A new quality theorem for tensor category and the equivarence of topological quantum field theories

对于每个有限分裂半单张量范畴C，标准Tannaka对偶给出了一个余模范畴等价于C的量子群(面代数)。这种对偶给出了数学和物理张量范畴的统一理解，也给出了普通群表示理论的一个新图景。通过应用量子群论中的对偶等技巧，我们得到了以下结果。我们证明了量子EJ-symbo是有限大小的ABF模的配分函数之和。给出了普通6J符号的一个简单求和公式。利用对偶性(或者更确切地说是标准纤维函子)，我们构造了二元二次曲面和二元立方体的不变量和半不变量的基。对于一般线性群的两个不可约表示的每个张量积，我们在模级上给出了它们的显式不可约分解。我们对每个量子经典、群和L5能级的A型张量范畴上的编织结构和带状结构进行了分类。对于每个线性约化矩阵群G，我们给出了G的每个元素的一个“经济”逆矩阵公式。

For each finite split semisimple tensor category C, the canonical Tannaka duality gives a quantum group (face algebra) whose comodule category is equivalent to C. The duality gives a unified understanding of tensor categories arising from mathematics and physics, and also, it gives a new picture of the representation theory of the ordinary groups. By applying the duality and other techniques in the quantum group theory, we obtained the following results.1. We showed that the quantum ej-symbo, is a sum of the partition functions of the ABF mode, of finite size. Also we gave a simple summation formula for the ordinary 6j-symbols.2. By using the duality (or rather the canonical fiber functor), we constructed a bases for invariants and semiinvariants of binary quadratics and binary cubics.3. For each tensor products of two irreducible representations of the general linear group, we gave their explicit irreducible decomposition at the module level.4. We classified the braiding and th'e ribbon structure on each quantum classica, groups and the tensor category of type A of level L.5. For each linearly reductive matrix group G, we give an "economical" inverse matrix formula for each elements of G.