14 1. TEMPERLEY-LIEB-JONES THEORIES Theorem 1.23. (1) VA is a semisimple algebroid. In particular, the quotients TLn(A)/ pr−1 of the TL algebras TLn(A), denoted as Jn(A), are semisimple algebras, and hence are direct sums of matrix algebras. Jn(A) will be called the Jones algebra at A. (2) The Kauffman bracket defines a representative of the braid groups as in the generic case for each A. These are the Jones representations ρA of braid groups. (3) The images of all braid generators σi have eigenvalues among {A, −A−3}. Hence all eigenvalues of ρ(σi) are roots of unity of order 4r. This theorem is a categorical version of Jones representations. Jones representations are reducible, which is important for applications in in- terferometric experiments in ν = 5 /2 FQH liquds. We will refer to each irreducible summand as a sector. The sectors for ρA on Bn are in 1-1 correspondence with i L, i = n mod 2. These Jones representations of Bn differ from the original Jones representations from von Neumann algebras by an abelian representation of Bn. See Sec. 1.4. As we saw in the generic case, in order to find the Jones rep- resentation explicitly, we introduce the trivalent bases of morphism spaces. Since our colors are now truncated to labels, we have to impose more conditions on the admissible labels. Definition 1.24. Three labels a, b, c are k-admissible if (1) a + b + c is even. (2) a + b c, b + c a, c + a b. (3) a + b + c 2k. A trivalent vertex is k-admissible if its three colors are k-admissible. Lemma 1.25. Let a, b, c be labels. Then the following are equivalent: (1) a, b, c are k-admissible. (2) Δ(a, b, c) = 0. (3) Hom(a b, c) = 0. The extra condition a+b+c 2k is very important. It has an origin in CFT as the positive energy condition. With the truncation of colors from natural numbers to labels L = {0,...,k} and the new positive energy condition, all formulas for the generic TLJ algebroids in Prop. 1.16 apply to Jones algebroids. The same is true for the F -matrices. Jones algebroids are our prototypical examples of RFCs, so let us describe their structures and introduce new terminology. First we have a label set L, which is the isomorphism classes of simple objects. The number of labels is called the rank of the theory. For the Jones algebroid VA, the label set is L = {0,...,k}, so it is of rank = k + 1 = r 1. The tensor product is given by horizontal juxtaposition of formal diagrams. The fusion rules are the tensor decomposition rules for a representative set of the simple objects. Jones algebroids have a direct sum on objects, denoted as ⊕. Therefore fusion rules for labels are written as a b = N c ab c, where N c ab are natural numbers representing the multiplicity of c in a b. By Lem. 1.25 N c ab is 1 if a, b, c are k-admissible and 0 otherwise. Note that 0 is the trivial label, and 0 a = a for any a.
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