论文标题
su（3）-skein代数和网络上的网
SU(3)-skein algebras and webs on surfaces
论文作者
论文摘要
表面$ f $的$ su_3 $ -skein代数由$ f \ times i $的某些框架图的同位素类别跨越，称为$ 3 $ -WEB，约为$ u_q（sl（3））$ - 表示的skein Ressanction。这些绞线代数是$ sl（3）$ - 表面的字符品种的量化。预计他们的理论与Kauffman支架绞线代数相似。我们通过证明有限类型的任何表面的减少$ su_3 $ -skein代数是有限生成的，我们迈出了开发该理论的第一步。 我们通过在表面中发展出范围的网络形式理论来实现这一结果。具体而言，我们表明，对于任何理想的三角剖分，每减少$ 3 $ -WEB都可以独特地分解为三角剖分的六角形和脱节弧的金字塔形成，并可能在理想的三角仪上连接边缘。我们表明，这种规范的位置是独特的，直到“横杆动作”。这使我们进入了三角形表面中网的相关坐标系统（对三角测量的边缘计数网络的交集及其在三角调节的脸部内部的旋转数字）独特地决定了网络的减少。 最后，我们将绞线代数与$ \ cal a $ varieties of fock-goncharov和$ \ text {loc} _ {sl（3）} $ - Goncharov-shen的品种。我们认为，我们的网坐标系统是Goncharov-Shen猜想的（量子）镜像对称性的表现。
The $SU_3$-skein algebra of a surface $F$ is spanned by isotopy classes of certain framed graphs in $F\times I$ called $3$-webs subject to the skein relations encapsulating relations between $U_q(sl(3))$-representations. These skein algebras are quantizations of the $SL(3)$-character varieties of surfaces. It is expected that their theory parallels that of the Kauffman bracket skein algebras. We make the first step towards developing that theory by proving that the reduced $SU_3$-skein algebra of any surface of finite type is finitely generated. We achieve that result by developing a theory of canonical forms of webs in surfaces. Specifically, we show that for any ideal triangulation of $F$ every reduced $3$-web can be uniquely decomposed into unions of pyramid formations of hexagons and disjoint arcs in the faces of the triangulation with possible additional "crossbars" connecting their edges along the ideal triangulation. We show that such canonical position is unique up to "crossbar moves". That leads us to an associated system of coordinates for webs in triangulated surfaces (counting intersections of the web with the edges of the triangulation and their rotation numbers inside of the faces of the triangulation) which determine a reduced web uniquely. Finally, we relate our skein algebras to $\cal A$-varieties of Fock-Goncharov and to $\text{Loc}_{SL(3)}$-varieties of Goncharov-Shen. We believe that our coordinate system for webs is a manifestation of a (quantum) mirror symmetry conjectured by Goncharov-Shen.
