2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70442We consider two families of categories. The first is the family of semisimple quotients of H. Andersen's tilting module categories for quantum groups of Lie type $B$ specialized at odd roots of unity. The second consists of categories constructed from a particular family of finite-dimensional quotients of the group algebra of Artin's braid group known as $BMW$-algebras of type $BC$. Our main result is to show that these families coincide as braided tensor categories using a recent theorem of Tuba and Wenzl. The morphism spaces in these categories can be equipped with a Hermitian form, and we are able to show that these categories are never unitary, and no braided tensor category sharing the Grothendieck semiring common to these families is unitarizable.25 pages, 1 figure. Final verstion to appear in Math. Z. Changes: expanded to include Lie type C, clarified/justified use of fusion rule result due to Andersen-Paradowski and to Sawin in the general case (reference added)Quantum AlgebraRepresentation Theory20G42,17B37(Primary);18D10,20F36,20C08(Secondary)text