Sliced skein algebras and geometric Kauffman bracket.(English)Zbl 07979655
Summary: 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 skein algebra \(\mathcal{S}_\xi(\mathfrak{S})\) corresponding to fixing the scalar values of its peripheral curves. We show that the sliced skein algebra of a finite type surface is a domain if the ground ring is a domain. When the quantum parameter \(\xi\) is a root of unity we calculate the center of the sliced skein algebra and its PI-degree. Among applications we show that any smooth point of a sliced character variety is an Azumaya point of the skein algebra \(\mathcal{S}_\xi(\mathfrak{S})\). For any \(S L_2(\mathbb{C})\)-representation \(\rho\) of the fundamental group of an oriented connected 3-manifold \(M\) and a root of unity \(\xi\) with the order of \(\xi^2\) odd, we introduce the \(\rho \)-reduced skein module \(\mathcal{S}_{\xi , \rho}(M)\). We show that \(\mathcal{S}_{\xi , \rho}(M)\) has dimension 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 a module over the skein algebra of its boundary, is simple and has the dimension equal to the PI-degree of the skein algebra of its boundary.
MSC:
| 57K31 | Invariants of 3-manifolds (including skein modules, character varieties) |
| 20G42 | Quantum groups (quantized function algebras) and their representations |
Cite
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