标题： 自旋链作为仿射Temperley-Lieb代数上的模 显示英文标题

标题： Spin chains as modules over the affine Temperley-Lieb algebra

摘要： 仿射Temperley-Lieb代数$\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(\beta)$是一个由数$\beta \in \mathbb{C}$和整数$N\in \mathbb{N}$参数化的无限维代数。 它自然作用于$(\mathbb{C}^2)^{\otimes N}$以产生由附加参数$z\in\mathbb C^\times$标记的一族表示。 这些表示的结构，最初由Pasquier和Saleur在研究自旋链时引入，此处被明确说明。 它们与Graham和Lehrer的细胞$\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(\beta)$-模具有相同的合成因子，但与后者表示的不同之处在于它们的Loewy图中大约一半的箭头方向相反。 这个陈述的证明使用了Morin-Duchesne和Saint-Aubin引入的一个同态，以及新的映射，这些映射在XXZ链上的各种$\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(\beta)$作用之间相互交织，并推广了Deguchi$\textit{et al}$研究的应用，之后由Morin-Duchesne和Saint-Aubin进一步研究。 显示英文摘要

摘要： The affine Temperley-Lieb algebra $\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(\beta)$ is an infinite-dimensional algebra parametrized by a number $\beta \in \mathbb{C}$ and an integer $N\in \mathbb{N}$. It naturally acts on $(\mathbb{C}^2)^{\otimes N}$ to produce a family of representations labeled by an additional parameter $z\in\mathbb C^\times$. The structure of these representations, which were first introduced by Pasquier and Saleur in their study of spin chains, is here made explicit. They share their composition factors with the cellular $\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(\beta)$-modules of Graham and Lehrer, but differ from the latter representations by the direction of about half of the arrows of their Loewy diagrams. The proof of this statement uses a morphism introduced by Morin-Duchesne and Saint-Aubin as well as new maps that intertwine various $\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(\beta)$-actions on the XXZ chain and generalize applications studied by Deguchi $\textit{et al}$ and after by Morin-Duchesne and Saint-Aubin.