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We consider the N-site U_{q}(gl(N)) integrable spin chain with periodic andopen diagonal soliton-preserving boundary conditions. By employing analyticalBethe ansatz techniques we are able to determine the spectrum and thecorresponding Bethe ansatz equations for the general case, where each site ofthe spin chain is associated to any representation of U_{q}(gl(N)).  In the case of open spin chain, we study finite dimensional representationsof the quantum reflection algebra, and prove in full generality that thepseudo-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 theSklyanin determinant for the affine U_{q}(gl(N)) and quantum reflectionalgebras. We also determine the symmetry algebra of these two types of spinchains, including general K and K^+ diagonal matrices for the open case.  The case of open spin chains with soliton non-preserving boundary conditionsis also presented in the framework of quantum twisted Yangians. The symmetryalgebra of this spin chains is studied. We also give an exhaustiveclassification of the invertible matricial solutions to the correspondingreflection equation.Comment: 48 pages; Bounds on parameters Mj corrected; References added;  Examples adde

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