Young Algebra Seminar 2001–2002 THIN LIE ALGEBRAS AND THIN PRO–p–GROUPS

The purpose of these notes is to give an overview on a class of (graded) Lie algebras, which satisfy certain narrowness conditions on the lattice of their homogeneous ideals. Narrowness conditions arise in the theory of p-groups and pro-p groups. In the group theoretic context, the best known condition for a group is to be of maximal class. Let p be a prime. A group G of order p is said to be a group of maximal class if its nilpotency class is (the maximum possible, that is) n − 1. The structure of groups of maximal class has been detailed studied by Blackburn (see [10]) in the fifties. From our point of view, the main feature of groups of maximal class is to exhibit a sort of ’global regularity’. This claim can be made precise in terms of pro-p-groups. We recall that a poset (I,≤) is said to be a direct poset if, for all i, j ∈ I there exists k ∈ I such that k ≥ i and k ≥ j. Suppose we have a family of groups {Gi}i∈I , where I is a direct poset, together with homomorphisms φj,i : Gj −→ Gi