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Let $V$ be the two-dimensional simple module and $M$ be a projective Vermamodule for the quantum group of $\mathfrak{sl}_2$ at generic $q$. We show thatfor any $r\ge 1$, the endomorphism algebra of $M\otimes V^{\otimes r}$ isisomorphic to the type $B$ Temperley-Lieb algebra $\rm{TLB}_r(q, Q)$ for anappropriate parameter $Q$ depending on $M$. The parameter $Q$ is determinedexplicitly. We also use the cellular structure to determine precisely for whichvalues of $r$ the endomorphism algebra is semisimple. A key element of ourmethod is to identify the algebras $\rm{TLB}_r(q,Q)$ as the endomorphismalgebras of the objects in a quotient category of the category of colouredribbon graphs of Freyd-Yetter or the tangle diagrams of Turaev and Reshitikhin.

Schur-Weyl duality for certain infinite dimensional $\rm{U}_q(\mathfrak{sl}_2)$-modules