1.1. GENERIC TEMPERLEY-LIEB-JONES ALGEBROIDS 3 We are using the so-called optimistic convention for diagrams: diagrams are drawn from bottom to top. A general morphism f Hom(x, y) is a linear combination of TL diagrams. We will call such f a formal diagram. Notice that all objects x of the same cardinality |x| are isomorphic. We will not speak of natural numbers as objects in TL(A) because they are used later to denote objects in Temperley-Lieb-Jones categories. We will denote the isomorphism class of objects x with |x| = n by 1n. By abuse of notation, 1n will be considered as an object. 1.1.2. Generic TL algebras. Definition 1.3. Given a natural number n N, the generic TL algebra TLn(A) is just the algebra Hom(1n, 1n) in the generic TL algebroid. Obviously TLn(A) is independent of our choice of the realization of 1n as an object x such that |x| = n. By definition Hom(0, 0) = F. Definition 1.4. The Markov trace of TLn(A) is an algebra homomorphism tr : TLn(A) F defined by a tracial closure: choosing n disjoint arcs outside the square R connecting the bottom n points with their corresponding top points, for a TL diagram D, after connecting the 2n boundary points with the chosen n arcs and deleting the boundary of R, we are left with a collection of disjoint simple closed loops in the plane. If there are m of them, we define tr(D) = dm (Fig. 1.3). For a formal diagram, we extend the trace linearly. tr = = d Figure 1.3. Markov trace. There is an obvious involution X X on TLn(A). Given a TL diagram D, let D be the image of D under reflection through the middle line I × 1 2 . Then X X is extended to all formal diagrams by the automorphism of F which takes A to A−1 and restricts to complex conjugation on C. The Markov trace then induces a sesquilinear inner product, called the Markov pairing, on TLn(A) by the formula X, Y = tr(XY ) for any X, Y TLn(A) (Fig. 1.4). , = = d3 Figure 1.4. Markov pairing. Define the nth Chebyshev polynomial Δn(d) inductively by Δ0 = 1, Δ1(d) = d, and Δn+1(d) = dΔn(d) Δn−1(d). Let cn = 1 n+1 ( 2n n ) be the Catalan number.
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