On the Hopf algebra structure of the Lusztig quantum divided power algebras

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.
Comment: 21 pages. v2: 22 pages; we give more details in the proof of Lemma 4.2; we change the convention on \ell' according with Lusztig's book and restate the formulas (2.3) and (2.4)