We propose a notion of a
quantum universal enveloping algebra for any 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 a monomial crystallization by
means of the Kashiwara idea. We describe all skew primitive elements of the
quantum universal enveloping algebras for the classical nilpotent algebras of the
infinite series defined by the Serre relations and prove that the above 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 the general
case.
