Spectrum and Bethe Ansatz Equations for the U_{q} {\left( {gl(\mathcal{N})} \right)} Closed and Open Spin Chains in any Representation

We consider the N-site \(U_{q} {\left( {gl(\mathcal{N})} \right)}\) 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 \(U_{q} {\left( {gl(\mathcal{N})} \right)}.\)