Tensor space representations of Temperley–Lieb algebra via orthogonal projections of rank r ≥ 1

Abstract

Unitary representations of the Temperley–Lieb algebra TLN(Q) on the tensor space (ℂn)⊗N are considered. Two criteria are given for determining when an orthogonal projection matrix P of a rank r gives rise to such a representation. The first of them is the equality of traces of certain matrices and the second is the unitary condition for a certain partitioned matrix. Some estimates are obtained on the lower bound of Q for a given dimension n and rank r. It is also shown that if 4r > n2, then Q can take only a discrete set of values determined by the value of n2/r. In particular, the only allowed value of Q for n = r = 2 is Q=2. Finally, properties of the Clebsch–Gordan coefficients of the quantum Hopf algebra Uq(su2) are used in order to find all r = 1 and r = 2 unitary tensor space representations of TLN(Q) such that Q depends continuously on q and P is the projection in the tensor square of a simple Uq(su2) module on the subspace spanned by one or two joint eigenvectors of the Casimir operator C and the generator K of the Cartan subalgebra.