The representations of Temperley-Lieb-Jones algebras

Abstract

Representations of the braid group obtained from rational conformal field theories can be used to obtain explicit representations of Temperley-Lieb-Jones algebras. The method is described in detail for SU(2)_k Wess-Zumino conformal field theories and its generalization to an arbitrary rational conformal field theory is outlined. Explicit definition of an associated linear trace operation in terms of a certain matrix element in the space of conformal blocks of such a conformal theory is presented. Further, for every primary field of a rational conformal field theory, there is a subfactor of the hyperfinite II₁ factor with trivial relative commutant. The index of the subfactor is given in terms of the identity-identity element of a certain duality matrix for the conformal blocks of the four-point correlators. Jones' formula for the index (< 4) for subfactors corresponds to the spin-1/2 representation of the SU(2)_k Wess-Zumino conformal field theory. The definition of the trace operation also provides a method of obtaining link invariants explicitly.