  A perturbative quantization procedure for Lie bialgebras is introduced and
used to classify all three dimensional complex quantum algebras compatible with
a given coproduct. The role of elements of the quantum universal enveloping
algebra that, analogously to generators in Lie algebras, have a distinguished
type of coproduct is discussed, and the relevance of a symmetrical basis in the
universal enveloping algebra stressed. New quantizations of three dimensional
solvable algebras, relevant for possible physical applications for their
simplicity, are obtained and all already known related results recovered. Our
results give a quantization of all existing three dimensional Lie algebras and
reproduce, in the classical limit, the most relevant sector of the complete
classification for real three dimensional Lie bialgebra structures given by X.
Gomez in J. Math. Phys. Vol. 41. (2000) 4939.
