Real Nullstellensatz for 2-step nilpotent Lie algebras

Abstract

We prove a noncommutative real Nullstellensatz for 2-step nilpotent Lie algebras that extends the classical, commutative real Nullstellensatz as follows: Instead of the real polynomial algebra $\mathbb R[x_1, \dots, x_d]$ we consider the universal enveloping *-algebra of a 2-step nilpotent real Lie algebra (i.e. the universal enveloping algebra of its complexification with the canonical *-involution). Evaluation at points of $\mathbb R^d$ is then generalized to evaluation through integrable *-representations, which in this case are equivalent to filtered *-algebra morphisms from the universal enveloping *-algebra to a Weyl algebra. Our Nullstellensatz characterizes the common kernels of a set of such *-algebra morphisms as the real ideals of the universal enveloping *-algebra.