The Associated Variety of a Poisson Prime Ideal

A Dixmier-Moeglin equivalence for Poisson algebras with torus actions

A Poisson analog of the Dixmier-Moeglin equivalence is established for any affine Poisson algebra $R$ on which an algebraic torus $H$ acts rationally, by Poisson automorphisms, such that $R$ has only finitely many prime Poisson $H$-stable ideals. In this setting, an additional characterization of the Poisson primitive ideals of $R$ is obtained -- they are precisely the prime Poisson ideals maximal in their $H$-strata (where two prime Poisson ideals are in the same $H$-stratum if the intersections of their $H$-orbits coincide). Further, the Zariski topology on the space of Poisson primitive ide…

Extensions of almost faithful prime ideals in virtually nilpotent mod-$p$ Iwasawa algebras

Let $G$ be a nilpotent-by-finite compact $p$-adic analytic group for some $p>2$, and $H = \mathbf{FN}_p(G)$ its finite-by-(nilpotent $p$-valuable) radical. Fix a finite field $k$ of characteristic $p$, and write $kG$ for the completed group ring of $G$ over $k$. We show that almost faithful $G$-stable prime ideals $P$ of $kH$ extend to prime ideals $PkG$ of $kG$.

Poisson algebras via model theory and differential-algebraic geometry

Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds for any complex affine Poisson algebra; that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are…

Toric Poisson Ideals in Cluster Algebras

This paper investigates the Poisson geometry associated to a cluster algebra over the complex numbers, and its relationship to compatible torus actions. We show, under some assumptions, that each Noetherian cluster algebra has only finitely many torus invariant Poisson prime ideals and we show how to obtain using the exchange matrix of an initial seed. In fact, these ideals are independent of the choice of compatible Poisson structure. In many interesting cases the ideals can be described more explicitly.

$A$型箭图表示范畴中的素理想

Prime ideals in categories of representations of quivers of type $A$

该研究聚焦箭图表示的张量阿贝尔范畴中的素张量理想，完成了顶点为整数集的有界路径长度之字形箭图表示范畴的素张量理想分类，证明其与整数幂集这一布尔代数的素理想存在典范双射。

Equivariant prime ideals for infinite dimensional supergroups

Let $A$ be a commutative algebra equipped with an action of a group $G$. The so-called $G$-primes of $A$ are the equivariant analogs of prime ideals, and of central importance in equivariant commutative algebra. When $G$ is an infinite dimensional group, these ideals can be very subtle: for instance, distinct $G$-primes can have the same radical. In previous work, the second author showed that if $G$ is ${\bf GL}$ and $A$ is a polynomial representation, then these pathologies disappear when working with super vector spaces; this leads to a geometric description of $G$-primes of $A$. In the pre…