Affinoid Dixmier modules and the deformed Dixmier-Moeglin equivalence

DOI10.1007/S10468-021-10084-4arXiv2102.03330MaRDI QIDQ6359976

Adam Jones

Publication date: 5 February 2021

Abstract: The affinoid envelope, $widehatU(mathcalL)$ of a free, finitely generated $mathbb{Z}_p$-Lie algebra $mathcalL$ has proven to be useful within the representation theory of compact $p$-adic Lie groups. Our aim is to further understand the algebraic structure of $widehatU(mathcalL)$, and to this end, we will define a Dixmier module over $widehatU(mathcalL)$, and prove that this object is generally irreducible in case where $mathcalL$ is nilpotent. Ultimately, we will prove that all primitive ideals in the affinoid envelope can be described in terms of the annihilators of Dixmier modules, and using this, we aim towards proving that these algebras satisfy a version of the classical Dixmier-Moeglin equivalence.

Mathematics Subject Classification ID

Valuations, completions, formal power series and related constructions (associative rings and algebras) (16W60) Solvable, nilpotent (super)algebras (17B30) Ideals in associative algebras (16D25) Chain conditions on annihilators and summands: Goldie-type conditions (16P60) Ideals and subalgebras (46H10) Complete rings, completion (13J10) Representation theory of associative rings and algebras (16Gxx)

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