Symmetries of integrable open boundaries in the Hubbard model and other spin chains

We proceed to study the symmetries of integrable open boundaries in the one dimensional Hubbard model, the Heisenberg XXX spin chain and Inozemtsev's hyperbolic spin chain. For the Hubbard model, we show that when placed on the left half-line, the known integrable open boundaries (a magnetic field and chemical potential) break the bulk Yangian symmetry to a twisted Yangian corresponding to the (sl2; u1) symmetric pair. Furthermore, we consider two additional boundaries, corresponding to the symmetric pairs (so4; sl2) and (sl2; sl2) and construct their twisted Yangian symmetries. This provides a step forward in the classification of integrable boundaries of the open Hubbard model. We conclude our study of this model by examining the symmetries of its bulk and open SU(n) generalisation. For the Heisenberg XXX spin chain and Inozemtsev's hyperbolic spin chain we construct a procedure to, given the integrable bulk models, systematically obtain their integrable boundaries and corresponding Yangian symmetries for the symmetric pairs (sl2; u1); (so4; sl2) and (sl2; sl2). We call this method `folding', and it is motivated by the wish to study integrable boundaries for long-range spin chains. We test this procedure by first applying it on the Heisenberg XXX spin chain and confirming it reproduces well known results. We then apply the folding to Inozemtsev's hyperbolic spin chain and classify its integrable open boundaries and their twisted Yangian symmetries.