Equivalence of a tangle category and a category of infinite dimensional 𝑈_{𝑞}(𝔰𝔩₂)-modules

It is very well known that if V V is the simple 2 2 -dimensional representation of U q ( s l 2 ) \mathrm {U}_q(\mathfrak {sl}_2) , the category of representations V ⊗ r V^{\otimes r} , r = 0 , 1 , 2 , … r=0,1,2,\dots , is equivalent to the Temperley-Lieb category T L ( q ) \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 ⊗ V ⊗ r M\otimes V^{\otimes r} , where M M is a projective Verma module of U q ( s l 2 ) \mathrm {U}_q(\mathfrak {sl}_2) and the type B B Temperley-Lieb category T L B ( q , Q ) \mathbb {TLB}(q,Q) , realised as a subquotient of the tangle category of Freyd, Yetter, Reshetikhin, Turaev and others.