Semisimplified Temperley–Lieb quotient at the BHMV root

Fix an even integer $p\ge 4$, a primitive $2p$-th root $A\in \mathbb{C}$, $r=p\mathrm{/}2$, and loop value $\delta =-A^2-A^{-2}$. Let ${\mathcal{T}}_{\delta }$ be the complex Temperley–Lieb diagram category with objects the nonnegative integers and morphisms finite linear combinations of perfect matchings of bottom and top boundary points with no intersecting arcs. Compose by stacking and remove each closed component with factor $\delta$. Define ${\mathcal{N}}_{\delta }(m,n)$ to be the subspace of morphisms $f:m\to n$ such that ${\mathrm{Tr}}_m(g\circ f)=0$ for every reverse morphism $g:n\to m$, where $\mathrm{Tr}$ is the unnormalized diagram-closure trace. Form the quotient category with the same objects and morphism spaces ${\mathrm{Hom}}_{{\mathcal{T}}_{\delta }}(m,n)\mathrm{/}{\mathcal{N}}_{\delta }(m,n)$. Its additive envelope adjoins finite formal direct sums, with matrix morphisms and matrix composition. Its idempotent completion adjoins objects $(X,e)$ for idempotents $e\in \mathrm{End}(X)$ with $e^2=e$, and morphisms $(X,e)\to (Y,f)$ are morphisms $a:X\to Y$ satisfying $a=fae$. Juxtaposition, planar cup-cap duality, and the induced trace descend to this completed quotient. This definition makes no assertion that it is semisimple, that its simple objects have a specified label set, or that any braid, twist, or modularity identities hold.

Root, loop, and quantum-integer conventions

Noncrossing matching diagrams and stacking

The exact closure-trace test defining negligible morphisms

Closure of negligible morphisms under composition and tensor product

Category and monoidal laws for the diagram operations