  We give an exact spectral equivalence between the quantum group invariant XXZ
chain with arbitrary left boundary term and the same XXZ chain with purely
diagonal boundary terms.
  This equivalence, and a further one with a link pattern Hamiltonian, can be
understood as arising from different representations of the one-boundary
Temperley-Lieb algebra. For a system of size L these representations are all of
dimension 2^L and, for generic points of the algebra, equivalent. However at
exceptional points they can possess different indecomposable structures.
  We study the centralizer of the one-boundary Temperley-Lieb algebra in the
'non-diagonal' spin-1/2 representation and find its eigenvalues and
eigenvectors. In the exceptional cases the centralizer becomes indecomposable.
We show how to get a truncated space of 'good' states. The indecomposable part
of the centralizer leads to degeneracies in the three mentioned Hamiltonians.
