Real Nullstellensatz for nilpotent Lie algebras

Abstract

We prove a real Nullstellensatz for the universal enveloping algebra U(g) of a nilpotent finite-dimensional real Lie algebra g: An ideal of U(g) is real if and only if it is the intersection of kernels of irreducible integrable *-representations of U(g) by (in general unbounded) operators on a Hilbert space. For this, we build upon some classical results about the complexification of U(g) that lead to a complex Nullstellensatz (every semiprime ideal is the intersection of primitive ideals), but focus only on the ``real points'', i.e., the *-representations on Hilbert spaces. On the technical side, this requires to take into account an involution of U(g), namely, the antipode of its Hopf algebra structure, which goes into the definition of real ideals of U(g). Our result is the first noncommutative real Nullstellensatz for *-algebras with unbounded irreducible *-representations, demonstrating how methods of real algebraic geometry can lead to results about operator *-algebras also in situations where C*-algebraic methods fail.