Interactions between representation theory, Poisson algebras and differential algebraic geometry

表示论是当今数学中最活跃的领域之一，它应用于许多科学，并与许多其他数学学科相互作用，如数论、组合学、几何学、概率论、量子力学和量子场论。这个美丽的主题起源于德德金德写给弗罗贝尼乌斯的一封信。粗略地说，这个想法是通过代数的对称性来研究它们。尽管取得了许多成功和应用，但许多基本问题仍然具有挑战性。例如，对给定代数的不可约表示进行分类通常是相当困难的(如果不是不可能的话)。迪克西米尔提出的解决这个问题的标准方法是研究不可约表示的零化子，即所谓的原始理想。对代数的本原理想进行分类可以看作是理解代数表示理论的第一个近似值。在有限维复李代数的包络代数的情况下，Dixmier和Moeglin证明了在素理想中，本原理想可以在代数上和拓扑上刻画。这些代数和拓扑准则也刻画了许多其他代数的素理想之间的本原理想，在这种情况下，我们说DixmierMoeglin等价成立。泊松代数最早出现在两个世纪前泊松的著作中，当时他正在研究天体力学中的三体问题。从那时起，泊松代数被证明与数学和物理的许多领域有关，因此，由于它们的广泛应用，它们的研究引起了数学家和理论物理学家的极大兴趣。目前，这门学科是数学和数学物理中最活跃的学科之一。逼近泊松代数的一种方法是通过量子化。在物理学中，量子化是从经典力学到量子力学的过渡。从数学上讲，(形变)量子化是从泊松代数/几何到非对易代数/几何的过渡。在形变量化的背景下，Poisson代数是非对易代数的半经典极限。粗略地说，“量子”空间的非对易代数几何与辛叶空间的几何密切相关。本着形变量子化的精神，人们被引导去研究迪克斯米尔-莫格林等价的泊松模拟，即所谓的泊松-迪克斯默-莫格林等价。Goodearl建立了具有适当环面作用的仿射Poisson代数的Poisson Dixmer-Moeglin等价，Brown和Gordon建立了具有有限多个Poisson本原理想的Poisson代数的Poisson Dixmer-Moeglin等价.鉴于这些成功，Brown和Gordon在2002年询问Poisson Dixmer-Moeglin等价是否对所有仿射复Poisson代数成立。在最近与Bell，Leon Sanchez和Moosa的一篇论文中，我们完全回答了这个问题，这要归功于一种基于微分代数几何和微分场模型理论的新方法。这个项目源于我希望继续这一新的研究路线，一方面是泊松几何和表示论，另一方面是微分代数几何和模型论。泊松几何/表示理论的这种非常新颖的方法已经解决了布朗和戈登12年来的一个问题。一如既往，将数学的不同领域联系起来将是产生深刻结果的源泉。本项目的目的是通过微分代数几何进一步研究Poisson几何/表示理论的这一新方法。这将导致Hopf代数、扭转齐次坐标环和Poisson代数的表示理论的进步，并为研究代数D-簇提供新的工具。

Representation theory is one of the most active fields of mathematics today with applications to many of the sciences and interactions with many other mathematical disciplines such as number theory, combinatorics, geometry, probability theory, quantum mechanics and quantum field theory. This beautiful subject originated in a letter to Frobenius by Dedekind. Roughly speaking, the idea is to study algebras through their symmetries. Despite many successes and applications, many basic questions remain challenging. For instance, it is often quite difficult (if not impossible) to classify the irreducible representations of a given algebra. A now standard approach to this problem, proposed by Dixmier, is to study the annihilators of the irreducible representations, the so-called primitive ideals. Classifying primitive ideals of an algebra can be seen as a first approximation towards understanding the representation theory of the algebra. In the case of enveloping algebras of finite dimensional complex Lie algebras, Dixmier and Moeglin proved that, among the prime ideals, primitive ideals can be characterized both algebraically and topologically. These algebraic and topological criteria also characterise primitive ideals among prime ideals in many other algebras, in which case we say that the Dixmier-Moeglin equivalence holds. Poisson algebras first appeared in the work of Poisson two centuries ago when he was studying the three-body problem in celestial mechanics. Since then, Poisson algebras have been shown to be connected to many areas of mathematics and physics, and so, because of their wide range of applications, their study is of great interest for both mathematicians and theoritical physicists. Currently, this subject is one of the most active in both mathematics and mathematical physics. One way to approach Poisson algebras is via quantisation. In physics, quantisation is the transition from classical to quantum mechanics. Mathematically, (deformation) quantisation is the transition from Poisson algebras/geometry to noncommutative algebras/geometry. In the context of deformation quantisation, Poisson algebras are the semiclassical limits of noncommutative algebras. Roughly speaking, the noncommutative algebraic geometry of the ``quantum'' spaces is closely related to the geometry of the space of symplectic leaves. In the spirit of deformation quantisation, one is led to study a Poisson analogue of the Dixmier-Moeglin equivalence, the so-called Poisson Dixmer-Moeglin Equivalence. The Poisson Dixmer-Moeglin Equivalence was established for affine Poisson algebras with suitable torus actions by Goodearl, and for Poisson algebras with only finitely many Poisson primitive ideals by Brown and Gordon. Given these successes, Brown and Gordon asked in 2002 whether the Poisson Dixmer-Moeglin Equivalence holds for all affine complex Poisson algebras. In a recent paper with Bell, Leon Sanchez and Moosa, we completely answered this question thanks to a novel approach based on tools from differential algebraic geometry and the model theory of differential fields. This project arises from my desire to continue this new line of research at the crossroad between Poisson geometry and representation theory on one hand, and differential algebraic geometry and model theory on the other hand. This highly novel approach of Poisson geometry/representation theory has already led to solving a 12-year old question of Brown and Gordon. As always, linking different areas of mathematics will be the source of deep results. The aim of this project is to further study this new approach of Poisson geometry/representation theory via differential algebraic geometry. This will lead to progress in the representation theory of Hopf algebras, twisted homogeneous coordinate rings and Poisson algebras, as well as to new tools to study algebraic D-varieties.