On the reality of spectra of Uq(sl2)-invariant XXZ Hamiltonians

Abstract

.A new inner product is constructed on each standard module over the Temperley–Lieb algebra for and . On these modules, the Hamiltonian is shown to be self-adjoint with respect to this inner product. This implies that its action on these modules is diagonalisable with real eigenvalues. A representation theoretic argument shows that the reality of spectra of the Hamiltonian extends to all other Temperley–Lieb representations. In particular, this result applies to the celebrated -invariant XXZ Hamiltonian, for all .