A Dixmier-Moeglin equivalence for Poisson algebras with torus actions

A strong Dixmier-Moeglin equivalence for quantum Schubert cells

Dixmier and Moeglin gave an algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the universal enveloping algebra of a finite-dimensional complex Lie algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the universal enveloping algebra of a finite-dimensional complex Lie algebra satisfies the Dixmier-Moeglin equivalence. We define quantities which measure how "close" an arbitrary prime ideal of a noetherian algebra is to being primitive, rational, and locally clos…

Affinoid Dixmier modules and the deformed Dixmier-Moeglin equivalence

The affinoid envelope, $\widehat{U(\mathcal{L})}$ of a free, finitely generated $\mathbb{Z}_p$-Lie algebra $\mathcal{L}$ 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 $\widehat{U(\mathcal{L})}$, and to this end, we will define a Dixmier module over $\widehat{U(\mathcal{L})}$, and prove that this object is generally irreducible in case where $\mathcal{L}$ is nilpotent. Ultimately, we will prove that all primitive ideals in the affinoid envelope can be described in terms of the annihilators of …

The Dixmier-Moeglin equivalence and a Gel'fand-Kirillov problem for Poisson polynomial algebras

The structure of Poisson polynomial algebras of the type obtained as semiclassical limits of quantized coordinate rings is investigated. Sufficient conditions for a rational Poisson action of a torus on such an algebra to leave only finitely many Poisson prime ideals invariant are obtained. Combined with previous work of the first-named author, this establishes the Poisson Dixmier-Moeglin equivalence for large classes of Poisson polynomial rings, such as semiclassical limits of quantum matrices, quantum symplectic and euclidean spaces, quantum symmetric and antisymmetric matrices. For a simila…

尖点Hopf代数、Dixmier-Moeglin等价性与诺特群代数

Pointed Hopf algebras, the Dixmier-Moeglin Equivalence and Noetherian group algebras

本文研究群代数及尖点Hopf代数的诺特性、有限Gelfand-Kirillov维数与Dixmier-Moeglin等价性之间的等价关系，并探讨群代数的有限性条件及一个相关猜想的最小反例。

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…

The Prime ideal Stratification and The Automorphism Group of $U^{+}_{r,s}(B_{2})$

Let ${\mathfrak g}$ be a finite dimensional complex simple Lie algebra, and let $r,s\in \mathbb{C}^{\ast}$ be transcendental over $\mathbb{Q}$ such that $r^{m}s^{n}=1$ implies $m=n=0$. We will obtain some basic properties of the two-parameter quantized enveloping algebra $U_{r,s}^{+}(\mathfrak g)$. In particular, we will verify that the algebra $U_{r,s}^{+}(\mathfrak g)$ satisfies many nice properties such as having normal separation, catenarity and Dixmier-Moeglin equivalence. We shall study a concrete example, the algebra $U_{r,s}^{+}(B_{2})$ in detail. We will first determine the normal ele…