  Integrable boundary conditions are studied for critical A-D-E and general
graph-based lattice models of statistical mechanics. In particular, using
techniques associated with the Temperley-Lieb algebra and fusion, a set of
boundary Boltzmann weights which satisfies the boundary Yang-Baxter equation is
obtained for each boundary condition. When appropriately specialized, these
boundary weights, each of which depends on three spins, decompose into more
natural two-spin edge weights. The specialized boundary conditions for the
A-D-E cases are naturally in one-to-one correspondence with the conformal
boundary conditions of sl(2) unitary minimal conformal field theories.
Supported by this and further evidence, we conclude that, in the continuum
scaling limit, the integrable boundary conditions provide realizations of the
complete set of conformal boundary conditions in the corresponding field
theories.
