Surveying the quantum group symmetries of integrable open spin chains

Using anisotropic R-matrices associated with affine Lie algebras gˆ (specifically, A_2n ⁽²⁾, A_2n−1 ⁽²⁾, B_n ⁽¹⁾, C_n ⁽¹⁾, D_n ⁽¹⁾) 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 gˆ. We show that these transfer matrices also have a duality symmetry (for the cases C_n ⁽¹⁾ and D_n ⁽¹⁾) and additional Z₂ symmetries that map complex representations to their conjugates (for the cases A_2n−1 ⁽²⁾, B_n ⁽¹⁾ and D_n ⁽¹⁾). 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.