  We define multi-colour generalizations of braid-monoid algebras and present
explicit matrix representations which are related to two-dimensional exactly
solvable lattice models of statistical mechanics. In particular, we show that
the two-colour braid-monoid algebra describes the Yang-Baxter algebra of the
critical dilute A-D-E models which were recently introduced by Warnaar,
Nienhuis, and Seaton as well as by Roche. These and other solvable models
related to dense and dilute loop models are discussed in detail and it is shown
that the solvability is a direct consequence of the algebraic structure. It is
conjectured that the Yang-Baxterization of general multi-colour braid-monoid
algebras will lead to the construction of further solvable lattice models.
