The deformed Dixmier-Moeglin equivalence for completed enveloping algebras

1 December 2020

An algebra $R$ is said to satisfy the Dixmier-Moeglin equivalence if a prime ideal $P$ of $R$ is primitive if and only if it is rational, if and only if it is locally closed, and a commonly studied problem in non-commutative algebra is to classify rings satisfying this equivalence, e.g. $U(\mathfrak g)$ for a finite dimensional Lie algebra $\mathfrak g$. We explore methods of generalising this to a $p$-adic setting, where we need to weaken the statement. Specifically, if $\hat R$ is the $p$-adic completion of a $\mathbb Q_p$-algebra $R$, rather than approaching the Dixmier-Moeglin equivalence for $\hat R$ directly, we instead compare the classes of primitive, rational and locally closed prime ideals of $\hat R$ within suitable "deformations". The case we focus on is where $R=U(L)$ for a $\mathbb Z_p$-Lie algebra $L$, and the deformations have the form $\hat U(p^n L)$, and we aim to prove a version of the equivalence in the instance where $L$ is nilpotent.

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