FINITE PRESENTABILITY OF SOME METABELIAN HOPF ALGEBRAS

The purpose of this paper is to try to unite some existing methods used in the classification results of metabelian Lie algebras and metabelian discrete groups of homological type FP2 via the language of Hopf algebras. This sheds more light on the similarities between the Lie and group cases and explains partially the differences. Still some of the results in the group case have homotopical flavour, using methods from covering spaces to establish that having homological type FP2 imposes strong condition on the first Σinvariant of the group ([4]). These methods do not have a purely algebraic counterpart. The Lie case was treated in [5, 6] with algebraic methods, and a Lie invariant (with a valuation flavour) for metabelian Lie algebras was proposed. This plays the same role in the Lie theory as the Bieri-Strebel Σ-invariant for metabelian groups. In this paper we do not suggest a new invariant but establish that the main result of [5] holds for some metabelian Hopf algebras. It is interesting to note that in both the Lie and group cases calculations with the second homology group of Abelian objects (Lie algebras or Abelian groups) viewed as modules over a commutative ring via the corresponding diagonal action was always quite helpful. The definition of the diagonal Lie and group actions can be united via the comultiplication map of Hopf algebras, and this was the starting point of our considerations. We study Hopf algebras H = U(L)#kG over a field k, that is, smash products of universal enveloping algebras U(L) of Lie algebras L over k by group rings kG, where G acts via conjugation on L and write X for the category of such Hopf algebras. This category is quite important. If char(k) = 0 it coincides with the category of cocommutative,