Stated $SL_n$-skein modules, roots of unity, and TQFT
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| Format: | Preprint |
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2024
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| _version_ | 1866908275625164800 |
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| author | Wang, Zhihao |
| author_facet | Wang, Zhihao |
| contents | For a pb surface $Σ$, two positive integers $m,n$ with $m\mid n$, and two invertible elements $v,ε$ in a commutative domain $R$ with $ε^{2m} = 1$, we construct an $R$-linear isomorphism between the stated $SL_n$-skein algebras $S_n(Σ,v)$ and $S_n(Σ,εv)$, which restricts to an algebraic ismorphism between subalgebras of $S_n(Σ,v)$ and $S_n(Σ,εv)$. Using this linear isomorphism, we prove the splitting map $Θ_{c}:S_n(Σ,v)\rightarrow S_n(\text{Cut}_c(Σ),v)$ for the pb surface $Σ$ 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 Lê's stated $SL_2$-TQFT theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09995 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stated $SL_n$-skein modules, roots of unity, and TQFT Wang, Zhihao Quantum Algebra For a pb surface $Σ$, two positive integers $m,n$ with $m\mid n$, and two invertible elements $v,ε$ in a commutative domain $R$ with $ε^{2m} = 1$, we construct an $R$-linear isomorphism between the stated $SL_n$-skein algebras $S_n(Σ,v)$ and $S_n(Σ,εv)$, which restricts to an algebraic ismorphism between subalgebras of $S_n(Σ,v)$ and $S_n(Σ,εv)$. Using this linear isomorphism, we prove the splitting map $Θ_{c}:S_n(Σ,v)\rightarrow S_n(\text{Cut}_c(Σ),v)$ for the pb surface $Σ$ 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 Lê's stated $SL_2$-TQFT theory. |
| title | Stated $SL_n$-skein modules, roots of unity, and TQFT |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/2401.09995 |