  We study the Hopf algebra structure of Lusztig's quantum groups. First we
show that the zero part is the tensor product of the group algebra of a finite
abelian group with the enveloping algebra of an abelian Lie algebra. Second we
build them from the plus, minus and zero parts by means of suitable actions and
coactions within the formalism presented by Sommerhauser to describe triangular
decompositions.
