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Abstract In a series of papers, the author jointly with J. Mondrag´on studied in a systematic way the structure of homogeneous Groebner basis algebras, or G-algebras, and in two recent papers the author applied these results to the homogenized enveloping algebra of the Lie algebra S`(2, C). The results obtained in the study of this particular algebra inspired our previous paper on the Hopf algebra structure of the homogenized enveloping algebra of finite-dimensional Lie algebras. In this ...