  We discuss generalizations of the Temperley-Lieb algebra in the Potts and XXZ
models. These can be used to describe the addition of different types of
integrable boundary terms.
  We use the Temperley-Lieb algebra and its one-boundary, two-boundary, and
periodic extensions to classify different integrable boundary terms in the 2,
3, and 4-state Potts models. The representations always lie at critical points
where the algebras becomes non-semisimple and possess indecomposable
representations. In the one-boundary case we show how to use representation
theory to extract the Potts spectrum from an XXZ model with particular boundary
terms and hence obtain the finite size scaling of the Potts models. In the
two-boundary case we find that the Potts spectrum can be obtained by combining
several XXZ models with different boundary terms. As in the Temperley-Lieb case
there is a direct correspondence between representations of the lattice algebra
and those in the continuum conformal field theory.
