Real Nullstellensatz for 2-step nilpotent Lie algebras
arxiv(2024)
摘要
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, …, 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.
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