论文标题
统一融合类别的绞线理论方法
Skein-Theoretic Methods for Unitary Fusion Categories
论文作者
论文摘要
由于与量子物理学的新兴联系，单一融合类别（UFC）引起了人们的关注。我们考虑$ q \ otimes q \ cong \ mathbf {1} \ oplus \ bigoplus^k_ {i = 1} x_ {i} $在UFC $ \ Mathcal {C} $中的融合规则，并使用图形计算提取信息。例如，当$ k = 1,2 $和$ \ mathcal {c} $ is corbbon时，我们将所有关联的绞线关系分类。特别是，我们还考虑了$ q $是反对称自我对齐的实例。我们的主要结果来自考虑旋转操作员在“规范基础上”的作用。假设汇总的$ x_ {i} $的自伴，进行了一些一般的观察，例如$ f $ -matrix $ f^{qqq} _q $的实数。然后，当$ k = 2 $和$ \ mathcal {c} $是功能区时，我们找到了$ f^{qqq} _q $的显式公式，并看到旋转操作员的频谱区分Kauffman和dubrovnik dolynomials。
Unitary fusion categories (UFCs) have gained increased attention due to emerging connections with quantum physics. We consider a fusion rule of the form $q\otimes q \cong \mathbf{1}\oplus\bigoplus^k_{i=1}x_{i}$ in a UFC $\mathcal{C}$, and extract information using the graphical calculus. For instance, we classify all associated skein relations when $k=1,2$ and $\mathcal{C}$ is ribbon. In particular, we also consider the instances where $q$ is antisymmetrically self-dual. Our main results follow from considering the action of a rotation operator on a "canonical basis". Assuming self-duality of the summands $x_{i}$, some general observations are made e.g. the real-symmetricity of the $F$-matrix $F^{qqq}_q$. We then find explicit formulae for $F^{qqq}_q$ when $k=2$ and $\mathcal{C}$ is ribbon, and see that the spectrum of the rotation operator distinguishes between the Kauffman and Dubrovnik polynomials.
