Temperley–Lieb quantum integers and quantum characteristic

Let $R$ be a commutative ring with unit and $\delta \in R$. Define the quantum integers by

$[0{]}_{\delta }=0,[1{]}_{\delta }=1,[r+1{]}_{\delta }=\delta [r{]}_{\delta }-[r-1{]}_{\delta }(r\ge 1)\mathrm{.}$

The quantum characteristic of $(R,\delta )$ is the least positive integer $\mathrm{\ell }$ with $[\mathrm{\ell }{]}_{\delta }=0$, if one exists, and $\mathrm{\ell }=\mathrm{\infty }$ otherwise. No element of $R$ is inverted in this definition.

For $q\in {\mathbb{C}}^{\times }$ and $\delta =q+q^{-1}$, write $[r{]}_q=[r{]}_{\delta }$. If $q\mathrm{\ne }\pm 1$, this agrees with $(q^r-q^{-r})\mathrm{/}(q-q^{-1})$; at $q=\pm 1$ the recurrence gives $[r{]}_q=r{q}^{r-1}$. For complex-parameter classification, write $\kappa =\mathrm{ord}(q^2)$, with $\kappa =\mathrm{\infty }$ when $q^2$ has infinite order. If $q\mathrm{\ne }\pm 1$, the quantum characteristic equals $\kappa$ when $\kappa <\mathrm{\infty }$ and is $\mathrm{\infty }$ otherwise. At $q=\pm 1$, $\kappa =1$ but the complex quantum characteristic is $\mathrm{\infty }$, because $[r{]}_q=r{q}^{r-1}\mathrm{\ne }0$ for every positive $r$.

For a field $K$ of positive characteristic $p$, set $p(0)=1$ and, when $\mathrm{\ell }<\mathrm{\infty }$, set $p(i)=\mathrm{\ell }{p}^{i-1}$ for $i\ge 1$. In characteristic zero and finite $\mathrm{\ell }$, the finite digit scale is $p(1)=\mathrm{\ell }$; the separate characteristic-zero criterion below uses $a\mathrm{\ell }-1$ for arbitrary positive integers $a$. If $\mathrm{\ell }=\mathrm{\infty }$, every finite index is already in the range $n<\mathrm{\ell }$.

The loop parameter $\delta$ is fixed independently of a choice of square root or of $q$. The complex semisimplicity theorem uses $\kappa =\mathrm{ord}(q^2)$, including its separate order-one case; Spencer's field criterion uses the recurrence quantum characteristic. These parameters agree for complex $q\mathrm{\ne }\pm 1$, but not at $q=\pm 1$.