1.1. GENERIC TEMPERLEY-LIEB-JONES ALGEBROIDS 9 1.1.6. Generic Temperley-Lieb-Jones categories. Generic TL(A) has a tensor product given by horizontal “stacking”: juxtaposition of diagrams. Using this tensor product, denoted as ⊗, we see that any object y with |y| = n is iso- morphic to a tensor power of an object x with |x| = 1, i.e., 1n = 1⊗n. For our applications to TQC, we would like to have x⊗m “collapsible” to a direct sum of finitely many “simple” objects for all sufficiently large m. To achieve this, we enlarge generic TL(A) to the generic Temperley-Lieb-Jones (TLJ) algebroid, then take a finite “quotient.” In this section, we describe the generic TLJ categories, which have generic TL(A) as subcategories. Let A be an indeterminant as before. The objects of TLJ(A) are objects of TL(A) with natural number colors: each point in I receives a natural number. A point colored by 0 can be deleted. A point without a color is understood to be colored by 1, hence TL(A)0 TLJ(A)0. Morphisms in Hom(x, y) for x, y TLJ(A)0 are formal F-linear combinations of uni-trivalent graphs connecting x, y with admissible compatible colorings. Again, an edge without a color is colored by 1, and an edge of color 0 can be deleted, along with its endpoints. TLJ(A) has a tensor product as in TL(A): horizontal juxtaposition of formal diagrams. The empty object is a tensor unit. Every object is self-dual. The involution X X, extended to TLJ(A), is the duality for morphisms. Theorem 1.13. TLJ(A) and TL(A) are ribbon tensor categories, but not ribbon fusion categories. Categories of tangles first appeared in [Y, Tu] to organize quantum invariants of links. For a detailed treatment of TLJ(A) and TL(A) as ribbon tensor categories, see [Tu]. We will define a ribbon fusion category in Chap. 4. Here we will list the properties of TLJ(A) that make it into a ribbon tensor category, which is essentially an abstraction of the pictures that we will draw. Generic TLJ(A) is not a fusion category because it has infinitely many simple object types, one for each JWP pn. Definition 1.14. An object x in an algebroid is simple if Hom(x, x) F. Let n denote the isomorphism class of a single point colored by n. By abuse of notation, we will treat n as an object of TLJ(A). Note that n 1n for n 1. For instance, while 3 is simple, 13 is not, because dim Hom(13, 13) = 5. Proposition 1.15. Let a, b, c N be objects of TLJ(A). (1) A JWP kills any turn-back: = 0. · · · · · · · · · · · · (2) Hom(a, b) = ® C if a = b, 0 otherwise. (3) Hom(a b, c) = ® C if a, b, c are admissible, 0 otherwise. (4) Let x be an object consisting of k points with colors a1,...,ak and y be an object consisting of l points with colors b1,...,bl. Let G be a (connected) uni-trivalent tree connecting x and y with compatible colorings. Then the admissible colorings of G form a basis of Hom(x, y) Hom(a1 · · · ak,b1 · · · bl).
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