Representations of Lie algebras by non-skewselfadjoint operators in Hilbert space

We study non-skewselfadjoint representations of a finite dimensional real Lie algebra g. To this end we embed a non-skewselfadjoint representation of g into a more complicated structure, that we call a g-operator vessel and that is associated to an overdetermined linear conservative input/state/output system on the corresponding simply connected Lie group G. We develop the frequency domain theory of the system in terms of representations of G, and introduce the joint characteristic function of a g-operator vessel which is the analogue of the classical notion of the characteristic function of a single non-selfadjoint operator. As the first non-commutative example, we apply the theory to the Lie algebra of the ax+b group, the group of affine transformations of the line.