1.2. JONES ALGEBROIDS 13 1.1.9. TLJ(A) at roots of unity. To be directly applicable to quantum com- putation, we need to work over C, not the field of rational functions in A. Therefore we will specialize A to a nonzero complex number. The structure of TLJ(A) is very sensitive to the choice of A. Given A ∈ C \ {0}, we call d = −A2 − A−2 the loop value Theorem 1.21. (1) If A ∈ C \ {0} such that the loop value d is not a root of any Chebyshev polynomial Δi, i = 1, 2,..., then the structure of TLJ(A) is the same as generic TLJ(A). (2) If A ∈ C \ {0} such that the loop value d is a root of some Chebyshev polynomial Δi, i = 1, 2,..., then some JWPs are undefined and for all suﬃciently large n, TLn(A)’s are not matrix algebras, i.e., not semisimple. This theorem is well-known to experts. The structure of TLn(A) at roots of unity is analyzed in [GW]. When A ∈ C \ {0} such that d is a root of some Δi, then A is a root of unity. The structure of TLJ(A) depends essentially on the order of A. We are interested in semisimple quotients of TLJ(A) in the next section, called Jones algebroids. 1.2. Jones algebroids When the Kauffman variable A is specialized to roots of unity, the Markov pairing becomes degenerate and some JWPs are undefined in TLJ(A). So TLJ(A) is not a semisimple algebroid anymore. Some semisimple quotients of the TLJ algebroids when A is a root of unity were discovered by V. Jones, so they will be called Jones algebroids. We assume A is either a primitive 4rth root of unity for arbitrary r ≥ 3 or a primitive 2rth root of unity for odd r ≥ 3. We will denote the Jones algebroid with a choice of A by VA,k, or just VA or Vk if no confusion arises, where k = r − 2 is called the level of the theory. Fix an A and a k as above. Then the loop value d becomes a root of some Chebyshev polynomial Δi. Since Δi appears in the denominator in the definition of the JWPs pn, some pn are undefined. The first JWP that is undefined for our choice of A is pr. Therefore we restrict our discussion to p0,...,pr−1. By convention attaching p0 to a strand is the same as coloring by 0. From eqn. (1.10) we have tr pi = Δi = (−1)i[i+1] = (−1)i A2i+2−A−2i−2 A2−A−2 . If A4r = 1, then tr pr−1 = 0. For our choice of A, tr pi = 0 for i = 0,...,r −2. (By convention tr p0 = 1.) Therefore pr−1 is a vector of norm 0 under the Markov pairing. Actually, any vector in the radical of the Markov pairing is a generalized annular consequence of pr−1 [Fr2, FNWW]. We denote the radical by pr−1 . Definition 1.22. (1) Given A as above, L = {0,...,k} is called the label set, and each i ∈ L is called a label. (2) The objects of VA are labeled points in I where the labels are from L. For morphisms, given two objects a, b ∈ V 0 A , VA(a, b) = Hom(a, b)/ pr−1 , where Hom(a, b) is the morphism space of TLJ(A) specialized to A and pr−1 is the radical above.

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