November 1993 OCR Output A270.ll93 for the non-simisimple contracted case, for all q.representations have their own interest. Moreover they lead to a consistent Hopf algebra and representations contracting the generator J 45 in the G-Z formalism. The contracted representations are given.The SO (5) is quite different, even at the classical level, from the familiar E(4), the 4 dim. Eucl. algebra, obtained on generators (e2, f2) leads to an easily obtained complete Hopf algebra. The structure of the contracted algebra restriction we show how to construct all representations for all q's. The contraction of the Chevalley irreducible representations labelled by 3 variable parameters, the maximal number being 4. Within this elsewhere, in q-deformations for orthogonal (though not for unitary) algebras. Our results are limited to sharp contrast to the classical canonical Gelfand-Zetlin representations which lead to problems, studied once obtained on this basis, are very simply deformed for arbitrary q, generic and root of unity. This is in Representations of SO (5) 'fractional part" discussed in detail in Sec.4. In fact, for q a root of unity, we have already presented OCR Output has been possible to unqfy the presentation including the root of unity case thanks to our formalism of chosen to present the representations directly in a form valid for all q (q=l, q¢1 and real, q = e). It *2"'N constant factors of (q+q·l)as in (2.13), (2.15) and (2.16). The q-deformation being simple, we have q-brackets defined in (2.1) along with [x] due to the existence of unequal roots. There are also external easy. The differences with the "minimal" q-deformation for the unitary case are in the appearences of [x]2Having constructed the classical solutions, the q-deformation, as anticipated, turned out to be the Appendix leading to (A.l5) and what follows. construct explicitly. But awareness of this relation, even if implicit, can be helpful. See the discussion in exhibit the equivalence of the GZ with our representations. The matrix diagonalizing J45 is too difficult to correspondance of the canonical and the Chevalley bases, the G Z generator J45 has to be diagonalized to detail in [8,9]. In our conventions, which we found to be the most convenient concerning the diagonalized. The problems due to a non-diagonalized Cartan generator in q-deformation is studied in particularly crucial since from hi we have to go over to qihi for q¢l. In the GZ formalism only hi = Ji2 is subset is diagonalized. Both these features are absent for SO(n). The question of diagonalisation is generators necessary for building the whole algebra, is thus the same in each formalism and the same hi to -the latter being already diagonalized in the GZ representations. The basic, minimal set of ' generators (eifi) correspond to (A'+f, Ai+i) of the canonical generators directly and the Cartan generators representations had to be constructed. For unitary algebras this problem does not exist. The Chevalley minimally on nearly so. This hopeful conjecture turned out to b...