B型网
Type $B$ Webs
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中文总结 AI 辅助
解决库珀伯格论文中B型开放问题，定义\(\mathbb{C}(q)\)-线性 pivotal 范畴\(\mathbf{Web}(\mathfrak{so}_{2n + 1})\)，证明其与特定表示子范畴等价，还为\(\iota\)量子群非经典表示构造辫子群对称性。
中文摘要 AI 辅助
我们解决了库珀伯格1996年论文《2阶李代数的蜘蛛》中主要开放问题的B型情况。即定义了一个\(\mathbb{C}(q)\)-线性 pivotal 范畴\(\mathbf{Web}(\mathfrak{so}_{2n + 1})\)，并证明它等同于由基本表示张量生成的\(U_q(\mathfrak{so}_{2n + 1})\)有限维表示的全子范畴。主要结果的一个推论是为\(\iota\)量子群\({U}^{\iota}_{-q^2}(\mathfrak{so}_m)\)的非经典有限维表示明确构造辫子群对称性，这可能具有独立的研究价值。
英文摘要
We solve the type $B$ case of the main open problem from Kuperberg's 1996 paper "Spiders for rank 2 Lie algebras." That is, we define a $\mathbb{C}(q)$-linear pivotal category $\mathbf{Web}(\mathfrak{so}_{2n+1})$ and prove that it is equivalent to the full subcategory of finite-dimensional representations of $U_q(\mathfrak{so}_{2n+1})$ tensor-generated by the fundamental representations. A consequence of our main result is an explicit construction of braid group symmetries for nonclassical finite-dimensional representations of the $ι$quantum group ${U}^ι_{-q^2}(\mathfrak{so}_m)$, which may be of independent interest.
