\neq 0$ for every $\rho \in \mathcal{R}'$, so $X(\mathbf{\Sigma})$ is smooth.

Let $x\in X(\mathbf{\Sigma})$ and write $\rho_x:= \overline{\Psi}_w^{-1}(x) \in \overline{R}_{\SL_2}(\mathbf{\Sigma})$. For $p\in \mathring{\mathcal{P}}$ an inner puncture, we say that $x$  is central at  $p$ if $\rho_x(\gamma_p)= \pm \id$.

Let $\mathbf{\Sigma}$ be an essential marked surface with at least two boundary arcs. Then the regular locus of $\widehat{X}(\mathbf{\Sigma})$ is the subset of elements $\widehat{x}=(x, h_p, h_{\partial})$ such that for every $p\in \mathring{\mathcal{P}}$ then either $x$ is not central at $p$ or $h_p=\pm 2$.

Consider the application $f: \SL_2\times \mathbb{C} \to \mathbb{C}$, $f : (g,z) \mapsto \tr(g) + T_N(z)$ and set $\widehat{X}:=f^{-1}(0)$. Then $(g,z)\in \widehat{X}$ is smooth if and only if either $g\neq \pm \mathds{1}$ or $g=\pm \mathds{1}$ and
$z=\mp 2$.

First, note that  for $z\in \mathbb{C}$ and $z^{1/N}$ an $N$-th root of $z$, the equation $T_N(X)=z+z^{-1}$ has solutions $X= z^{n/N} + z^{-n/N}$ for $n\in \{0, \ldots, N-1\}$ counted with multiplicity, i.e. $T_N(X)-(z+z^{-1})= \prod_{n=0}^{N-1} (X- z^{n/N}-z^{-n/N})$. Therefore, for $c\in \mathbb{C}$, the polynomial $P_{N,c}(X):= T_N(X)-c$ has roots of multiplicity $\geq 2$ if and only if $c=\pm 2$ in which case the multiple roots are $X= \pm (q^n+q^{-n})$ with   $n\in \{1, \ldots, (N-3)/2\}$.
For $g=\begin{pmatrix} g_{++} & g_{+-} \\ g_{-+} & g_{--} \end{pmatrix} \in \SL_2$, let us consider the tangent space

$$T_g\SL_2=   \left\{ X= \begin{pmatrix} X_{++} & X_{+-} \\ X_{-+} & X_{--} \end{pmatrix}, \mbox{ s.t. } g_{--}X_{++} + g_{++}X_{--} = g_{-+}X_{+-} + g_{+-}X_{-+} \right\}$$

as before. Then the differential at $(g,z)\in \widehat{X}$ is computed as

$$ D_{(g,z)}f : T_g\SL_2 \oplus \mathbb{C} \to \mathbb{C}, \quad D_{(g,z)}f: (X, z_0) \mapsto \tr(X) + T_N'(z)z_0.$$

In order to find at which point $f$ is a submersion, we need to find when this differential vanishes. This will happen if and only if we have the two conditions: $(i)$ $\tr(X)=0$ for all $X\in T_g\SL_2$ and $(ii)$ $T_N'(z)=0$. By the preceding discussion, $T_N'(z)=0$ if and only if $\tr(g)=\pm 2$ and $z\neq \mp 2$. Let us analyse $(i)$.
First suppose that $g_{+-}\neq 0$. Then for every $x\in \mathbb{C}$ we have $X:= \begin{pmatrix} x & 0 \\ x \frac{g_{--}}{g_{+-}} & 0 \end{pmatrix} \in T_g\SL_2$ and $\tr(X)=x$ so $(i)$ is not satisfied. Similarly, if $g_{-+}\neq 0$, we can use $X:= \begin{pmatrix} x & x \frac{g_{--}}{g_{-+}}  \\ 0& 0 \end{pmatrix} \in T_g\SL_2$ and $\tr(X)=x$ so $(i)$ is not satisfied. If $g=\begin{pmatrix} x & 0 \\ 0 & x^{-1} \end{pmatrix}$ then every $X\in T_g\SL_2$ satisfies $\tr(X)=(1-x^{-2})X_{++}$ so $(i)$ is satisfied if and only if $g= \pm \mathds{1}$ in which case $(ii)$ is satisfied if and only if $z=\mp (q^n +q^{-n})$ for $n\in \{1, \ldots, (N-3)/2\}$. This concludes the proof.

Write $\mathring{\mathcal{P}}=\{p_1, \ldots, p_s\}$ and consider the composition

$$ g : \overline{R}_{\SL_2}(\mathbf{\Sigma}) \times \mathbb{C}^{s} \xrightarrow{h} (\SL_2\times \mathbb{C})^{s} \xrightarrow{ (f, \ldots, f)} \mathbb{C}^s$$

where $g(\rho, \{h_{p_i}\}_i) := (\rho(\gamma_{p_i}), h_{p_i})_{p_i}$ and $f$ is defined in Lemma <ref>. Let $X^0(\mathbf{\Sigma}):=g^{-1}(0, \ldots, 0)$ and consider the covering $\pi: \widehat{X}(\mathbf{\Sigma})\to X^0(\mathbf{\Sigma})$ defined by $\pi(x, h_{p_i}, h_{\partial}):=(\rho_x, h_{p_i})$ which consists in forgetting the boundary invariants $h_{\partial}$. For $(\rho_x, h_p)\in X^0(\mathbf{\Sigma})$, set $c_{\partial}:= \rho_x(\alpha_{\partial})$. The fiber $\pi^{-1}(\rho_x, h_{p_i})$ consists in the set of elements $(x, h_{p_i}, h_{\partial})$ where
the $h_{\partial}$ satisfy $h_{\partial}^N = c_{\partial}$ so are zeroes of the polynomials $X^N-c_{\partial}$. Since the polynomials $X^N-c_{\partial}$ have $N$ zeroes each with multiplicity one, $\pi$ is a regular covering so the smooth locus of $\widehat{X}(\mathbf{\Sigma})$ is the pull-back by $\pi$ of the smooth locus of $X^0(\mathbf{\Sigma})$.
Since  $h$ is a submersion on $X^0(\mathbf{\Sigma})$, by the chain rule $g$ is a submersion at $(\rho, \{h_{p_i}\}_i)$ if and only if for every $i$ $f$ is a submersion at $(\rho(\gamma_{p_i}), h_{p_i})$. By Lemma <ref>, this condition is equivalent to the condition that for every $p\in \mathring{\mathcal{P}}$, either $\rho(\gamma_p)\neq \pm \mathds{1}$ or $h_p=\mp 2$.  This concludes the proof.

§.§ Behavior for the gluing operation

The goal of this subsection is to prove the

Let  $\mathbf{\Sigma}_1$ and $\mathbf{\Sigma}_2$ be connected marked surfaces and for $i=1,2$ let $a_i$ be a boundary edges of $\mathbf{\Sigma}_i$ such that the connected component of $\partial \Sigma_i$ which contains $a_i$ also contains at least another boundary edge.
Then the maps $p_{a_1\# a_2}: {X}(\mathbf{\Sigma}_1)\times  {X}(\mathbf{\Sigma}_2) \to {X}(\mathbf{\Sigma}_1\cup_{a_1\# a_2} \mathbf{\Sigma}_2)$ and $\widehat{p}_{a_1\# a_2}: \widehat{X}(\mathbf{\Sigma}_1)\times  \widehat{X}(\mathbf{\Sigma}_2) \to \widehat{X}(\mathbf{\Sigma}_1\cup_{a_1\# a_2} \mathbf{\Sigma}_2)$ are surjective.

To prove Theorem <ref>, it is useful to introduce an alternative geometric interpretation of $X(\mathbf{\Sigma})$ defined in [33]. Let $\mathbf{\Sigma}$ be a connected essential surface and fix a Riemannian metric on $\Sigma$. Let $\pi: U\Sigma \to \Sigma$ be the unitary tangent bundle, so a point $\overrightarrow{v}\in U\Sigma$ is given by a pair $\overrightarrow{v}=(p,v)$ where $p\in \Sigma$ and $v\in T_p\Sigma$ has norm one.
For such a  $\overrightarrow{v}=(p,v)$, write $-\overrightarrow{v}:= (p, -v)$.
Recall that the orientation of $\Sigma$ induces an orientation of $\partial \Sigma$.
For $p \in \partial \Sigma$, let $\overrightarrow{p} \in U\Sigma$ be $\overrightarrow{p}:=(p,v)$ where $v\in T_p \partial \Sigma \subset T_p\Sigma$ is the unitary tangent vector oriented in the direction of $\partial \Sigma$. Said differently, if $n\in T_p\Sigma$ is a vector normal to $\partial \Sigma$ which points outside of $\Sigma$, then $(n,v)$ is a positive basis of $T_p\Sigma$. Write $\widehat{\mathbb{V}} := \{ \pm \overrightarrow{v_a}, a\in \mathcal{A} \}$ (recall that $v_a$ is a point in the middle of the boundary edge $a$).
For $\overrightarrow{v} \in U\Sigma$ with $p=\pi(\overrightarrow{v})$,  let $\theta_{\overrightarrow{v}} \in \pi_1(U\Sigma, \overrightarrow{v})$ be the homotopy class of the path in $U\Sigma$ which lies inside the circle fiber $\pi^{-1}(p)=T_p\Sigma$ and makes a single positive turn. Also denote by $\theta_{\overrightarrow{v}}^{1/2}$ be the homotopy class of the path starting at $\overrightarrow{v}$ and ending at $-\overrightarrow{v}$ which makes a positive half turn in the fiber $T_p\Sigma$.
Let $\alpha$ be an oriented arc of $\mathbf{\Sigma}$ with endpoints in some boundary edges $a$ and $b$ such that $\alpha$ is oriented from $a$ to $b$. Let $c_{\alpha}: [0,1]\to \Sigma$ be an oriented embedding such that $(1)$ $c_{\alpha}([0,1])$ is isotopic to $\alpha$ with the same orientation, $(2)$ $c_{\alpha}'(0)= - \overrightarrow{v_a}$ and $c_{\alpha}'(1)=\overrightarrow{v_b}$ and $(3)$ $c_{\alpha}'(t)$ has norm $1$ for all $t\in [0,1]$. We denote by $\widehat{\alpha}$ the homotopy class of the path $[0,1] \to U\Sigma$, $t\mapsto (c_{\alpha}(t), c'_{\alpha}(t))$.

* The twisted relative representation variety $\mathcal{R}^{tw}_{\SL_2}(\mathbf{\Sigma})$  is the set of functors $\widehat{\rho}: \Pi_1(U\Sigma, \widehat{\mathbb{V}}) \to \SL_2$ such that $\widehat{\rho}(\theta^{1/2}_{\overrightarrow{v}})= \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix}$ for all $\overrightarrow{v} \in \widehat{\mathbb{V}}$. It admits a natural structure of affine variety as in Definition <ref>

* The  reduced twisted relative representation variety $\overline{\mathcal{R}}^{tw}_{\SL_2}(\mathbf{\Sigma})$ is the subvariety of  $\mathcal{R}^{tw}_{\SL_2}(\mathbf{\Sigma})$  of functors $\widehat{\rho}$ such that $\widehat{\rho}(\widehat{\alpha(p)})$ is upper triangular for all $p \in \mathcal{P}^{\partial}$.

(<cit.>) There exists an isomorphism $\Psi_{\mathbf{\Sigma}} : X(\mathbf{\Sigma}) \xrightarrow{\cong} {\mathcal{R}}^{tw}_{\SL_2}(\mathbf{\Sigma})$ sending a character $\chi : \mathcal{S}_{+1}(\mathbf{\Sigma}) \to \mathbb{C}$ to the functor $\widehat{\rho}: \Pi_1(U\Sigma, \widehat{\mathbb{V}}) \to \SL_2$ characterized by the fact that for every arc $\alpha$, then

$$ \widehat{\rho}(\widehat{\alpha}) = \begin{pmatrix}\chi(\alpha_{++}) & \chi(\alpha_{+-}) \\ \chi(\alpha_{-+}) & \chi(\alpha_{--})\end{pmatrix}.$$

In particular, the ring isomorphism $\Psi_{\mathbf{\Sigma}}^{*}$ sends a bad arc $\alpha(p)_{-+}$ to the function $X_{-+}^{\widehat{\alpha(p)}}$ which sends $\widehat{\rho}$ to the lower left matrix coefficient of $\widehat{\rho}(\widehat{\alpha(p)})$, so we get the

The isomorphism $\Psi_{\mathbf{\Sigma}}$ restricts to an isomorphism (still denoted by the same letter) $\Psi_{\mathbf{\Sigma}}: \overline{X}(\mathbf{\Sigma}) \xrightarrow{\cong} \overline{\mathcal{R}}^{tw}_{\SL_2}(\mathbf{\Sigma})$.

Now consider $\mathbf{\Sigma}_1$ and $\mathbf{\Sigma}_2$ with $a_1$ and $a_2$ as in Theorem <ref> and write $\mathbf{\Sigma}:=  \mathbf{\Sigma}_1\cup_{a_1\# a_2} \mathbf{\Sigma}_2$.
For $i=1,2$, let $\iota_i : \Sigma_i \hookrightarrow \Sigma$ be the inclusion map.
Fix a Riemannian metric on $\Sigma$ which induces, by restriction, some Riemannian metrics on $\Sigma_1$ and $\Sigma_2$. Let $\overrightarrow{w} \in U\Sigma$ be the image of $\overrightarrow{v_{a_1}}\in U\Sigma_1 \xrightarrow{\subset} U\Sigma$ by $(\iota_1)_*$. Note that the image of $\overrightarrow{v_{a_2}}\in U\Sigma_2$ in $U\Sigma$ by $(\iota_2)_*$ is $-\overrightarrow{v}$.
Write $\Pi_1:= \Pi_1(U\Sigma_1, \mathbb{V}_1)$, $\Pi_2:= \Pi_1(U\Sigma_2, \mathbb{V}_2)$ and $\Pi:= \Pi_1(U\Sigma, \mathbb{V} \cup \{\overrightarrow{w}\})$ and consider the variety:

$$ \mathcal{R}^{w}:= \{ \widehat{\rho}\in \Hom(\Pi, \SL_2), \widehat{\rho}(\theta^{1/2}_{\overrightarrow{v}})= \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix} \mbox{, for all }\overrightarrow{v}\in \widehat{\mathbb{V}} \}.$$

For $i=1,2$, the embedding $\iota_i : \Sigma_i \hookrightarrow \Sigma$ induces a fully faithful functor $(\iota_i)_*: \Pi_i \hookrightarrow \Pi$ so we can consider $\Pi_i$ as a full subcategory of $\Pi$. We are in the following situation:

Let $G$ be a groupoid and $G_1, G_2\subset G$ be two full subcategories of $G$ and $v\in G$ an object such that

* one has $\Ob(G_1)\cup \Ob(G_2)= \Ob(G)$ and $\Ob(G_1)\cap \Ob(G_2)=\{ v\}$;

* for $v_1\in G_1$ and $v_2\in G_2$, the composition map $G_1(v_1, v)\times G_2(v, v_2) \to G(v_1,v_2)$ is a bijection.

Then the restriction map $\res: \Hom(G, \SL_2) \to \Hom(G_1, \SL_2) \times \Hom(G_2, \SL_2)$ sending $\rho$ to the pair $(\restriction{\rho}{G_1}, \restriction{\rho}{G_2})$, is a bijection.

Consider the map
$$F: \Hom(G_1, \SL_2) \times \Hom(G_2, \SL_2) \to \Hom(G, \SL_2)$$
which sends a pair $(\rho_1, \rho_2)$ to the functor $\rho$ such that for $v_1, v_2 \in G$, the restriction $\rho: G(v_1,v_2) \to \SL_2(\mathbb{C})$ is equal to $(i)$ the restriction $\rho_1: G_1(v_1,v_2) \to \SL_2(\mathbb{C})$ if $v_1, v_2 \in \Pi_1$, $(ii)$ to the restriction $\rho_2: G_2(v_1,v_2) \to \SL_2(\mathbb{C})$ if $v_1,v_2\in G_2$, $(iii)$ to the composition $G(v_1,v_2) \cong G_1(v_1, v) \times G_2(v, v_2) \xrightarrow{(\rho_1, \rho_2)}\SL_2(\mathbb{C}) \times \SL_2(\mathbb{C}) \xrightarrow{ \times} \SL_2(\mathbb{C})$ if $v_1\in G_1$, $v_2\in G_2$, $(iv)$ to a similar composition map with $1$ and $2$ exchanged if $v_1\in G_2$ and $v_2\in G_1$. $F$ is clearly the inverse of  $\res$.

The morphism $F$ of the proof, obtained by replacing $(G,G_1,G_2, v) $  by $(\Pi, \Pi_1, \Pi_2, \overrightarrow{w})$,   induces by restriction an isomorphism (still denoted by the same letter):

$$ F :\mathcal{R}^{tw}_{\SL_2}(\mathbf{\Sigma}_1) \times \mathcal{R}^{tw}_{\SL_2}(\mathbf{\Sigma}_2) \xrightarrow{\cong} \mathcal{R}^w.$$

The inclusion $\Pi_1(U\Sigma, \widehat{\mathbb{V}}) \subset \Pi_1(U\Sigma, \widehat{\mathbb{V}} \cup \{\overrightarrow{w}\})$ induces a regular (restriction) map $G: \mathcal{R}^w \to \mathcal{R}^{tw}_{\SL_2}(\mathbf{\Sigma})$.

The composition $G\circ F : \mathcal{R}^{tw}_{\SL_2}(\mathbf{\Sigma}_1) \times \mathcal{R}^{tw}_{\SL_2}(\mathbf{\Sigma}_2) \to  \mathcal{R}^{tw}_{\SL_2}(\mathbf{\Sigma})$ is surjective and induces by restriction a surjective map
$\varphi: \overline{\mathcal{R}}^{tw}_{\SL_2}(\mathbf{\Sigma}_1) \times \overline{\mathcal{R}}^{tw}_{\SL_2}(\mathbf{\Sigma}_2) \to  \overline{\mathcal{R}}^{tw}_{\SL_2}(\mathbf{\Sigma})$.

In order to prove Lemma <ref>, we first prove the following:

Let $B^+\subset \SL_2(\mathbb{C})$ be the subgroup of upper triangular matrices and let $A,B,C,D \in \SL_2(\mathbb{C})$ be such that $AB \in B^+$ and $CD\in B^+$. Then there exists $g \in \SL_2(\mathbb{C})$ such that $Ag$, $g^{-1}B$, $Dg^{-1}$ and $gC$ are in $B^+$.

Write $A=\begin{pmatrix} a_{++} & a_{+-} \\ a_{-+} & a_{--} \end{pmatrix}$ with similar notations for $B,C,D,g$. Then

$$ \left\{ \begin{array}{l} Ag \in B^+ \\ g^{-1}B \in B^+ \end{array} \right. \Leftrightarrow \begin{pmatrix} a_{-+} & a_{--} \\ b_{-+} & -b_{++} \end{pmatrix} \begin{pmatrix} g_{++} \\ g_{-+} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}.$$

Since $AB \in B^+ \Leftrightarrow \det \begin{pmatrix} a_{-+} & a_{--} \\ b_{-+} & -b_{++} \end{pmatrix} =0$, one can find $(g_{++}, g_{-+})\neq (0,0)$ which satisfies this equation. Similarly,

$$ \left\{ \begin{array}{l} Dg^{-1} \in B^+ \\ gC \in B^+ \end{array} \right. \Leftrightarrow \begin{pmatrix} c_{++} & c_{-+} \\ -d_{--} & d_{-+} \end{pmatrix} \begin{pmatrix} g_{-+} \\ g_{--} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}.$$

Since $CD\in B^+ \Leftrightarrow \det  \begin{pmatrix} c_{++} & c_{-+} \\ -d_{--} & d_{-+} \end{pmatrix} =0$, one can find $(g_{-+}, g_{--})\neq (0,0)$ which satisfies this equation. Up to multiplying $(g_{-+}, g_{--})$ by a non zero scalar, one can further suppose that $g_{++}g_{--}-g_{+-}g_{-+}=1$, so the matrix $g= \begin{pmatrix} g_{++} & g_{+-} \\ g_{-+} & g_{--} \end{pmatrix}$ satisfies the conclusion of the lemma.

Since $F$ is a bijection, we need to prove that $G$ is surjective. Let us first define an action of $\SL_2(\mathbb{C})$ on $\mathcal{R}^w$ as follows. For $g\in \SL_2(\mathbb{C})$ and $\rho \in \mathcal{R}^w$, we define $g\bullet \rho \in \mathcal{R}^w$ as the functor which sends a path $\beta: v_1 \to v_2$ in  $\Pi_1(U\Sigma, \widehat{\mathbb{V}} \cup \{ \overrightarrow{w}\})$ to

$$ (g\bullet \rho) (\beta) := \left\{ \begin{array}{ll}
\rho(\beta) & \mbox{, if }v_1 \neq \overrightarrow{w}, v_2 \neq \overrightarrow{w}; \\
\rho(\beta)g & \mbox{, if }v_1 \neq \overrightarrow{w}, v_2 = \overrightarrow{w}; \\
g^{-1}\rho(\beta) & \mbox{, if }v_1 = \overrightarrow{w}, v_2 \neq \overrightarrow{w}; \\
g^{-1}\rho(\beta)g & \mbox{, if }v_1 = v_2 = \overrightarrow{w}.
\end{array} \right. $$

Clearly $G(g\bullet \rho) = G(\rho)$ so $G$ induces a map $G': \quotient{ \mathcal{R}^w}{\SL_2(\mathbb{C})} \to \mathcal{R}^{tw}_{\SL_2}(\mathbf{\Sigma})$. Let us prove that $G'$ is surjective (it is not difficult to prove that it is an isomorphism).
Let $c$ be a boundary edge of $\mathbf{\Sigma}$ (which exists by the  hypotheses made in Theorem <ref>) and consider an arbitrary path $\beta: \overrightarrow{v_c} \to \overrightarrow{w}$. Let $\rho \in  \mathcal{R}^{tw}_{\SL_2}(\mathbf{\Sigma})$ and define an extension $\widehat{\rho} \in \mathcal{R}^w$ as follows. First, the restriction of $\widehat{\rho}$ to $\Pi_1(U\Sigma, \widehat{\mathbb{V}})$ is $\rho$ and $\widehat{\rho}(\beta)= \mathds{1}_2$. Next for a path $\gamma: \overrightarrow{w} \to \overrightarrow{v}$ with $\overrightarrow{v}\in \widehat{V}$, set $\widehat{\rho}(\gamma):= \rho(\gamma \beta^{-1})$. Similarly for a path $\gamma': \overrightarrow{v} \to \overrightarrow{w}$ set $\widehat{\rho}(\gamma'):= \rho(\beta^{-1}\gamma' )$. Clearly  $\widehat{\rho} \in \mathcal{R}^w$ and $G(\widehat{\rho})=\rho$ so $G$ and $G'$  are surjective.
Let us prove that the restriction $\varphi$ of $G\circ F$ is surjective as well. Let $p_1, p_1'$ be the two punctures adjacent to $a_1$ and $p_2, p_2'$ the two punctures adjacent to $a_2$ such that while gluing $a_1$ with $a_2$, $p_1$ and $p_2$ have the same image $p\in \partial \Sigma$ and $p_1', p_2'$ have the same image $p'\in \partial \Sigma$ (see Figure <ref>). By assumption, $p_1\neq p_1'$ and $p_2\neq p_2'$.
Let $\rho \in \overline{\mathcal{R}}^{tw}_{\SL_2}(\mathbf{\Sigma})$. By surjectivity of $G$, there exists $\widehat{\rho} \in \mathcal{R}^w$ such that $G(\widehat{\rho})=\rho$. Then $\widehat{\rho}$ is the image by $F$ of an element of
$ \overline{\mathcal{R}}^{tw}_{\SL_2}(\mathbf{\Sigma}_1) \times \overline{\mathcal{R}}^{tw}_{\SL_2}(\mathbf{\Sigma}_2) $ if and only if for all $p\in \mathcal{P}^{\partial_1} \cup \mathcal{P}^{\partial_2}$, then $\widehat{\rho}(\widehat{\alpha(p)}) \in B^+$. Since  $G(\widehat{\rho})=\rho \in  \overline{\mathcal{R}}^{tw}_{\SL_2}(\mathbf{\Sigma})$, $\widehat{\rho}(\widehat{\alpha(p)}) \in B^+$ for all $p\notin \{p_1,p_1', p_2, p_2'\}$. Write $A:= \widehat{\rho}(\widehat{\alpha(p_1)})$, $B:=  \widehat{\rho}(\widehat{\alpha(p_2)})$, $C:=  \widehat{\rho}(\widehat{\alpha(p_2')})$, $D:=  \widehat{\rho}(\widehat{\alpha(p_1')})$. Then $AB= \rho (\widehat{\alpha(p)}) \in B^+$ and $CD=\rho (\widehat{\alpha(p')})\in B^+$ so by Lemma <ref>, there exists $g\in \SL_2(\mathbb{C})$ such that  $Ag$, $g^{-1}B$, $Dg^{-1}$ and $gC$ are in $B^+$. Therefore, writing  $\rho':= g\bullet \widehat{\rho}$ one has $\restriction{\rho'}{\Pi_1}\in \overline{\mathcal{R}}^{tw}_{\SL_2}(\mathbf{\Sigma}_1)$, $\restriction{\rho'}{\Pi_2} \in \overline{\mathcal{R}}^{tw}_{\SL_2}(\mathbf{\Sigma}_2)$ and $\varphi(\restriction{\rho'}{\Pi_1}, \restriction{\rho'}{\Pi_2)} = \rho$. So $\varphi$ is surjective.

The surjectivity of $p_{a_1\# a_2}$ follows from the commutativity of the following diagram

$$ \begin{tikzcd}
X(\mathbf{\Sigma}_1) \times X(\mathbf{\Sigma}_2)
\ar[r, "p_{a_1\# a_2}"] \ar[d, "\Psi_{\mathbf{\Sigma}_1}\times \Psi_{\mathbf{\Sigma}_2}", "\cong"'] &
X(\mathbf{\Sigma}) \ar[d, "\Psi_{\mathbf{\Sigma}}", "\cong"'] \\
\overline{\mathcal{R}}^{tw}_{\SL_2}(\mathbf{\Sigma}_1) \times \overline{\mathcal{R}}^{tw}_{\SL_2}(\mathbf{\Sigma}_2)
\ar[r, "\varphi"] &
\overline{\mathcal{R}}^{tw}_{\SL_2}(\mathbf{\Sigma})
\end{tikzcd} $$

together with the surjectivity of $\varphi$ proved in Lemma <ref>. Let us prove that  $\widehat{p}_{a_1\# a_2}$ is surjective as well. Let $\widehat{x}=({x}, h_p, h_{\partial})_{p, \partial} \in \widehat{X}(\mathbf{\Sigma})$ and fix $(x_1,x_2) \in X(\mathbf{\Sigma}_1)\times X(\mathbf{\Sigma}_2)$ such that $p_{a_1\# a_2}(x_1,x_2)=x$. Let $\partial_1$ be the boundary component of $\Sigma_1$ containing $a_1$ and $\partial_2$ the boundary component of $\Sigma_2$ containing $a_2$ and $\partial'$ the boundary component of $\Sigma$ obtained by $1$-surgery on $\partial_1\cup \partial_2$. By definition $h_{\partial'}^N=\chi_x(\alpha_{\partial})$. Since $\theta_{a_1\# a_2}(\alpha_{\partial_1}\otimes \alpha_{\partial_2})=\alpha_{\partial}$, one can find (and do fix) $h_{\partial_1}, h_{\partial_2} \in \mathbb{C}^*$ such that $h_{\partial}=h_{\partial_1}h_{\partial_2}$ and $h_{\partial_1}^N=\chi_{x_1}(\alpha_{\partial_1})$, $h_{\partial_2}^N= \chi_{x_2}(\alpha_{\partial_2})$. Define $(\widehat{x}_1, \widehat{x}_2)\in \widehat{X}(\mathbf{\Sigma}_1)\times \widehat{X}(\mathbf{\Sigma}_2)$ by, for $i=1,2$,  $\widehat{x}_i= (x_i, h_p, h_{\partial})_{p \in \mathring{\mathcal{P}}_i, \partial \in \Gamma_i^{\partial}}$ where for $p\in \mathring{\mathcal{P}}_i  \ \subset \mathring{\mathcal{P}}$, $h_p$ is the puncture invariant of $\widehat{x}$, for $\partial \in \Gamma_i^{\partial} \setminus \{ \partial_i\}$ then $h_{\partial}$ is the boundary invariant of $\widehat{x}$ and $h_{\partial_i}$ is the previously chosen scalar. Then $\widehat{p}_{a_1\# a_2}(\widehat{x}_1, \widehat{x}_2)= \widehat{x}$, thus $\widehat{p}_{a_1\# a_2}$ is surjective.

§ REPRESENTATIONS OF $ \OVERLINE{\MATHCAL{S}}_{A}(\MATHBB{D}_1)$

Let $\mathbb{D}_1=(\Sigma_{0,2}, \{a_L, a_R\})$ be an annulus with two boundary edges in the same boundary component (i.e. a punctured bigon). It admits a presenting graph with two edges $\alpha, \beta$ represented in Figure <ref> (note that $\alpha$ and $\beta^{-1}$ are corner arcs). The algebra $ \overline{\mathcal{S}}_{A}(\mathbb{D}_1)$ is isomorphic to the Drinfeld double of the quantum Borel algebra $\Dq$ described as follows.

Two oriented arcs $\alpha$ and $\beta$ in $\mathbb{D}_1$.

Write $q:= A^2$. The algebra $\Dq$ is presented by the generators $K^{\pm 1/2}, L^{\pm 1/2}, E$ and $F$ together with the following relations:

\begin{align*}
&EK^{1/2}= q^{-1}K^{1/2}E; \quad EL^{1/2}= qL^{1/2} E; \quad  FK^{1/2}=qK^{1/2}F; \quad FL^{1/2}= q^{-1}L^{1/2}F; \\
&xy=yx \mbox{, for all }x,y\in \{K^{\pm 1/2}, L^{\pm 1/2} \}; \quad K^{1/2}K^{-1/2}=L^{1/2}L^{-1/2}= 1; \\
&EF-FE= \frac{K-L}{q-q^{-1}}.
\end{align*}

There is an isomorphism of algebras $\Psi:  \Dq \xrightarrow{\cong}  \overline{\mathcal{S}}_{A}(\mathbb{D}_1)$ defined by

\begin{align*}
{} & \Psi(K^{1/2})= \alpha_{--}, \quad \Psi(K^{-1/2})= \alpha_{++}, \quad \Psi(L^{1/2})= \beta_{--}, \quad \Psi(L^{-1/2})=\beta_{++} \\
{} & \Psi(E)=\frac{-A}{q-q^{-1}} \alpha_{+-}\alpha_{--}, \quad \Psi(F)= \frac{A^{-1}}{q-q^{-1}}\beta_{--} \beta_{-+}
\end{align*}

Note that $H_{\partial}:=K^{-1/2}L^{-1/2}$ is a central element such that $\Psi(H_{\partial})=h_{\partial}$ for $\partial$ the connected component containing $a_L$ and $a_R$. Also, if $p$ denotes the only inner puncture of $\mathbb{D}_1$, then

\begin{equation}\label{eq_Casimir}
C:= - \frac{\Psi^{-1}(\gamma_p)H_{\partial}^{-1}}{(q-q^{-1})^2} = EF+\frac{qL +q^{-1}K}{(q-q^{-1})^2} = FE+ \frac{qK+q^{-1}L}{(q-q^{-1})^2}
\end{equation}

is the Casimir element of $\Dq$. Let $\widetilde{U}_q \mathfrak{sl}_2:= \quotient{\Dq}{(H_{\partial} -1)}$ and $U_q\mathfrak{sl}_2 \subset \widetilde{U}_q \mathfrak{sl}_2$ the subalgebra generated by $K^{\pm 1}, E$ and $F$. The simple modules of  $U_q\mathfrak{sl}_2$ have been classified in [35, 7] (see also  <cit.>) whereas many indecomposable semi-weight $U_q\mathfrak{sl}_2$ modules were found and studied in [8]. We now provide a classification of the indecomposable (semi) weight $\Dq$ modules as follows.

We define three families of indecomposable weight $\Dq$-modules as follows. For $n\in \mathbb{Z}$, we write $[n]:=\frac{q^n-q^{-n}}{q-q^{-1}}$.

* For $\lambda, \mu\in \mathbb{C}^*$ and $a,b\in \mathbb{C}$, the module $V(\lambda, \mu, a,b)$ has canonical basis $(v_0, \ldots, v_{N-1})$ and module structure given by:

\begin{align*}
& K^{1/2}v_i = \lambda q^{-i}v_i; \quad L^{1/2}v_i = \mu q^i v_i \mbox{, for }i\in \{0, \ldots, N-1\}; \\
& Fv_{N-1}=bv_0; \quad Fv_i=v_{i+1} \mbox{, for }i \in \{0, \ldots, N-2\}; \\
& Ev_0=a v_{N-1}; \quad Ev_{i}= \left( \frac{q^{-i+1}\lambda^2 - q^{i-1}\mu^2}{q-q^{-1}} [i] +ab \right) v_{i-1}\mbox{, for }i\in \{1, \ldots, N-1\}.
\end{align*}

* For $\lambda, \mu \in \mathbb{C}^*$, $c\in \mathbb{C}$, the module $\widetilde{V}(\lambda, \mu, c)$ has canonical basis $(w_0, \ldots, w_{N-1})$  and module structure given by:

\begin{align*}
& K^{1/2}w_i= \lambda q^{i}w_i; \quad L^{1/2}w_i= \mu q^{-i}w_i \mbox{, for }i \in \{0, \ldots, N-1\}; \\
& Ew_{N-1}= cw_0; \quad Ew_i= w_{i+1} \mbox{, for }i\in \{0, \ldots, N-2\}; \\
& F w_0= 0; \quad Fw_{i} = \left( \frac{ \mu^2q^{-i+1} - \lambda^2 q^{i-1} }{q-q^{-1}}[i] \right) w_i \mbox{, for }i\in \{1, \ldots, N-1\}.
\end{align*}

* For $\mu \in \mathbb{C}^*$, $\varepsilon \in \{-1,+1\}$, $0\leq n \leq N-1$, the module $S_{\mu, \varepsilon, n}$ has canonical basis $(e_0, \ldots, e_n)$ and module structure given by:

\begin{align*}
&K^{1/2}e_i = \varepsilon \mu A^{n-2i} e_i; \quad L^{1/2}e_i= \mu A^{2i-n}e_i \mbox{, for }i\in \{0, \ldots, n\}; \\
&Fe_n=0; \quad Fe_i = e_{i+1} \mbox{, for }i\in \{0,\ldots, n-1\}; \\
&Ee_0=0; \quad Ee_i= \mu^2 [i][n-i+1]e_{i-1}\mbox{, for }i\in \{1,\ldots, n\}.
\end{align*}

We define three families of indecomposable semi-weight representations as follows.

* For $\lambda, \mu, b \in \mathbb{C}^*$ and $a\in \mathbb{C}$ such that

$$ h_p:= -(q-q^{-1})^2(\lambda\mu)^{-1}ab-\lambda \mu^{-1}q - \mu\lambda^{-1}q^{-1} = \pm (q^n+q^{-n}) \quad \mbox{ for some }n\in \{1, \ldots, (N-1)/2\}$$

the module $P(\lambda, \mu, a,b)$ has canonical basis $(x_0, \ldots, x_{N-1}, y_0, \ldots, y_{N-1})$ and module structure given by:

\begin{align*}
& K^{1/2}x_i=\lambda q^{-i}x_i, K^{1/2}y_i=\lambda q^{-i}y_i, L^{1/2}x_i= \mu q^i x_i, L^{1/2}y_i=\mu q^i y_i \mbox{ for }i \in \{0, \ldots, N-1\}; \\
& Fx_{N-1}=b x_0, Fy_{N-1}=by_0, Fx_i= x_{i+1}, Fy_i=y_{i+1} \mbox{ for }i\in \{0, \ldots, N-2\}; \\
& Ex_0=ax_{N-1}, Ex_i=\left( \frac{q^{-i+1}\lambda^2 - q^{i-1}\mu^2}{q-q^{-1}} [i] +ab \right)x_{i-1} \mbox{ for }i\in \{1, \ldots, N-1\}; \\
& Ey_0=a y_{N-1}+b^{-1}x_{N-1}, E y_i= \left( \frac{q^{-i+1}\lambda^2 - q^{i-1}\mu^2}{q-q^{-1}} [i] +ab \right)y_{i-1} + x_{i-1}  \mbox{ for }i\in \{1, \ldots, N-1\}.
\end{align*}

* For $\lambda, \mu, c \in \mathbb{C}^*$ such that $\lambda \mu^{-1}= \pm q^n$ for some $n\in \{1, \ldots, (N-1)/2\}$, the module $\widetilde{P}(\lambda, \mu, c)$ has canonical basis $(x_0, \ldots, x_{N-1}, y_0, \ldots, y_{N-1})$ and module structure given by:

\begin{align*}
& K^{1/2}x_i= \lambda q^i x_i, K^{1/2}y_i= \lambda q^i y_i, L^{1/2}x_i= \mu q^{-i}x_i, L^{1/2}y_i= \mu q^{-i} y_i  \mbox{ for }i \in \{0, \ldots, N-1\}; \\
& Ex_{N-1} = cx_0, Ey_{N-1}=cy_0, Ex_i= x_{i+1}, Ey_i= y_{i+1} \mbox{ for }i\in \{0, \ldots, N-2\}; \\
& Fx_0= 0, Fx_i= \left( \frac{ \mu^2q^{-i+1} - \lambda^2 q^{i-1} }{q-q^{-1}}[i] \right) x_{i-1}  \mbox{ for }i\in \{1, \ldots, N-1\}; \\
& Fy_0=x_0, Fy_i=  \left( \frac{ \mu^2q^{-i+1} - \lambda^2 q^{i-1} }{q-q^{-1}}[i] \right) y_{i-1} + x_i  \mbox{ for }i\in \{1, \ldots, N-1\}.
\end{align*}

* For $\mu \in \mathbb{C}^*$, $\varepsilon \in \{-1,+1\}$, $0\leq n \leq N-2$, the module $P_{\mu, \varepsilon, n}$ has canonical basis $(x_0, \ldots, x_{N-1}, y_0, \ldots, y_{N-1})$ and module structure given by:

\begin{align*}
& K^{1/2} x_i = \varepsilon \mu A^{-2-n-2i}x_i, K^{1/2}y_i= \varepsilon \mu A^{n-2i}y_i, \quad L^{1/2} x_i= \mu A^{n+2i+2} x_i, L^{1/2}y_i=\mu A^{2i-n}y_i \mbox{, for }i\in \{0, \ldots, N-1\}; \\
& F x_i = x_{i+1}, \quad Fy_i= y_{i+1} \quad \mbox{( where }y_N=x_N=0)\mbox{, for }i\in \{0, \ldots, N-1\}; \\
& Ex_0=0, \quad Ex_i= -\mu^2[i][n+i+1] x_{i-1} \mbox{, for }i\in \{1, \ldots, N-1\}; \\
& Ey_i= \mu^2[i][n-i+1]y_{i-1} + x_{N-n+i-2} \mbox{, for }i\in \{0, \ldots, n\}, \quad E y_i = \mu^2[i] [n-i+1]y_{i-1}   \mbox{, for }i\in \{n+1, \ldots, N-1\}.
\end{align*}

For $0\leq n \leq N-1$, write $\overline{n}:=N-2-n$.

* For $\mu \in \mathbb{C}^*$, $\varepsilon\in \{-1, +1\}$, $0\leq n \leq N-2$, $k\geq 1$, the module $\Omega^{-k}_{\mu, \varepsilon, n}$ has basis $(e_i^j, \overline{e}_{i'}^{j'}, 0\leq i \leq n, 0\leq i' \leq \overline{n}, 1\leq j \leq k, 1\leq j' \leq k+1)$ and module structure given by:

\begin{align*}
& K^{1/2}e_i^j = \varepsilon \mu A^{n-2i}e_i^j, K^{1/2}\overline{e}_i^j =\varepsilon \mu A^{\overline{n}-2i}\overline{e}_i^j, L^{1/2}e_i^j = \mu A^{2i-n}e_i^j, L^{1/2}\overline{e}_i^j= \mu A^{2i-\overline{n}}\overline{e}_i^j; \\
& F e_n^j = \overline{e}_0^{j+1},  F\overline{e}_{\overline{n}}^j = 0, Fe_i^j = e_{i+1}^j \mbox{ for }i\neq n , F \overline{e}_i^j = \overline{e}_{i+1}^j \mbox{ for } i\neq \overline{n}; \\
& E e_0^j= \overline{e}_{\overline{n}}^j ,  E \overline{e}_0^j = 0,  Ee_i^j = \mu^2 [i][n-i+1] e_{i-1}^j, E \overline{e}_i^j = \mu^2 [i][\overline{n}-i+1]\overline{e}_{i-1}^j \mbox{ for } i\neq 0.
\end{align*}

* For $\mu \in \mathbb{C}^*$, $\varepsilon\in \{-1, +1\}$, $0\leq n \leq N-2$, $k\geq 1$, the module $\Omega^{k}_{\mu, \varepsilon, n}$ has basis $(e_i^j, \overline{e}_{i'}^{j'}, 0\leq i \leq n, 0\leq i' \leq \overline{n}, 1\leq j \leq k+1, 1\leq j' \leq k)$ and module structure given by:

\begin{align*}
& K^{1/2}e_i^j = \varepsilon \mu A^{n-2i}e_i^j, K^{1/2}\overline{e}_i^j =\varepsilon \mu A^{\overline{n}-2i}\overline{e}_i^j, L^{1/2}e_i^j = \mu A^{2i-n}e_i^j, L^{1/2}\overline{e}_i^j= \mu A^{2i-\overline{n}}\overline{e}_i^j; \\
& Fe_n^{k+1}=0, F e_n^j = \overline{e}_0^{j} \mbox{ for }j\neq k+1,  F\overline{e}_{\overline{n}}^j = 0, Fe_i^j = e_{i+1}^j \mbox{ for }i\neq n , F \overline{e}_i^j = \overline{e}_{i+1}^j \mbox{ for } i\neq \overline{n}; \\
& Ee_0^1=0, E e_0^j= \overline{e}_{\overline{n}}^{j-1} \mbox{ for }j\neq 1 ,  E \overline{e}_0^j = 0,  Ee_i^j = \mu^2 [i][n-i+1] e_{i-1}^j, E \overline{e}_i^j = \mu^2 [i][\overline{n}-i+1]\overline{e}_{i-1}^j \mbox{ for } i\neq 0.
\end{align*}

* For $\mu \in \mathbb{C}^*$, $\varepsilon\in \{-1, +1\}$, $0\leq n \leq N-2$, $k\geq 1$ and $\lambda= \in \mathbb{CP}^1= \mathbb{C} \cup \{\infty \}$, the module  $M^k_{\mu, \varepsilon, n}(\lambda)$ has basis $(e_i^j, \overline{e}_{i'}^{j'}, 0\leq i \leq n, 0\leq i' \leq \overline{n}, 1\leq j \leq k, 1\leq j' \leq k)$ and module structure given as follows.
If $\lambda \in \mathbb{C}$ then

\begin{align*}
& K^{1/2}e_i^j = \varepsilon \mu A^{n-2i}e_i^j, K^{1/2}\overline{e}_i^j =\varepsilon \mu A^{\overline{n}-2i}\overline{e}_i^j, L^{1/2}e_i^j = \mu A^{2i-n}e_i^j, L^{1/2}\overline{e}_i^j= \mu A^{2i-\overline{n}}\overline{e}_i^j; \\
& Fe_n^k = \lambda \overline{e}_0^k,  F e_n^j = \lambda \overline{e}_0^{j} + \overline{e}_0^{j+1} \mbox{ for }j\neq k,  F\overline{e}_{\overline{n}}^j = 0, Fe_i^j = e_{i+1}^j \mbox{ for }i\neq n , F \overline{e}_i^j = \overline{e}_{i+1}^j \mbox{ for } i\neq \overline{n}; \\
&  E e_0^j= \overline{e}_{\overline{n}}^j,  E \overline{e}_0^j = 0,  Ee_i^j = \mu^2 [i][n-i+1] e_{i-1}^j, E \overline{e}_i^j = \mu^2 [i][\overline{n}-i+1]\overline{e}_{i-1}^j \mbox{ for } i\neq 0.
\end{align*}

If $\lambda= \infty \in \mathbb{CP}^1$, then

\begin{align*}
& K^{1/2}e_i^j = \varepsilon \mu A^{n-2i}e_i^j, K^{1/2}\overline{e}_i^j =\varepsilon \mu A^{\overline{n}-2i}\overline{e}_i^j, L^{1/2}e_i^j = \mu A^{2i-n}e_i^j, L^{1/2}\overline{e}_i^j= \mu A^{2i-\overline{n}}\overline{e}_i^j; \\
&  F e_n^j =  \overline{e}_0^{j},  F\overline{e}_{\overline{n}}^j = 0, Fe_i^j = e_{i+1}^j \mbox{ for }i\neq n , F \overline{e}_i^j = \overline{e}_{i+1}^j \mbox{ for } i\neq \overline{n}; \\
&  E e_0^k= 0, E e_0^j= \overline{e}_{\overline{n}}^{j+1} \mbox{ for }j \neq k,  E \overline{e}_0^j = 0,  Ee_i^j = \mu^2 [i][n-i+1] e_{i-1}^j, E \overline{e}_i^j = \mu^2 [i][\overline{n}-i+1]\overline{e}_{i-1}^j \mbox{ for } i\neq 0.
\end{align*}

We postpone the proof of the following theorem to the next subsection. Let $\mathcal{C}$ be the category of weight $\Dq$ modules and $\overline{\mathcal{C}}$ the category of semi-weight $\Dq$ modules.

* Every modules $V(\lambda, \mu, a,b), \widetilde{V}(\lambda, \mu, c), S_{\mu, \varepsilon, n}$ are weight indecomposable and every weight indecomposable $\Dq$ module is isomorphic to one of them.

* Every modules $P(\lambda, \mu, a,b), \widetilde{P}(\lambda, \mu, c), P_{\mu, \varepsilon, n}, \Omega^k_{\mu, \varepsilon, n}, \Omega^{-k}_{\mu, \varepsilon, n}, M_{\mu, \varepsilon, n}^k(\lambda)$ are semi weight indecomposable and every indecomposable semi-weight module is either isomorphic to one of them or is a weight module.

* The modules $S_{\mu, \varepsilon, n}$ are simple. The module $\widetilde{V}(\lambda, \mu, c)$ is simple if and only if either $c\neq 0$ or $\lambda \mu^{-1} \neq \pm q^{n-1}$ for all $n\in \{1, \ldots, N-1\}$. The module $V(\lambda, \mu, a, b)$ is simple if and only if either $\prod_{i\in \mathbb{Z}/N\mathbb{Z}} (ab+\frac{q^{1-i}\lambda^2 - q^{i-1}\mu^2}{q-q^{-1}}[i])\neq 0$ or $\lambda \mu^{-1} \neq \pm q^{n-1}$ for all $n\in \{1, \ldots, N-1\}$. Any simple $\Dq$-module is isomorphic to one of these modules.

* The weight representations $S_{\mu, \epsilon, N-1}$, $V(\lambda, \mu, a,b)$ and $\widetilde{V}(\lambda, \mu, c)$ are projective objects in $\mathcal{C}$ and any indecomposable projective object of $\mathcal{C}$ is isomorphic to one of them.

* The semi weight representations $S_{\mu, \epsilon, N-1}$, $P_{\mu, \epsilon, n}$, $P(\lambda, \mu, a,b)$ and $\widetilde{P}(\lambda, \mu, c)$ are projective objects in $\overline{\mathcal{C}}$ and any indecomposable projective object of $\overline{\mathcal{C}}$ is isomorphic to one of them.

Let us identify $X(\mathbb{D}_1)$ with the set of pairs $g=(g_+, g_-)$ where $g_+ \in B^+$ and $g_-\in B^-$  are upper and lower triangular matrices of $\SL_2(\mathbb{C})$. More precisely, for $x\in X(\mathbb{D}_1)$, we set

\begin{align*}
{} & g_+:= \begin{pmatrix} \chi_x(\alpha_{++}^N) &  \chi_x(\alpha_{+-}^N) \\ 0 & \chi_x(\alpha_{--}^N) \end{pmatrix} = \chi_x\begin{pmatrix} K^{-N/2} & -(q-q^{-1})^N E^N K^{-N/2} \\ 0 & K^{N/2} \end{pmatrix}, \quad
\mbox{ and } \\
{}&  g_-:= \begin{pmatrix} \chi_x(\beta_{++}^N) &  0\\ \chi_x(\beta_{-+}^N)  & \chi_x(\beta_{--}^N) \end{pmatrix}= \chi_x \begin{pmatrix} L^{-N/2} & 0 \\ (q-q^{-1})^N L^{-N/2}F^N & L^{N/2} \end{pmatrix}
\end{align*}

Write $\varphi(g):=g_{-}^{-1}g_+$. $\widehat{X}(\mathbb{D}_1)$ is identified with the set of elements $(g, h_p, h_{\partial})$ with $g=(g_+, g_-) \in X(\mathbb{D}_1)$, $T_N(\gamma_p)=-\tr (\varphi(g))$ and $h_{\partial}^{-N}$ is the left upper matrix coefficient of $g_-g_+$. An indecomposable representation of weight   $(g, h_p, h_{\partial})$ is called cyclic/ semi-cyclic/ diagonal/ central if $\varphi(g)\in \SL_2(\mathbb{C})$ is not triangular/ triangular not diagonal/ diagonal not central/ central (i.e. $\varphi(g)=\pm \mathds{1}_2$) respectively.

Write $\Delta_+(\widehat{\mathfrak{sl}}_2):= \{ (n,m) \in \mathbb{N}^2 | (n-m)^2 \leq 1\}$ and decompose it as

$\Delta_+(\widehat{\mathfrak{sl}}_2)= \{(0,1), (1,0)\} \sqcup \Delta^{\Re}_{++}(\widehat{\mathfrak{sl}}_2)\sqcup \Delta^{Im}_+(\widehat{\mathfrak{sl}}_2)$ where

$$  \Delta^{\Re}_{++}(\widehat{\mathfrak{sl}}_2)= \{ (k, k+1), k\geq1\} \cup \{(k+1, k), k\geq 1\}, \quad \Delta^{Im}_+(\widehat{\mathfrak{sl}}_2)=\{ (k,k), k\geq 1\}.$$

Further write

$$ \Delta:= \{S, P\} \sqcup   \Delta^{\Re}_{++}(\widehat{\mathfrak{sl}}_2) \sqcup  \Delta^{Im}_+(\widehat{\mathfrak{sl}}_2)\times \mathbb{CP}^1.$$

Given $\mu \in \mathbb{C}^*$, $\varepsilon \in \{-1, +1\}$, $n\in \{0, \ldots, N-2\}$, define a map $\underline{\sigma}_{\mu, \varepsilon, n}: \Delta \to \Indecomp(U_q\mathfrak{gl}_2)$ by

$$ \underline{\sigma}_{\mu, \varepsilon, n}: S \mapsto S_{\mu, \varepsilon, n}, P\mapsto P_{\mu, \varepsilon, n}, (k+1, k)\mapsto \Omega^k_{\mu, \varepsilon, n}, (k, k+1) \mapsto \Omega^{-k}_{\mu, \varepsilon, n}, ((k,k), \Lambda) \mapsto M^k_{\mu, \varepsilon, n}(\Lambda).$$

Consider the set $\Delta \sqcup \overline{\Delta}$ made of two copies of $\Delta$; said differently $\Delta \sqcup \overline{\Delta}:= \Delta \times \{0,1\}$ and for $\alpha \in \Delta$ we write $\alpha:= (\alpha, 0) \in \Delta \bigsqcup \overline{\Delta}$ and $\overline{\alpha}:= (\alpha, 1) \in \Delta \sqcup \overline{\Delta}$. For $\mu \in \mathbb{C}^*$, $\varepsilon \in \{-1, +1\}$, $n\in \{0, \ldots, (N-3)/2\}$, recall that $\overline{n}=N-n-2$ and  consider the map

$$\sigma_{\mu, \varepsilon, n}:= \underline{\sigma}_{\mu, \varepsilon, n}\sqcup \underline{\sigma}_{\mu, \varepsilon, \overline{n}} : \Delta \sqcup \overline{\Delta} \to  \Indecomp(\Dq).$$

As a consequence of Theorem <ref>, we obtain the

Let $\overline{\mathcal{C}}$ be the category of finite dimensional semi-weight $D_q$ modules.

* The Azumaya locus of $\Dq \cong \overline{\mathcal{S}}_A(\mathbb{D}_1)$ is the set of elements $(g, h_p, h_{\partial})$ such that either $\varphi(g)$ is not central or $\varphi(g)=\pm \mathds{1}_2$ and $h_p=\mp 2$ so it is equal to the regular locus $\widehat{X}^{reg}(\mathbb{D}_1)$.

* The fully Azumaya locus of $\Dq \cong \overline{\mathcal{S}}_A(\mathbb{D}_1)$ is the set of elements $g\in X(\mathbb{D}_1)$ such that $\varphi(g)\neq \pm \mathds{1}$. For such a $g$, the isomorphism classes of semi-weight indecomposable $D_q$ modules with classical shadow $g$ are completely determined by their full shadow.

* For $\widehat{g}=(g, h_p, h_{\partial})\in \widehat{X}(\mathbb{D}_1)$ with $\varphi(g)\neq \pm \mathds{1}$ then

* If $\tr(\varphi(g))\neq \pm 2$, then up to isomorphism there is a unique indecomposable semi-weight module representation $S_{\widehat{g}}$ with maximal shadow $\widehat{g}$: it is the Azumaya representation which is both simple and projective in $\overline{\mathcal{C}}$.

* If $\tr(\varphi(g))=\pm 2$, then up to isomorphism there exist two indecomposable semi-weight representations $S_{\widehat{g}}, P_{\widehat{g}}$ with maximal shadow $\widehat{g}$. $S_{\widehat{g}}$ is the Azumaya representation and is simple not projective. $P_{\widehat{g}}$ is projective in $\overline{\mathcal{C}}$ not simple and we have a non split exact sequence

$$ 0 \to S_{\widehat{g}} \to P_{\widehat{g}} \to S_{\widehat{g}} \to 0.$$

* Let $\widehat{x}=(g,h_p, h_{\partial})\in \widehat{X}(\mathbb{D}_1)$ which is not in the Azumaya locus, so one has $g=\left( \varepsilon \begin{pmatrix} \Lambda & 0 \\ 0 & \Lambda^{-1} \end{pmatrix},  \begin{pmatrix} \Lambda & 0 \\ 0 & \Lambda^{-1} \end{pmatrix} \right)$ for some $\varepsilon = \pm 1$ and $\Lambda \in \mathbb{C}^*$ and $h_p=-\varepsilon (q^{n+1} + q^{-(1+n)})$ for some $n\in \{0, \ldots, (N-3)/2\}$. Let $\mu \in \mathbb{C}^*$ be the unique scalar such that $\mu^{-N} = \Lambda$ and $h_{\partial}= \varepsilon \mu^{-2}$. Then $\sigma_{\mu, \varepsilon, n}$ is a bijection between $\Delta \sqcup \overline{\Delta} $ and the set of isomorphism classes of indecomposable $\Dq$ modules with classical shadow $\widehat{x}$.

In particular, Theorem <ref> is verified for $\mathbb{D}_1$.

Being a projective object in the category $\overline{\mathcal{C}}$ of semi-weight modules is different from being a projective module (i.e. from being a projective object in $\Dq-\Mod$). As we shall see, for $\widehat{g}=(g,h_p,h_{\partial})\in \widehat{X}(\mathbb{D}_1)$ with $\tr(\varphi(g))\neq \pm 2$, the Azumaya representation $S_{\widehat{g}}$ is a proper submodule of an indecomposable module of the form $P(\lambda, \mu, a, b)$ which is not semi-weight. So  $S_{\widehat{g}}$
is not a projective module though it is a projective object in $\overline{\mathcal{C}}$.

§.§ Classification of semi-weight $\Dq$ modules

In this subsection, we prove Theorem  <ref>. Denote by $\mathcal{C}$ the category of  weight $\Dq$ modules and by $\overline{\mathcal{C}}$ its category of semi-weight modules. For $0\leq n \leq N-1$, we write $\overline{n}:= N-n-2$.

§.§.§ Representations in the fully Azumaya locus

* For $0\leq n \leq N-2$, we have a non-split exact sequence

\begin{equation}\label{suitexacte}
0 \rightarrow S_{\mu, \varepsilon, \overline{n}} \xrightarrow{i} V(\varepsilon \mu A^n, \mu A^{-n}, 0, 0) \xrightarrow{p} S_{\mu, \varepsilon, n} \rightarrow 0,
\end{equation}

where the equivariant maps $i$ and $p$ are defined by $i(e_j):= v_{j+n+1}$ for $0\leq j  \leq N-n-2$ and $p(v_j):= \left\{ \begin{array}{ll} e_j &\mbox{, if }0\leq i \leq n; \\ 0 & \mbox{, else.} \end{array} \right.$

* One has non-split  exact sequences:

$$ 0 \to V(\varepsilon \mu A^{-2-n}, \mu A^{2+n}, 0, 0) \xrightarrow{\iota} P_{\mu, \varepsilon, n} \xrightarrow{p} V(\varepsilon \mu A^n, \mu A^{-n}, 0, 0) \to 0, $$

$$ 0 \to V(\lambda, \mu, a,b)  \xrightarrow{\iota} P(\lambda, \mu, a,b)  \xrightarrow{p}  V(\lambda, \mu, a,b) \to 0,$$

$$0 \to \widetilde{V}(\lambda, \mu, c)  \xrightarrow{\iota} \widetilde{P}(\lambda, \mu, c)  \xrightarrow{p}  \widetilde{V}(\lambda, \mu, c)\to 0, $$

where $\iota(v_i):= x_i$ and $p(x_i):=0$, $p(y_i):=v_i$ (resp. $\iota(w_i)=x_i$, $p(x_i)=0$, $p(y_i)=w_i$ in the third case).

The proof is a straightforward verification left to the reader.