4 1. TEMPERLEY-LIEB-JONES THEORIES There are cn different TL diagrams {Di} in TLn(A) consisting only of n disjoint arcs up to isotopy in R connecting the 2n boundary points of R. These cn diagrams span TLn(A) as a vector space. Let Mc n ×cn = (mij) be the matrix of the Markov pairing of {Di} in a certain order, i.e., mij = tr(DiDj). Then (1.5) Det(Mc n ×cn ) = ± n i=1 Δi(d)an,i where an,i = ( 2n n−i−2 ) + ( 2n n−i ) − 2 ( 2n n−i−1 ) . Formula (1.5) is derived in [DGG]. Let {Ui}, i = 0, 1,...,n − 1, be the TL diagrams in TLn(A) shown in Fig. 1.5. Figure 1.5. Generators of TL. Theorem 1.6. (1) The diagrams {Di}, i = 1, 2,..., 1 n+1 ( 2n n ) , form a basis of TLn(A) as a vector space. (2) TLn(A) has a presentation as an abstract algebra with generators {Ui}i=01−n and relations Ui2 = dUi UiUi±1Ui = Ui UiUj = UjUi if |i − j| ≥ 2. (3) Generic TLn(A) is a direct sum of matrix algebras over F. Proof. (1) It suﬃces to show that every basis diagram Di is a monomial in the generators Ui. Fig. 1.6 should convince the reader to construct his/her own proof. The dimension of the underlying vector space of TLn(A) is the number of isotopic diagrams without loops, which is one of the many equivalent definitions of the Catalan number. = = U2 · U1 Figure 1.6. Decomposition into Ui’s. (2) By drawing diagrams, we can easily check the three TL relations above. Therefore there is a surjective algebra map φ from TLn(A) onto the ab- stract algebra with generators {Ui} and those relations. Injectivity of φ follows from a dimension count: the dimensions of the underlying vector spaces of both algebras are given by the Catalan number. (3) Formula (1.5) can be used to deduce that generic TLn(A) is a semisimple algebra, hence a direct sum of matrix algebras. An explicit proof is given in Sec. 1.1.6.

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