Invariant orderings in solvable Lie groups (Q1823329)

The author defines the concepts of oscillator Lie algebra and Lie group. An invariant cone in the Lie algebra \({\mathfrak g}\) is a cone C with acute vertex, closed, and invariant with respect to inner automorphisms of \({\mathfrak g}\). Theorem 1.1. Let \({\mathfrak g}\) be a solvable real Lie algebra. If \({\mathfrak g}\) admits an invariant cone C with int \(C\neq \emptyset\), then \({\mathfrak g}\) can be equipped with the structure of an oscillator Lie algebra. Every oscillator Lie algebra has an invariant cone C with int \(C\neq \emptyset.\) A continuous ordering in a Lie group G is a closed subsemigroup P which is invariant with respect to inner automorphisms, \(P\cap P^{-1}=\{e\}\), P (topologically) can be generated by any neighborhood of e, and the cone \(\{\) \(x\in {\mathfrak g}:\) \((\forall t>O)\) \((\exp(tx)\in P)\}\) has non-empty interior. Here \({\mathfrak g}\) is Lie algebra of G. Theorem 1.3-1.4. For every invariant cone C with int \(C\neq \emptyset\) in an oscillator Lie algebra there exists a unique continuous ordering in the corresponding simply connected Lie group. - A connected solvable Lie group admits a continuous ordering if and only if when its maximal compact subgroup is contained in the center of the group, and the Lie algebra admits an invariant cone C with int \(C\neq \emptyset\) having trivial intersection with the subalgebra that corresponds to the maximal compact subgroup. Every such cone corresponds to a unique continuous ordering.