Temperley–Lieb planar diagram data and endomorphism modules

Let $R$ be a commutative ring with unit (Algebras over a commutative ring, central structure maps, and algebra homomorphisms) and let $\delta \in R$. The object labels are the nonnegative integers $n=0,1,2,\dots$. For labels $m,n$, place bottom points $b_1,\dots ,b_m$ and top points $t_1,\dots ,t_n$ along the rectangle from left to right on their respective edges. Their cyclic order is $t_1,\dots ,t_n,b_m,\dots ,b_1$. When $m=n=0$, the empty matching is included.

An $(m,n)$ diagram is a perfect matching of these $m+n$ boundary labels which is noncrossing: there are no four labels $a,b,c,d$ in cyclic order whose matched pairs are $\{a,c\}$ and $\{b,d\}$. A picture consists of disjoint arcs realizing the matching; equality is equality of the matching, so planar deformation that fixes the boundary does not change a diagram. If $m+n$ is odd, there are no such diagrams. Write $\mathcal{D}(m,n)$ for this set and define the $R$-module

$M_R(m,n):=R^{(\mathcal{D}(m,n))},$

the free $R$-module on the set of diagrams (The free module on a set and its standard basis). In particular, its diagram basis is empty when $m+n$ is odd.

For basis diagrams $D:m\to n$ and $E:n\to p$, stack $E$ above $D$ and identify the $n$ matching boundary points. The resulting finite graph has degree two at each identified point, where degree means the number of incident arcs. Each component is a path joining two exterior boundary points or a closed cycle. Remove every cycle and multiply the remaining exterior matching by ${\delta }^c$, where $c$ is the number of removed cycles. Denote this output by $E\circ D$ and extend the rule $R$-bilinearly to a map $M_R(n,p)\times M_R(m,n)\to M_R(m,p)$. A following result proves closure of these operations on the diagram modules and the category axioms.

The matching $1_n:n\to n$ pairs each bottom point to the top point directly above it. Horizontal juxtaposition, with the left diagram placed first, defines $m\otimes m^{\mathrm{\prime }}=m+m^{\mathrm{\prime }}$ on object labels and $D\otimes D^{\mathrm{\prime }}$ on basis diagrams; extend it $R$-bilinearly to the modules. A following result proves that these operations give a strict monoidal category.

For $1\le i<n$, let $d_i\in M_R(n,n)$ be the diagram with a cap joining top points $i,i+1$, a cup joining bottom points $i,i+1$, and vertical through strands at every other position. Define ${\mathrm{T}\mathrm{L}}_n(R,\delta )$ to be the $R$-module $M_R(n,n)$ with bilinear multiplication $xy:=x\circ y$ and distinguished element $1_n$. A following result proves that it is a unital associative $R$-algebra and identifies it with the endomorphism algebra at object $n$.

The left factor is stacked above the right factor. The loop value is $\delta$; no invertibility or nonzero condition on $\delta$ is part of these data.

This TikZ picture shows $d_1$ on three strands. The top and bottom labels increase left to right; the cap is on top, the cup is on the bottom, and the third strand passes through. The algebraic definition is the matching on labeled boundary points, not the drawing.

These data are over a commutative ring; associativity and the $R$-algebra structure are proved on this page. They make no semisimplicity, trace, cellularity, or Hecke-quotient assertion. The published localized Hecke quotient The Temperley-Lieb quotient and the Jones specialization uses idempotent generators and a separate parameter conversion; the diagram/presentation isomorphism is proved locally below.