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The sliced skein algebra of a closed surface of genus $g$ with $m$ punctures,$\mathfrak{S}=\Sigma_{g,m}$, is the quotient of the Kauffman bracket skeinalgebra $\mathcal{S}_\xi(\mathfrak{S})$ corresponding to fixing the scalarvalues of its peripheral curves. We show that the sliced skein algebra of afinite type surface is a domain if the ground ring is a domain. When thequantum parameter $\xi$ is a root of unity we calculate the center of thesliced skein algebra and its PI-degree. Among applications we show that anysmooth point of a sliced character variety is a fully Azumaya point of theskein algebra $\mathcal{S}_\xi(\mathfrak{S})$. For any$SL_2(\mathbb{C})$--representation $\rho$ of the fundamental group of anoriented connected 3-manifold $M$ and a root of unity $\xi$ with odd$ord(\xi^2)$, we introduce the $\rho$-reduced skein module$\mathcal{S}_{\xi,\rho}(M)$. We show that $\mathcal{S}_{\xi,\rho}(M)$ hasdimension 1 when $M$ is closed and $\rho$ is irreducible. We also show that if$\rho$ is irreducible the $\rho$-reduced skein module of a handlebody, as amodule over the skein algebra of its boundary, is simple and has the dimensionequal to the PI-degree of the skein algebra of its boundary.

Sliced skein algebras and geometric Kauffman bracket