Stated \(\mathrm{SL}(n)\)-skein modules and algebras (Q6646207)

A marked 3-manifold is a pair \((M,\mathcal N)\), where \(M\) is an oriented 3-manifold and \(\mathcal N\) consists of embedded oriented open intervals in \(\partial M\). The authors define the stated \(\mathrm{SL}(n)\)-skein module \(\mathcal S_n(M,\mathcal N)\) of \((M,\mathcal N)\) using properly embedded oriented graphs in \((M,\mathcal N)\). They prove that their stated \(\mathrm{SL}(n)\)-skein theory is the generalization of the existing stated \(\mathrm{SL}(3)\)-skein theory in [\textit{V. Higgins}, Quantum Topol. 14, No. 1, 1--63 (2023; Zbl 1548.57039)], stated \(\mathrm{SL}(2)\)-skein theory in [\textit{T. T. Q. Lê}, Quantum Topol. 9, No. 3, 591--632 (2018; Zbl 1427.57011)], and \(\mathrm{SL}(n)\)-skein theory in [\textit{A. S. Sikora}, Algebr. Geom. Topol. 5, 865--897 (2005; Zbl 1087.57008)].\N\NA pb surface \(\Sigma\) is obtained from a compact surface by removing finitely many points such that every boundary component of \(\Sigma\) is an open interval. The authors define the stated \(\mathrm{SL}(n)\)-skein algebra \(\mathcal S_n(\Sigma)\) to be the stated \(\mathrm{SL}(n)\)-skein module \(\mathcal S_n(\Sigma\times (-1,1))\), where \(\Sigma\times (-1,1)\) is a marked 3-manifold obtained by thickening \(\Sigma\). The algebra structure of \(\mathcal S_n(\Sigma)\) is given by stacking the properly embedded oriented graphs in \((M,\mathcal N)\). They prove that \(\mathcal S_n(\Sigma)\) is a free module when every component of \(\Sigma\) contains nonempty boundaries. Let \(\mathcal B\) be the pb surface obtained from the closed disk by removing two points in its boundary. The authors show that \(\mathcal S_n(\mathcal B)\) has a Hopf algebra structure and \[\mathcal S_n(\mathcal B)\simeq\mathcal O_q(\mathrm{SL}(n))\] as Hopf algebras, where \(\mathcal O_q(\mathrm{SL}(n))\) is the quantization of the function ring of \(\mathrm{SL}_n(\mathbb C)\).\N\NFor an ideal arc \(c\) of \(\Sigma\), the authors define an algebra homomorphism \[\Theta_c\colon \mathcal S_n(\Sigma)\rightarrow \mathcal S_n(\Sigma')\quad \text{(called the splitting homomorphism)},\] where \(\Sigma'\) is the pb surface obtained from \(\Sigma\) by cutting along the ideal arc \(c\). They show that \(\Theta_c\) is injective when every component of \(\Sigma\) contains nonempty boundaries. The splitting homomorphism is very important for the stated \(\mathrm{SL}(n)\)-skein theory. For example, some questions can be reduced to the triangle case by cutting the triangulable pb surfaces into a collection of triangles; for any ideal arc \(c\) isotopic to a boundary component of \(\Sigma\), the authors show that the splitting homomorphism \(\Theta_c\) defines an \(\mathcal O_q(\mathrm{SL}(n))\)-comodule structure on \(\mathcal S_n(\Sigma)\).\N\NWhen every component of \(\Sigma\) contains nonempty boundaries, the authors show that \(\mathcal S_n(\Sigma)\) is isomorphic to a certain number of tensor copies of \(\mathcal O_q(\mathrm{SL}(n))\), which offers a basis of \(\mathcal S_n(\Sigma)\).