The Representations of Temperley - Lieb - Jones Algebras ̄

Representations of 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 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 hyperfinite II 1 factor with trivial relative commutant. The index of the subfactor is given in terms of identity-identity element of certain duality matrix for conformal blocks of four-point correlators. Jones formula for index (< 4) for subfactors corresponds to spin 1 2 representation of SU(2) k Wess-Zumino conformal field theory. Definition of the trace operation also provides a method of obtaining link invariants explicitly.