1.1. GENERIC TEMPERLEY-LIEB-JONES ALGEBROIDS 7 p5 = 5 = + d2 1 d4 3d2 + 1 Ä + + + ä + d2 1 d6 5d4 + 7d2 2 Ä + + + + + ä + d4 3d2 + 3 d6 5d4 + 7d2 2 d d6 5d4 + 7d2 2 Ä + + + ä + d2 d4 3d2 + 1 Ä + ä + d4 d2 d6 5d4 + 7d2 2 Ä + ä + −d3 + d d6 5d4 + 7d2 2 Ä + + + + + + + ä + −d3 + 2d d4 3d2 + 1 Ä + ä + d2 d6 5d4 + 7d2 2 Ä + ä + 1 d4 3d2 + 1 Ä + ä d d4 3d2 + 1 Ä + + + ä + −d3 + d d4 3d2 + 1 Ä + ä + 1 d6 5d4 + 7d2 2 Ä + ä 1.1.5. Trivalent graphs and bases of morphism spaces. To realize each TL diagram as a matrix, we study representations of TLn(A) = Hom(1n, 1n). If |y| = n, then for any object x, Hom(x, y) is a representation of TLn(A) by compo- sition of morphisms: TLn(A) × Hom(x, y) Hom(x, y). Therefore we begin with the analysis of the morphism spaces of the TL algebroid. To analyze these morphism spaces, we introduce colored trivalent graphs to represent some special basis elements. Let G be a uni-trivalent graph in the square R, possibly with loops and multi-edges, such that all trivalent vertices are in the interior of R and all uni-vertices are on the bottom and/or top of R. The univalent vertices together with the bottom or top of R are objects in TL(A). A coloring of G is an assignment of natural numbers to all edges of G such that edges with uni-vertices are colored by 1. An edge colored by 0 can be dropped, and an edge without a color is colored by 1. A coloring is admissible for a trivalent vertex v of G if the three colors a, b, c incident to v satisfy (1) a + b + c is even. (2) a + b c, b + c a, c + a b. Let G be a uni-trivalent graph with an admissible coloring whose bottom and top objects are x, y. Then G represents a formal diagram in Hom(x, y) as follows. Split each edge of color l into l parallels held together by a Jones-Wenzl projector pl. For each trivalent vertex v with colors a, b, c, admissibility furnishes unique natural numbers m, n, p such that a = m + p, b = m + n, c = n + p, allowing us to smooth v into a formal diagram as in Fig. 1.7. To simplify drawing and notation, 3 2 1 = p1 p3 p2 Figure 1.7. Trivalent vertex.
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