The abstract Temperley–Lieb generator presentation

Let $R$ be a commutative ring with unit and $\delta \in R$. For each $n\ge 0$, let $F_n:=R⟨u_1,\dots ,u_{n-1}⟩$ be the free associative unital $R$-algebra on the indicated symbols (The free associative R-algebra on a set and descent of relations). When $n=0$ or $n=1$, the symbol set is empty and $F_n=R$.

Let $I_n$ be the two-sided ideal generated by

$u_i^2-\delta u_i(1\le i<n),u_i{u}_j-u_j{u}_i(\mathrm{\mid }i-j\mathrm{\mid }>1),$

and

$u_i{u}_{i+1}{u}_i-u_i,u_{i+1}{u}_i{u}_{i+1}-u_{i+1}(1\le i\le n-2)\mathrm{.}$

The abstract rank-$n$ Temperley–Lieb algebra is the presented $R$-algebra

$P_n(R,\delta ):=F_n\mathrm{/}{I}_n\mathrm{.}$

Thus its defining relations are $u_i^2=\delta u_i$, $u_i{u}_{i\pm 1}{u}_i=u_i$, and $u_i{u}_j=u_j{u}_i$ for $\mathrm{\mid }i-j\mathrm{\mid }>1$. The definition of a presented $R$-algebra and its universal property are those of The free associative R-algebra on a set and descent of relations.

The parameter convention here is fixed by the loop rule: a closed component contributes $\delta$. No rescaling by $\delta$, division by $\delta$, or field hypothesis occurs in the presentation.