Finite Hilbert boundary representation of Temperley-Lieb

Fix an integer $n\ge 1$ and a real loop parameter $\delta$. Define $[0{]}_{\delta }=0$, $[1{]}_{\delta }=1$, and $[r+1{]}_{\delta }=\delta [r{]}_{\delta }-[r-1{]}_{\delta }$ for $r\ge 1$. Assume $[k{]}_{\delta }\mathrm{\ne }0$ for $2\le k\le n$; this list is empty when $n=1$. The abstract generator algebra $P_n(\mathbb{C},\delta )$ is the unital complex algebra with generators $u_1,\dots ,u_{n-1}$ and relations $u_i^2=\delta u_i$, $u_i{u}_{i\pm 1}{u}_i=u_i$, and $u_i{u}_j=u_j{u}_i$ when $\mathrm{\mid }i-j\mathrm{\mid }>1$. The diagram algebra ${\mathrm{T}\mathrm{L}}_n(\mathbb{C},\delta )$ has basis the noncrossing pairings of $n$ ordered bottom and $n$ ordered top points; its product stacks the left factor above the right, deletes each closed loop, and multiplies by $\delta$ per loop. Its elementary diagram $d_i$ has a cap joining top points $i,i+1$, a cup joining bottom points $i,i+1$, and vertical strands elsewhere. The isomorphism ${\mathrm{\Phi }}_n:P_n(\mathbb{C},\delta )\to {\mathrm{T}\mathrm{L}}_n(\mathbb{C},\delta )$ sends $u_i$ to $d_i$.

For every $1\le i\le n-2$, the adjacent relations in the presentation mean both $u_i{u}_{i+1}{u}_i=u_i$ and $u_{i+1}{u}_i{u}_{i+1}=u_{i+1}$. For matching the imported recursion's notation, write $U_i:=d_i$.

Set $p_1=1_1$. For $1\le r<n$, put $Q_r=p_r\otimes 1_1$ by adjoining one rightmost vertical strand and define

$p_{r+1}:=Q_r-\frac{[r{]}_{\delta }}{[r+1{]}_{\delta }}{Q}_r{d}_r{Q}_r\mathrm{.}$

The denominator hypotheses make this recursion defined. For each $1\le r\le n$, let ${\widehat{p}}_r:={\mathrm{\Phi }}_r^{-1}(p_r)$. The cited Jones-Wenzl result gives $p_n^2=p_n$ under these unit-denominator hypotheses. For $1\le r\le n$, let ${\iota }_{r,n}:P_r(\mathbb{C},\delta )\to P_n(\mathbb{C},\delta )$ send each generator $u_i$ to the same-named generator, and let ${\mathrm{ȷ}}_{r,n}:{\mathrm{T}\mathrm{L}}_r(\mathbb{C},\delta )\to {\mathrm{T}\mathrm{L}}_n(\mathbb{C},\delta )$ append $n-r$ rightmost vertical strands. The maps commute with $\mathrm{\Phi }$: on each generator, both composites send $u_i$ to $d_i$ with those added strands, and both send the unit to the identity diagram. Since generators and unit generate $P_r$, the composites agree on every element.

Let $H$ be a nonzero finite-dimensional complex inner-product space whose inner product is linear in its first argument and conjugate-linear in its second. A unital complex-algebra representation is a complex-linear map $\pi :P_n(\mathbb{C},\delta )\to {\mathrm{End}}_{\mathbb{C}}(H)$ such that $\pi (xy)=\pi (x)\pi (y)$ for all $x,y\in P_n(\mathbb{C},\delta )$ and $\pi (1)=I_H$. Here ${\mathrm{End}}_{\mathbb{C}}(H)$ is the algebra of complex-linear endomorphisms, with composition as multiplication and $I_H$ as unit. Require each generator image to be self-adjoint, meaning $⟨\pi (u_i)x,y⟩=⟨x,\pi (u_i)y⟩$ for all $x,y\in H$. Define $P:=\pi ({\widehat{p}}_n)$, define $\mathrm{rank}P:={\dim }_{\mathbb{C}}\mathrm{im}P$, and restrict to data with $\mathrm{rank}P>0$. The algebraic representation as a physical boundary state space is an additional model identification and is not part of this definition.

The abstract Temperley–Lieb generator presentation

Temperley–Lieb planar diagram data and endomorphism modules

The generator presentation is the planar diagram algebra over every commutative ring

Temperley–Lieb quantum integers and quantum characteristic

Jones–Wenzl recursion over unit quantum integers

Jones–Wenzl projectors: annihilation, uniqueness, partial trace, and closure

Kernel and image of a linear map

Real and complex inner product spaces, with the inner product linear in the first argument

Joshua Chen, The Temperley-Lieb Categories and Skein Modules

Zhenghan Wang, Topological Quantum Computation

K. Iohara, G. I. Lehrer, and R. B. Zhang, The Jones Quotients of the Temperley-Lieb Algebras

For $r<n$, the presentation tower map ${\iota }_{r,n}$ is well-defined because the defining relations are unchanged and it preserves the unit. For basis diagrams $E,D\in {\mathrm{T}\mathrm{L}}_r$, appending $n-r$ rightmost vertical strands before stacking gives the same matching as stacking $E$ above $D$ first and then appending those strands; the appended strands create no closed loops. Hence ${\mathrm{ȷ}}_{r,n}(E\circ D)={\mathrm{ȷ}}_{r,n}(E)\circ {\mathrm{ȷ}}_{r,n}(D)$, and ${\mathrm{ȷ}}_{r,n}(1_r)=1_n$. Bilinearity then makes ${\mathrm{ȷ}}_{r,n}$ a unital algebra homomorphism. The two unital algebra homomorphisms ${\mathrm{\Phi }}_n\circ {\iota }_{r,n}$ and ${\mathrm{ȷ}}_{r,n}\circ {\mathrm{\Phi }}_r$ agree on every generator and the unit, so they agree on all of $P_r$; this identity transfers the unit-denominator recursion on diagrams to the presentation used by $\pi$.

The box is notation for the three-strand algebra element $p_3$; strands are ordered from left to right and continue through the box. For general $n$, the same box notation has $n$ strands. The picture records no physical apparatus, and no proof below relies on it.

Only $[2{]}_{\delta },\dots ,[n{]}_{\delta }$ are required to be nonzero; $[n+1{]}_{\delta }$ may vanish. The item asserts no root-of-unity quotient, modular representation classification, trace-preserving realization, or physical area law.