Loop symmetry of integrable vertex models at roots of unity

Abstract

It has been recently discovered in the context of the six vertex or XXZ model in the fundamental representation that new symmetries arise when the anisotropy parameter \((q+q^{-1})/2\) is evaluated at roots of unity \(q^{N}=1\). These new symmetries have been linked to an \(U(A^{(1)}_1)\) invariance of the transfer matrix and the corresponding spin-chain Hamiltonian.In this paper these results are generalized for odd primitive roots of unity to all vertex models associated with trigonometric solutions of the Yang-Baxter equation by invoking representation independent methods which only take the algebraic structure of the underlying quantum groups \(U_q(\hat g)\) into account. Here \(\hat g\) is an arbitrary Kac-Moody algebra. Employing the notion of the boost operator it is then found that the Hamiltonian and the transfer matrix of the integrable model are invariant under the action of \(U(\hat{g})\). For the simplest case \(\hat g=A_1^{(1)}\) the discussion is also extended to even primitive roots of unity.