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Polycyclic groups, Lie algebras and algebraic groups.

It is our purpose here to generalise the construction of Jennings to the class of polycyclic-by-finite groups and to study the basic properties of this construction. To a polycyclic-by-finite group G and a field k of characteristic zero we canonically associate a fmite dimensional Lie algebra Lk(G) and, when k is algebraically closed, an affine algebraic group G over k, of which Lk(G) is the Lie algebra. Both Lk(G) and G are defined by the subHopf algebra, of the Hopf algebra rQ(kG) dual to kG, generated by the principal block component of r0(kG).