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 $R[x_1,\dots ,x_d]$ 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 $R^d$ 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.