Jones–Wenzl recursion over unit quantum integers

Let $R$ be a commutative ring with unit and let $\delta \in R$. Use the quantum integers $[0]=0$, $[1]=1$, $[r+1]=\delta [r]-[r-1]$ of Temperley–Lieb quantum integers and quantum characteristic. Put $p_0=1_0$ and $p_1=1_1$. For $r\ge 1$, if $p_r$ is defined and $[r+1]$ is a unit, put

$P_r=p_r\otimes 1_1\in {\mathrm{T}\mathrm{L}}_{r+1}(R,\delta ),p_{r+1}=P_r-\frac{[r]}{[r+1]}{P}_r{U}_r{P}_r,$

where $U_r$ is the cap-cup generator on the final two of the $r+1$ strands in Temperley–Lieb planar diagram data and endomorphism modules. Thus $p_n$ is defined by this recursion only when every $[2],\dots ,[n]$ is a unit. The definition makes no assertion that failure of a denominator condition prevents every other normalized projector from existing.

The box labels stand for the corresponding idempotents and the right-hand cup-cap is $U_r$; the picture is only a convention aid. All algebraic identities are proved from the labeled matching composition and the relations of The elementary diagram generators satisfy the local Temperley–Lieb relations.