ADE $SU(3)$图的路空间上算子代数的另一种表述
Alternative formulation for the operator algebra over the space of paths in a ADE $SU(3)$ graph
摘要：
在这项工作中，我们讨论了所需的元素的建设的运营商代数的道路空间上的一个简单的花边$SU(3)$图。这些操作是一个重要的一步，在双代数的建设需要找到一些模块不变的CFTs的配分函数。我们定义了与来回序列相关的杯和帽算子，并将它们添加到算子代数中的创建和湮灭算子中，因为它们是计算融合代数规定的基本路径的完整空间所必需的。这些运营商需要倒塌的三角形细胞，没有发现在以前的作品，在这里，我们提供了明确的价值观，这些细胞，并显示其重要性，以满足Kuperberg关系的细胞系统$SU(3)$缠结。我们还发现，要求我们的运营商满足Temperley-Lieb代数导致人们自然地考虑运营商，创建和消灭封闭的三角序列，这反过来又提供了一个替代的杯和帽运营商，因为它们允许一个替换来回序列与封闭的三角序列。最后，我们证明了用闭三角形得到的基本路径与原来用来回序列得到的基本路径是等价的。
英文摘要：
In this work we discuss the elements required for the construction of the operator algebra for the space of paths over a simply laced $SU(3)$ graph. These operators are an important step in the construction of the bialgebra required to find the partition functions of some modular invariant CFTs. We define the cup and cap operators associated with back-and-forth sequences and add them to the creation and annihilation operators in the operator algebra as they are required for the calculation of the full space of essential paths prescribed by the fusion algebra. These operators require collapsed triangular cells that had not been found in previous works; here we provide explicit values for these cells and show their importance in order for the cell system to fulfill the Kuperberg relations for $SU(3)$ tangles. We also find that demanding that our operators satisfy the Temperley-Lieb algebra leads one naturally to consider operators that create and annihilate closed triangular sequences, which in turn provides an alternative the cup and cap operators as they allow one to replace back-and-forth sequences with closed triangular ones. We finally show that the essential paths obtained by using closed triangles are equivalent to those obtained originally using back-and-forth sequences.
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