Holonomy and (stated) skein algebras in combinatorial quantization
Holonomy and (stated) skein algebras in combinatorial quantization
复制标题
组合量化中的完整性和（规定的）绞纱代数
作者：
Matthieu Faitg
The algebra $mathcal{L}_{g,n}(H)$ was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and quantizes the character variety of the Riemann surface $Sigma_{g,n}!setminus! D$ ($D$ is an open disk). In this article we define a holonomy map in that quantized setting, which associates a tensor with components in $mathcal{L}_{g,n}(H)$ to tangles in $(Sigma_{g,n}!setminus!D) imes [0,1]$, generalizing previous works of Buffenoir-Roche and Bullock-Frohman-Kania-Bartoszynska. We show that holonomy behaves well for the stack product and the action of the mapping class group; then we specialize this notion to links in order to define a generalized Wilson loop map. Thanks to the holonomy map, we give a geometric interpretation of the vacuum representation of $mathcal{L}_{g,0}(H)$ on $mathcal{L}_{0,g}(H)$. Finally, the general results are applied to the case $H=U_{q^2}(mathfrak{sl}_2)$ in relation to skein theory and the most important consequence is that the stated skein algebra of a compact oriented surface with just one boundary edge is isomorphic to $mathcal{L}_{g,n}ig( U_{q^2}(mathfrak{sl}_2) ig)$. Throughout the paper we use a graphical calculus for tensors with coefficients in $mathcal{L}_{g,n}(H)$ which makes the computations and definitions very intuitive.
The algebra $mathcal{L}_{g,n}(H)$ was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and quantizes the character variety of the Riemann surface $Sigma_{g,n}!setminus! D$ ($D$ is an open disk). In this article we define a holonomy map in that quantized setting, which associates a tensor with components in $mathcal{L}_{g,n}(H)$ to tangles in $(Sigma_{g,n}!setminus!D) imes [0,1]$, generalizing previous works of Buffenoir-Roche and Bullock-Frohman-Kania-Bartoszynska. We show that holonomy behaves well for the stack product and the action of the mapping class group; then we specialize this notion to links in order to define a generalized Wilson loop map. Thanks to the holonomy map, we give a geometric interpretation of the vacuum representation of $mathcal{L}_{g,0}(H)$ on $mathcal{L}_{0,g}(H)$. Finally, the general results are applied to the case $H=U_{q^2}(mathfrak{sl}_2)$ in relation to skein theory and the most important consequence is that the stated skein algebra of a compact oriented surface with just one boundary edge is isomorphic to $mathcal{L}_{g,n}ig( U_{q^2}(mathfrak{sl}_2) ig)$. Throughout the paper we use a graphical calculus for tensors with coefficients in $mathcal{L}_{g,n}(H)$ which makes the computations and definitions very intuitive.
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规定的表面绞纱代数
影响因子：
2.6
作者：
Costantino, Francesco;Lê, Thang T.
通讯作者：
Lê, Thang T.
标记的 3 流形/表面的绞纱模块，一项调查
影响因子：
0.5
作者：
Lê, Thang T.;Yu, Tao
通讯作者：
Yu, Tao
影响因子：
3.1
作者：
Gunningham, Sam;Jordan, David;Safronov, Pavel
通讯作者：
Safronov, Pavel
影响因子：
1.1
作者：
Lê, Thang T.
通讯作者：
Lê, Thang T.