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arXiv 1802.09203math-phmath.CTmath.MP
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Temperley-Lieb范畴中的融合与单值

Fusion and monodromy in the Temperley-Lieb category

Jonathan Belletête, Yvan Saint-Aubin

备注：
32 pages, 4 figures
发表信息：
SciPost Phys. 5, 041 (2018)
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摘要：

Graham和Lehrer（1998）引入了一个Temperley-Lieb范畴$\mathsf{\widetilde{TL}}$，它的对象是非负整数，$\mathsf{Hom}(n,m)$中的态射是从$n$到$m$节点的连接图.将Temperley-Lieb代数$\mathsf{TL}_{n}$等同于$\mathsf{Hom}(n,n)$。范畴$\mathsf{\widetilde{TL}}$被证明是monoidal的。通过构造一个交换子，我们证明了它也是一个辫子范畴。扭曲也在$\mathsf{\widetilde{TL}}$上定义。我们引入了一个模范畴${\text{ Mod}_{\mathsf{\widetilde{TL}}}}$，它的对象是从$\mathsf{\widetilde{TL}}$到$\mathsf{Vect}_{\mathbb C}$的函子，并在它上面定义了一个融合双函子，扩展了Read和Saleur（2007）引入的融合双函子.当$q$不是单位根时，我们利用为$\mathsf{\widetilde{TL}}$构造的自然态射导出了${\text{ Mod}_{\mathsf{\widetilde{TL}}}}(β=-q-q^{-1})$上的带状范畴的结构.讨论了$\mathsf{\widetilde{TL}}$上的编织与统计模型的可积性之间的关系。本文还讨论了这些结构在稀释Temperley-Lieb代数族中的推广。

英文摘要：

Graham and Lehrer (1998) introduced a Temperley-Lieb category $\mathsf{\widetilde{TL}}$ whose objects are the non-negative integers and the morphisms in $\mathsf{Hom}(n,m)$ are the link diagrams from $n$ to $m$ nodes. The Temperley-Lieb algebra $\mathsf{TL}_{n}$ is identified with $\mathsf{Hom}(n,n)$. The category $\mathsf{\widetilde{TL}}$ is shown to be monoidal. We show that it is also a braided category by constructing explicitly a commutor. A twist is also defined on $\mathsf{\widetilde{TL}}$. We introduce a module category ${\text{ Mod}_{\mathsf{\widetilde{TL}}}}$ whose objects are functors from $\mathsf{\widetilde{TL}}$ to $\mathsf{Vect}_{\mathbb C}$ and define on it a fusion bifunctor extending the one introduced by Read and Saleur (2007). We use the natural morphisms constructed for $\mathsf{\widetilde{TL}}$ to induce the structure of a ribbon category on ${\text{ Mod}_{\mathsf{\widetilde{TL}}}}(β=-q-q^{-1})$, when $q$ is not a root of unity. We discuss how the braiding on $\mathsf{\widetilde{TL}}$ and integrability of statistical models are related. The extension of these structures to the family of dilute Temperley-Lieb algebras is also discussed.

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