1.2. JONES ALGEBROIDS 15 The next structure for a RFC is rigidity: a dual object for each object. The axioms for rigidity are to ensure that we can straighten out zigzags: = = It demands the existence of special morphisms called births and deaths. In VA, they are represented by Then zigzags can always be straightened out. It follows that all simple objects of VA are self-dual. Another structure in a RFC is braiding. In VA, for objects a, b, it is simply a formal diagram in Hom(a ⊗ b, b ⊗ a): b b a a Braidings should be compatible with the other structures. When compatibility holds, we have a RFC. The first nice thing about a RFC is that we can define the quantum trace of any f ∈ VA(x, x). In VA, the Markov trace is the quantum trace. Now more terminologies: (1) quantum dimension of a label: di = i = Δi (2) S-matrix: Let D2 = ∑ i∈L d2, i ˜ij = i j = (−1)i+j[(i + 1)(j + 1)], and sij = 1 D ˜ij. Then S = (sij) is called the modular S-matrix. There are two choices of D. We usually choose the positive D, but sometimes we need the negative D (see Sec. 1.3). (3) Twist: i = θi i θi = (−1)iAi(i+2) (4) Braiding eigenvalues: j k i = Rjk i j k i Rjk i = (−1) i+j+k 2 A i(i+2)−j(j+2)−k(k+2) 2

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