Sliced skein algebras and geometric Kauffman bracket
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866910325274574848 |
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| author | Frohman, Charles Kania-Bartoszynska, Joanna Lê, Thang |
| author_facet | Frohman, Charles Kania-Bartoszynska, Joanna Lê, Thang |
| contents | The sliced skein algebra of a closed surface of genus $g$ with $m$ punctures, $\mathfrak{S}=Σ_{g,m}$, is the quotient of the Kauffman bracket skein algebra $\mathcal{S}_ξ(\mathfrak{S})$ 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 $ξ$ 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 a fully Azumaya point of the skein algebra $\mathcal{S}_ξ(\mathfrak{S})$. For any $SL_2(\mathbb{C})$--representation $ρ$ of the fundamental group of an oriented connected 3-manifold $M$ and a root of unity $ξ$ with odd $ord(ξ^2)$, we introduce the $ρ$-reduced skein module $\mathcal{S}_{ξ,ρ}(M)$. We show that $\mathcal{S}_{ξ,ρ}(M)$ has dimension 1 when $M$ is closed and $ρ$ is irreducible. We also show that if $ρ$ is irreducible the $ρ$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_06189 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sliced skein algebras and geometric Kauffman bracket Frohman, Charles Kania-Bartoszynska, Joanna Lê, Thang Geometric Topology Quantum Algebra 57K31 The sliced skein algebra of a closed surface of genus $g$ with $m$ punctures, $\mathfrak{S}=Σ_{g,m}$, is the quotient of the Kauffman bracket skein algebra $\mathcal{S}_ξ(\mathfrak{S})$ 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 $ξ$ 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 a fully Azumaya point of the skein algebra $\mathcal{S}_ξ(\mathfrak{S})$. For any $SL_2(\mathbb{C})$--representation $ρ$ of the fundamental group of an oriented connected 3-manifold $M$ and a root of unity $ξ$ with odd $ord(ξ^2)$, we introduce the $ρ$-reduced skein module $\mathcal{S}_{ξ,ρ}(M)$. We show that $\mathcal{S}_{ξ,ρ}(M)$ has dimension 1 when $M$ is closed and $ρ$ is irreducible. We also show that if $ρ$ is irreducible the $ρ$-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. |
| title | Sliced skein algebras and geometric Kauffman bracket |
| topic | Geometric Topology Quantum Algebra 57K31 |
| url | https://arxiv.org/abs/2310.06189 |