  For a pb surface $\Sigma$, two positive integers $m,n$ with $m\mid n$, and
two invertible elements $v,\epsilon$ in a commutative domain $R$ with
$\epsilon^{2m} = 1$, we construct an $R$-linear isomorphism between the stated
$SL_n$-skein algebras $S_n(\Sigma,v)$ and $S_n(\Sigma,\epsilon v)$, which
restricts to an algebraic ismorphism between subalgebras of $S_n(\Sigma,v)$ and
$S_n(\Sigma,\epsilon v)$. Using this linear isomorphism, we prove the splitting
map $\Theta_{c}:S_n(\Sigma,v)\rightarrow S_n(\text{Cut}_c(\Sigma),v)$ for the
pb surface $\Sigma$ and the ideal arc $c$ is injective when $v^{2m} = 1$ and
$m\mid n$.
  We generalize Barrett's work to the $SL_n$-skein space and stated
$SL_n$-skein space. As an application, we prove the splitting map for the
marked 3-manifolds is always injective when the quantum parameter $v=-1$.
  Let $(M,\mathcal{N})$ be a connected marked 3-manifold with
$\mathcal{N}\neq\emptyset$, and let $(M,\mathcal{N}')$ be obtained from
$(M,\mathcal{N})$ by adding one extra marking. When $v^4 =1$, we prove the
$R$-linear map from $S_n(M,\mathcal{N},v)$ to $S_n(M,\mathcal{N}',v)$ induced
by the embedding $(M,\mathcal{N})\rightarrow (M,\mathcal{N}')$ is injective and
$S_n(M.\mathcal{N}',v) = S_n(M,\mathcal{N},v)\otimes_{R}O_{q_v}(SL_n)$, where
$O_{q_v}(SL_n)$ is the quantization of the regular function ring of $SL_n$.
This shows the splitting map for $S_n(M,\mathcal{N},v)$ is always injective.
  We formulate the stated $SL_n$-TQFT theory, which generalizes the Costantino
and L\^e's stated $SL_2$-TQFT theory.
