  We propose a notion of a quantum universal enveloping algebra for an
arbitrary Lie algebra defined by generators and relations which is based on the
quantum Lie operation concept. This enveloping algebra has a PBW basis that
admits the Kashiwara crystalization. We describe all skew primitive elements of
the quantum universal enveloping algebra for the classical nilpotent algebras
of the infinite series defined by the Serre relations and prove that the set of
PBW-generators for each of these enveloping algebras coincides with the
Lalonde-Ram basis of the ground Lie algebra with a skew commutator in place of
the Lie operation. The similar statement is valid for Hall-Shirshov basis of
any Lie algebra defined by one relation, but it is not so in general case.
