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Any deformation of a Weyl or Clifford algebra can be realized through achange of generators in the undeformed algebra. q-Deformations of Weyl orClifford algebrae that were covariant under the action of a simple Lie algebrag are characterized by their being covariant under the action of the quantumgroup $U_q g$. We present a systematic procedure for determining all possiblecorresponding changes of generators, together with the correspondingrealizations of the $U_q g$-action. The intriguing relation betweeng-invariants and $U_q g$-invariants suggests that these changes of generatorsmight be employed to simplify the dynamics of some g-covariant quantum physicalsystems.Comment: latex file, 35 pages, no figures. Final version to appear in Rev.  Math. Phy

DRINFEL'D TWIST AND q-DEFORMING MAPS FOR LIE GROUP COVARIANT HEISENBERG ALGEBRAE