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

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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.