Lie semialgebras are real phenomena

Abstract

In recent year there has been considerable interest in local semigroups in Lie groups, partly from the view point of geometric control theory, partly from the view point of differential geometry (for the former aspect see [Br, Hi, JS,l, for the latter [Lo, O11 and 2, Pa, Ro, Vi]). Lie theory relates local subgroups of a Lie group with subalgebras of its Lie algebra; in just the same spirit it relates certain local subsemigroups with Lie semialgebras in its Lie algebra. Here a Lie semialgebra in a finite dimensional real Lie algebra is a wedge (i.e., a topologically closed subset which is stable under addition and multiplication by non-negative scalars) for which there is a neighborhood B of 0 in L such that for all X, Y~ B the CampbellHausdorff product X * Y = X + Y+ 89 Y-l+ ... is well defined and in addition stays in W for X, Ye W. If W is a Lie semialgebra in L, then W W is a Lie subalgebra [HL 2,l. If W is any wedge in L and if W contains the Lie algebra [W,W] generated by all commutators w,w" with w,w'~W, then W is a Lie semialgebra. Lie semialgebras arising in this fashion we will call trivial. We shall prove the following