Spin-half exchange realizes Temperley-Lieb at loop two

Let $N\ge 1$ and let ${\mathcal{H}}_N=({\mathbb{C}}^2{)}^{\otimes N}$ have the standard tensor-product Hermitian inner product, linear in its first argument, with $\mathrm{\mid }0⟩,\mathrm{\mid }1⟩$ orthonormal on each factor. For $1\le i<N$, let $s_i$ be the swap of tensor factors $i,i+1$, let $\mathrm{\mid }\omega ⟩=(\mathrm{\mid }01⟩-\mathrm{\mid }10⟩)\mathrm{/}\sqrt{2}$, and let $P_i=I^{\otimes (i-1)}\otimes \mathrm{\mid }\omega ⟩⟨\omega \mathrm{\mid }\otimes I^{\otimes (N-i-1)}$ be the projection onto this singlet on those two factors. Put $e_i:=I-s_i$. Then these are the loop-weight-two Temperley-Lieb generators. In the Hecke normalization $g_i(q):=qI-e_i$, at $q=1$ one has $g_i(1)=s_i$. For arbitrary real coefficients ${\mathcal{J}}_i$, the finite exchange sum $X_N(\mathcal{J}):={\sum }_{i=1}^{N-1}{\mathcal{J}}_i{s}_i$ satisfies $X_N(\mathcal{J})=\left({\sum }_{i=1}^{N-1}{\mathcal{J}}_i\right)I-{\sum }_{i=1}^{N-1}{\mathcal{J}}_i{e}_i\mathrm{.}$ The equality $g_i(1)=s_i$ is internal to this spin-half representation; it does not identify its state space with the distinct $Q=4$ Potts representation.

Spin-half swaps realize the Temperley–Lieb algebra at loop weight two

Potts local algebra and braid quotient

No nonzero-$q$ deformation is identified with this Hermitian swap representation. For the Potts $Q=2,3$ values, $q$ is non-real and the relation $g_i=qI-e_i$ alone supplies no star-preserving or unitary representation.