On Finiteness Theorems and Porcupine varieties in Lie algebras

This is an abridged diary of some of thèmathematical adventures' we encountered on our way towards a proof of the Divisibility Conjecture, formulated below. In this section we give a brief account of how Porcupine Varieties enter the stage naturally in connection with the Divisibility Problem. Let us start with some basic concepts and the appropriate notation from 5]. Let G be a connected Lie group, and let S be a closed subsemigroup containing the identity. We say that S is divisible if each of its elements has roots in S of arbitrary order, i.e., (8s 2 S; n 2 N)(9s n 2 S) (s n) n = s: We say that S is exponential if every point of S lies on a one-parameter semigroup lying entirely in S. One sees immediately that an exponential semigroup is divisible, and it is a principal result of 10] that the converse holds. The property that a semigroup be divisible is quite restrictive, and it reasonable to expect that one could achieve a classiication of such semigroups. As a beginning point one would like to pull the problem back to the Lie algebra level. The Lie theory of semigroups provides the machinery for such a transfer. Indeed, let g be the Lie algebra of a connected Lie group G containing a closed subsemigroup S and let exp: g ! G denote the exponential function. Recall rst that we denote the set of all subtangent vectors at 0 of a subset A of g with L(A); for B G we let L (B) = L(exp ?1 B). Then the tangent wedge W = L (S) of S satisses W = fx 2 g : exp R + x Sg; it is a closed wedge in g, invariant under the inner automorphisms induced by the elements of its edge W = e ad H(W) W is called a Lie wedge. The closed subsemigroup S is said to be innnitesimally generated if it coincides with the smallest closed subsemigroup containing the exponential image of its Lie wedge W , i.e., if S = hexp Wi; it is said to be weakly exponential if S = exp W , and one sees at once that S is exponential in the sense of the deenition above if and only if S = exp W .