Quantum analysis in tensor categories and their free products

传统上，科学上的对称性是用一个叫做群的数学概念来描述的。由于近年来数学物理领域对称性研究的结果，我们面临着一个新的对称性阶段，它超出了普通群概念的范围，被称为量子对称。在对这个问题进行更深入的研究之后，描述量子对称性的相关概念原来是张量范畴的概念。在这个研究项目中，我们专注于平面图，并揭示了它们的矩阵表示与单个生成的张量类别是一一对应的。

Traditionally, scientific symmetry is described in terms of a mathematical notion called group. As a consequence of recent studies on symmetry in the field of mathematical physics, we are faced with a new phase of symmetry, which is out of coverage in the ordinary group notion and referred to as quantum symmetry. After closer investigations on the subject, the relevant notion in describing quantum symmetry turned out to be that of tensor category.In this research project, we focus on planar diagrams and have revealed that their matrix representations are in one-to-one correspondence with singly generated tensor categories.