Loop symmetry of integrable vertex models at roots of unity

TL;DR

This paper generalizes the discovery of new symmetries at roots of unity in integrable vertex models, showing invariance under quantum group actions for a broad class of models and algebraic structures.

Contribution

It extends the symmetry results at roots of unity from the six-vertex model to all vertex models linked with trigonometric solutions, using algebraic methods.

Findings

Hamiltonian and transfer matrix are invariant under quantum group actions.

Symmetries are generalized to all vertex models with trigonometric R-matrices.

Results include both odd and even primitive roots of unity for specific cases.

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.