  Braid groups may be defined for every Coxeter diagram. Artin's braid group is
of type A. Analogs of Temperley-Lieb, Hecke and Birman-Wenzl algebras exist for
B-type. Our general hypothethis is that the braid group of B-type replaces
Artin's braid group in most physical applications if the model is equipped with
a nontrivial boundary. Solutions of a Potts model with a boundary and the
reflection equation illustrate this principle. Braided tensor categories of
B-type and dually Coxeter-B braided Hopf algebras are introduced. The
occurrence of such categories in QFT on a half plane is discussed.
