Surveying the quantum group symmetries of integrable open spin chains

Using anisotropic R-matrices associated with affine Lie algebras $\hat{g}$ (specifically, $A_{2n}^{\left(2\right)}$, $A_{2n-1}^{\left(2\right)}$, $B_n^{\left(1\right)}$, $C_n^{\left(1\right)}$, $D_n^{\left(1\right)}$) and suitable corresponding K-matrices, we construct families of integrable open quantum spin chains of finite length, whose transfer matrices are invariant under the quantum group corresponding to removing one node from the Dynkin diagram of $\hat{g}$. We show that these transfer matrices also have a duality symmetry (for the cases $C_n^{\left(1\right)}$ and $D_n^{\left(1\right)}$) and additional $Z_2$ symmetries that map complex representations to their conjugates (for the cases $A_{2n-1}^{\left(2\right)}$, $B_n^{\left(1\right)}$ and $D_n^{\left(1\right)}$). A key simplification is achieved by working in a certain "unitary" gauge, in which only the unbroken symmetry generators appear. The proofs of these symmetries rely on some new properties of the R-matrices. We use these symmetries to explain the degeneracies of the transfer matrices.