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The transfer matrix of the XXZ open spin-1/2 chain with general integrableboundary conditions and generic anisotropy parameter (q is not a root of unityand |q|=1) is diagonalized using the representation theory of the q-Onsageralgebra. Similarly to the Ising and superintegrable chiral Potts models, thecomplete spectrum is expressed in terms of the roots of a characteristicpolynomial of degree d=2^N. The complete family of eigenstates are derived interms of rational functions defined on a discrete support which satisfy asystem of coupled recurrence relations. In the special case of linear relationsbetween left and right boundary parameters for which Bethe-type solutions areknown to exist, our analysis provides an alternative derivation of the resultsby Nepomechie et al. and Cao et al.. In the latter case the complete family ofeigenvalues and eigenstates splits in two sets, each associated with acharacteristic polynomial of degree $d< 2^N$. Numerical checks performed forsmall values of $N$ support the analysis.Comment: 21 pages; LaTeX file with amssymb; v2: typos corrected, references  added, more details, to appear in JSTA

Exact spectrum of the XXZ open spin chain from theq-Onsager algebra representation theory