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.

We study a two-boundary extension of the Temperley-Lieb algebra which has recently arisen in statistical mechanics. This algebra lies in a quotient of the affine Hecke algebra of type C and has a natural diagrammatic representation. The algebra has three parameters and, for generic values of these, we determine its representation theory. We use the action of the centre of the affine Hecke algebra to show that all irreducible representations lie within a finite dimensional diagrammatic quotient. These representations are fully characterised by an additional parameter related to the action of the centre. For generic values of this parameter there is a unique representation of dimension 2 N and we show that it is isomorphic to a tensor space representation. We construct a basis in which the Gram matrix is diagonal and use this to discuss the irreducibility of this representation.

We show that from the spectra of the Uq(sl(2)) symmetric XXZ spin-1/2 finite quantum chain at ∆ = −1/2 (q = e πi/3 ) one can obtain the spectra of certain XXZ quantum chains with diagonal and non-diagonal boundary conditions. Similar observations are made for ∆ = 0 (q = e πi/2 ). In the finite-size scaling limit the relations among the various spectra are the result of identities satisfied by known character functions. For the finite chains the origin of the remarkable spectral identities can be found in the representation theory of one and two boundaries Temperley-Lieb algebras at exceptional points. Inspired by these observations we have discovered other spectral identities between chains with different boundary conditions.

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.