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Tensor space representations of Temperley-Lieb algebra via orthogonal projections of rank $r \geq 1$

Unitary representations of the Temperley-Lieb algebra $TL_N(Q)$ on the tensor space $({\mathbb C^n})^{\otimes N}$ 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 $4r>n^2$, then $Q$ can take only a discrete set of values determined by the value of $n^2/r$. In particular, the only allowed value of $Q$ for $n=r=2$ is $Q=\sqrt{2}$. Finally, properties of the Clebsch-Gordan coefficients of the quantum Hopf algebra $U_q(su_2)$ are used in order to find all $r=1$ and $r=2$ unitary tensor space representations of $TL_N(Q)$ such that $Q$ depends continuously on $q$ and $P$ is the projection in the tensor square of a simple $U_q(su_2)$ module on the subspace spanned by one or two joint eigenvectors of the Casimir operator $C$ and the generator $K$ of the Cartan subalgebra.