Equivalence of a tangle category and a category of infinite dimensional $\mathrm {U}_q(\mathfrak {sl}_2)$-modules

It is very well known that if $V$ is the simple $2$-dimensional representation of $\mathrm {U}_q(\mathfrak {sl}_2)$, the category of representations $V^{\otimes r}$, $r=0,1,2,\dots$, is equivalent to the Temperley-Lieb category $\mathrm {TL}(q)$. Such categorical equivalences between tangle categories and categories of representations are rare. In this work we give a family of new equivalences by extending the above equivalence to one between the category of representations $M\otimes V^{\otimes r}$, where $M$ is a projective Verma module of $\mathrm {U}_q(\mathfrak {sl}_2)$ and the type $B$ Temperley-Lieb category $\mathbb {TLB}(q,Q)$, realised as a subquotient of the tangle category of Freyd, Yetter, Reshetikhin, Turaev and others.