Spectrum and Bethe ansatz equations for the U_ {q}(gl(N)) closed and open spin chains in any representation

TL;DR

This paper derives the spectrum and Bethe ansatz equations for U_q(gl(N)) spin chains with various boundary conditions and representations, using analytical Bethe ansatz and fusion techniques, and explores their symmetry algebras.

Contribution

It provides a comprehensive analytical Bethe ansatz solution for general representations and boundary conditions of U_q(gl(N)) spin chains, including fusion procedures and symmetry analysis.

Findings

Derived spectrum and Bethe equations for general representations.

Established fusion procedures for quantum contraction and Sklyanin determinant.

Analyzed symmetry algebras for different boundary conditions.

Abstract

We consider the N-site U_{q}(gl(N)) integrable spin chain with periodic and open diagonal soliton-preserving boundary conditions. By employing analytical Bethe ansatz techniques we are able to determine the spectrum and the corresponding Bethe ansatz equations for the general case, where each site of the spin chain is associated to any representation of U_{q}(gl(N)). In the case of open spin chain, we study finite dimensional representations of the quantum reflection algebra, and prove in full generality that the pseudo-vacuum is a highest weight of the monodromy matrix. For these two types of spin chain, we study the (generalized) "algebraic" fusion procedures, which amount to construct the quantum contraction and the Sklyanin determinant for the affine U_{q}(gl(N)) and quantum reflection algebras. We also determine the symmetry algebra of these two types of spin chains, including general K and K^+ diagonal matrices for the open case. The case of open spin chains with soliton non-preserving boundary conditions is also presented in the framework of quantum twisted Yangians. The symmetry algebra of this spin chains is studied. We also give an exhaustive classification of the invertible matricial solutions to the corresponding reflection equation.