论文标题
在量子类型$ c $中的网络上
On webs in quantum type $C$
论文作者
论文摘要
我们研究量子类型$ c $的网，重点介绍了第三个案例。我们通过生成器和关系来定义一个线性关键类别$ \ mathbf {web}（\ mathfrak {sp} _6）$ diagrammatication，并猜想它等于$ \ m m m m m mathbf {funderrep}（funderrep}（u_q（sp））由基本表示形式生成，用于参数$ q $的通用值。我们证明了许多结果支持此猜想，最值得注意的是，有一个完整的，本质上是滤波的函数$ \ Mathbf {web}（\ Mathfrak {\ Mathfrak {sp} _6）\ rightarrow \ rightArrow \ Mathbf {fundRep} $ \ mathbf {web}（\ mathfrak {sp} _6）$是有限维度的，并且在$ \ mathbf {web}（\ mathfrak {sp} _6）$ $ 1 $ dimensional中，单型单位的内态代数为$ \ mathfrak {\ mathfrak {sp} _6 _6）。后者对应于可以使用本地关系将所有封闭网络评估为标量的陈述。因此，我们获得了一种新的方法，用于量子$ \ mathfrak {sp} _6 $链接不变，类似于琼斯多项式的Kauffman支架描述。
We study webs in quantum type $C$, focusing on the rank three case. We define a linear pivotal category $\mathbf{Web}(\mathfrak{sp}_6)$ diagrammatically by generators and relations, and conjecture that it is equivalent to the category $\mathbf{FundRep}(U_q(\mathfrak{sp}_6))$ of quantum $\mathfrak{sp}_6$ representations generated by the fundamental representations, for generic values of the parameter $q$. We prove a number of results in support of this conjecture, most notably that there is a full, essentially surjective functor $\mathbf{Web}(\mathfrak{sp}_6) \rightarrow \mathbf{FundRep}(U_q(\mathfrak{sp}_6))$, that all $\mathrm{Hom}$-spaces in $\mathbf{Web}(\mathfrak{sp}_6)$ are finite-dimensional, and that the endomorphism algebra of the monoidal unit in $\mathbf{Web}(\mathfrak{sp}_6)$ is $1$-dimensional. The latter corresponds to the statement that all closed webs can be evaluated to scalars using local relations; as such, we obtain a new approach to the quantum $\mathfrak{sp}_6$ link invariants, akin to the Kauffman bracket description of the Jones polynomial.
