The Temperley–Lieb negligible quotient and TLJ completion

Let $\mathrm{\ell }\ge 3$ be an integer, put $q=e^{\pi i\mathrm{/}\mathrm{\ell }}$ and $\delta =q+q^{-1}=2\cos (\pi \mathrm{/}\mathrm{\ell })\mathrm{\ne }0$, and use the category ${\mathcal{T}}_{\delta }$ and tensor ideal ${\mathcal{N}}_{\delta }$ from Negligible morphisms in the Temperley–Lieb category and Negligible Temperley–Lieb morphisms form a tensor ideal. The negligible quotient category ${\mathcal{T}}_{\delta }\mathrm{/}{\mathcal{N}}_{\delta }$ has the same objects $n\in \mathbb{N}$ and morphism spaces ${\mathrm{Hom}}_{{\mathcal{T}}_{\delta }\mathrm{/}{\mathcal{N}}_{\delta }}(m,n)={\mathrm{Hom}}_{{\mathcal{T}}_{\delta }}(m,n)\mathrm{/}{\mathcal{N}}_{\delta }(m,n)\mathrm{.}$ Composition and tensor product are induced from ${\mathcal{T}}_{\delta }$; the tensor-ideal result makes both operations well-defined on these quotient spaces.

For a $\mathbb{C}$-linear monoidal category $\mathcal{C}$, its additive envelope has finite formal direct sums of objects and matrices of morphisms, with matrix composition and tensor product distributing over direct sums. Its idempotent completion has objects $(X,e)$ where $e\in {\mathrm{End}}_{\mathcal{C}}(X)$ and $e^2=e$; morphisms $(X,e)\to (Y,f)$ are morphisms $a:X\to Y$ satisfying $a=fae$, with inherited composition and tensor product. The Temperley–Lieb–Jones category at order $\mathrm{\ell }$ in this text means the idempotent completion of the additive envelope of ${\mathcal{T}}_{\delta }\mathrm{/}{\mathcal{N}}_{\delta }$.

This is a definition of the quotient and completion only. It does not assert that the quotient is semisimple, that the completion is a fusion category, or that its simple objects have labels $0,\dots ,\mathrm{\ell }-2$.