The representations of Temperley-Lieb-Jones algebras

Details The representations of Temperley-Lieb-Jones algebras The representations of Temperley-Lieb-Jones algebras

1993-12-31

arxiv.org/abs/hep-th/9312214v1

Nucl.Phys. B417 (1994) 267-288

Physics

High Energy Physics

High Energy Physics - Theory

24 pages

Scientific paper

10.1016/0550-3213(94)90547-9

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 ${\frac{1}{2}}$ representation of SU(2)$_k$ Wess-Zumino conformal field theory. Definition of the trace operation also provides a method of obtaining link invariants explicitly.

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