On Bethe vectors for the XXZ model at roots of unity

Recently in a series of papers [DFM], [FM1], [FM2] the Bethe ansatz equations and Bethe vectors for the six-vertex model with the anisotropy commensurable with π or, as usually said, at roots of unity, were studied. In that case the spectrum of the transfer-matrix, which is a generating function of the standard commuting conservation laws in the model, becomes highly degenerate. In [FM2] a construction of creation operators, responsible for appearance of the Bethe vectors with the same eigenvalues of the transfer-matrix, was suggested in the framework of the algebraic Bethe ansatz. In the note we extend that construction to the case of the inhomogeneous arbitrary spin XXZ model. Even for the case of six-vertex model the proof of the main formulae given in the note is simpler than the original proof in [FM2]. The detailed exposition of the algebraic Bethe ansatz method can be found in [KBI]. The notation used in the note does not coincide with those of [FM2] and [KBI], however a reader can easily establish the correspondence.