1.1. GENERIC TEMPERLEY-LIEB-JONES ALGEBROIDS 5 The generic TL algebras TLn(A) first appeared in physics, and were rediscov- ered by V. Jones [Jo3]. Our diagrammatic definition is due to L. Kauffman [Kau]. 1.1.3. Generic representation of the braid groups. The most important and interesting representation of the braid group Bn is the Jones representation discovered in 1981 [Jo2, Jo4], which led to the Jones polynomial, and the earlier Burau representation, related to the Alexander polynomial. The approach pioneered by Jones is to study finite-dimensional quotients of the group algebra F[Bn], which is an infinite-dimensional representation of the braid group: b Bn, b( cigi) = ci(bgi). If a finite-dimensional quotient is given by an algebra homomorphism, then the regular representation on F[Bn] descends to the quotient, yielding a finite-dimensional representation of Bn. TLn(A) is obtained as a quotient of F[Bn] through the Kauffman bracket: = A + A−1 Recall the n-strand braid group Bn has a presentation with generators {σi | i = 1, 2,...,n 1} and relations σiσj = σjσi if |i j| 2, σiσi+1σi = σi+1σiσi+1. The Kauffman bracket induces a map , : F[Bn] TLn(A) by the formula σi = A · id + A−1Ui. Proposition 1.7. The Kauffman bracket , : F[Bn] TLn(A) is a surjective algebra homomorphism. The proof is a straightforward computation. Definition 1.8. Since generic TLn(A) is isomorphic to a direct sum of matrix algebras over F, the Kauffman bracket , maps Bn to nonsingular matrices over F, yielding a representation ρA of Bn called the generic Jones representation. It is a difficult open question to determine whether ρA sends nontrivial braids to the identity matrix, i.e., whether the Jones representation is faithful. Next we will use Jones-Wenzl projectors to describe the Jones representation explicitly. 1.1.4. Jones-Wenzl projectors. In this section we show the existence and uniqueness of the Jones-Wenzl projectors. Theorem 1.9. Generic TLn(A) contains a unique pn characterized by: (1) pn = 0. (2) pn 2 = pn. (3) Uipn = pnUi = 0 for all 1 i n 1. Furthermore pn can be written as pn = 1 + U, where U = cjmj, where mj are nontrivial monomials of Ui’s, 1 i n 1, and cj F. Proof. For uniqueness, suppose pn exists and can be expanded as pn = c1+U. Then pn 2 = pn(c1 + U) = pn(c1) = cpn = c21 + cU, so c = 1. Let pn = 1 + U and p n = 1 + V , both having the properties above, and expand pnpn from both sides: pn = 1 · pn = (1 + U)pn = pnpn = pn(1 + V ) = pn · 1 = pn.
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