18 1. TEMPERLEY-LIEB-JONES THEORIES Since 6j symbols are given in terms of da and θ(a, b, c), any formula in the old normalization is easily rewritten in the unitary normalization. One nice property of the unitary normalization is that the norm of m a b · · · e n is √ dmdadb · · · dedn, depending only on m, a, b, . . ., e, n, i.e., independent of the interior colors. Next we will explicitly describe the unitary Jones representations ρA of braid groups. To do so, we apply the Kauffman bracket to a braid generator σi ∈ Bn: ρA(σi) = A · id + A−1ρA(Ui) where Ui is a TLn(A) generator. Hence to compute the Jones representation ρA(σi) it suﬃces to compute ρA(Ui), which acts on the vector space Hom(t, 1n) spanned by {eC t } in Lem. 1.19. For unitary representations, we will use the normalized basis {eC U t } obtained from {eC t } by modifying each trivalent vertex as in eqn. (1.29). ρA(Ui)(eC U t ) = ρA(Ui) 1 i − 1 a i a ± 1 i + 1 a i + 2 n · · · · · · = i − 1 a i a ± 1 i + 1 a i + 2 For 0 a k, let ea+ = 1 » eC U t , eC U t i − 1 a i a + 1 i + 1 a i + 2 ea− = 1 » eU C t , eU C t i − 1 a i a − 1 i + 1 a i + 2 If a = 0 or a = k, then ρA(Ui) is the scalar δaa · d on ea+ or ea− respectively. If 0 a k, then ρA(Ui) restricted to the 2-dimensional subspace {ea+,ea−} is Ñ Δa+1 Δa √ Δa+1Δa−1 Δa √ Δa+1Δa−1 Δa Δa−1 Δa é Therefore ρA(Ui) consists of 2 × 2 or 1 × 1 blocks, as does ρA(σi). The original Jones representations from von Neumann algebras are given in terms of projectors {ei} for the Jones algebras. The generators {ei} are related to

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