Model theory of D-large fields and connections to representation theory.

这个项目的目的是在代数的模型理论和表示理论之间建立进一步的联系。这是由五年前由PI(与Bell、Launois和Moosa一起)发起的一系列有希望的研究推动的，该研究利用模型理论的几何稳定性机制提供了一种新的方法来获得Dixmier-Moeglin等价性--一个对Notherian代数的不可约表示进行分类的程序。更详细地说，本项目旨在进一步探索和统一Tame场与一般算符的模型理论。我们研究了配备了遵循一定乘法和交换规则的一般加法算符(记为D)的大场的模型论性质。我们的结果自然而然地引出了D-大域的概念，D-大域在D-算符集合中的类比，我们探索了它们在D-域算术和D-Galois逆问题中的作用。然后将这些发展应用到Noether代数的表示理论中。也就是说，我们使用模型论机制刻画了一大类Notherian Hopf代数的本原理想，这些本原理想粗略地从纯拓扑和代数的角度对不可约表示进行了分类。模型论代数(或者更确切地说，具有算子场的模型理论)特别研究了配备交换导子的环的代数结构以及多次解析结构。经典的例子是光滑函数环和亚纯函数域(含多个变量)，它们配备了通常的微分算子。大多数微分场理论可以与其经典的代数理论平行地进行探索。例如，有代数闭域、实闭域和p闭域的微分类比。此外，本着伽罗瓦多项式方程理论的精神，发展了一种美丽的线性微分方程组的伽罗瓦微分理论，并将其应用于函数超越问题。另一方面，表示论是纯数学中最有影响力的领域之一。它的发展是由具有挑战性的、但非常基本的问题推动的。特别地，基本问题之一是对给定的Noether代数的不可约表示进行分类(这通常是相当困难的)。现在解决这个问题的一个标准方法是研究不可约表示的核心--所谓的原始理想。在有限维复李代数的包络代数的情况下，Dixmier和Moeglin证明了本原理想可以在纯代数和拓扑学上刻画。这些特征引发了人们对现在所知的DixmierMoeglin等价的兴趣。广义地说，这个项目被两个广泛的愿景所指导：(1)导子是由对偶数诱导的简单的加性算子(局部有限代数的一个特例)，我们的目标是将所有算子的模型理论和伽罗瓦理论与从任何局部有限代数(在特定类型的大域上)导出的乘法和交换规则统一起来。这包括Hasse-Schmidt导子的重要情况(例如，在代数和实闭域上)。(2)利用上述模型理论结果(特别是几何稳定性工具)解决Notherian代数的不可约表示的分类问题。更确切地说，揭示了Bell-Leung猜想中所有有限生成的有限Gelfand-Kirillov维Notherian Hopf代数满足DixmierMoeglin等价。我们的目标是证明一大类重要情形(迭代Hopf-Ore扩张)及其Poisson形式的完全一般性的等价性。

The aim of this project is to build further connections between model theory and representation theory of algebras. This is driven by a promising line of research initiated five years ago, by the PI (together with Bell, Launois, and Moosa), that exploits the geometric-stability machinery from model theory to provide a new approach to the Dixmier-Moeglin equivalence -- a program to classify irreducible representations of noetherian algebras. In more detail, this project aims at exploring further and unifying the model theory of tame fields with generic operators. We investigate the model-theoretic properties of large fields equipped with generic additive operators (denoted by D) obeying certain multiplicative and commutative rules. Our results naturally lead to the notion of D-large field, the analogue of large fields in the D-operators setting, and we explore their role in D-field arithmetic and Inverse D-Galois questions. These developments are then deployed in the representation theory of noetherian algebras. Namely, we use the model-theoretic machinery to characterize primitive ideals, which roughly classify irreducible representations, in purely topological and algebraic terms for a wide class of noetherian Hopf algebras.Model theoretic algebra (or rather, the model theory of fields with operators) studies in particular the algebraic, and also many times analytic, structure of rings equipped with commuting derivations. Classical examples are rings of smooth functions and fields of meromorphic functions (in several variables), equipped with the usual differentiation operators. Most of the differential field theory can be explored in parallel to its classical algebraic counterpart. For instance, there are differential analogues of algebraically closed, real closed, and p-adically closed fields. Furthermore, in the spirit of Galois theory for polynomial equations, a beautiful differential Galois theory for linear differential equations has been developed and used in functional transcendence questions.Representation theory, on the other hand, is one of the most influential fields of pure mathematics. Its development has been driven by challenging, yet very basic problems. In particular, one of the fundamental questions is to classify the irreducible representations of a given noetherian algebra (which is often quite difficult). A now standard approach to this problem is to study the kernels of irreducible representations -- the so-called primitive ideals. In the case of enveloping algebras of finite dimensional complex Lie-algebras, Dixmier and Moeglin proved that primitive ideals can be characterised purely algebraically and topologically. These characterisations initiated the interest in what is nowadays known as the Dixmier-Moeglin equivalence.In broad terms, this project is guided by two broad visions:(1) Derivations are simply additive operators induced by the dual numbers (a special case of a local finite algebra), we aim to unify the model theory and Galois theory of all operators with multiplicative and commutative rules induced from ANY local finite algebra (on specific classes of large fields). This includes the important case of Hasse-Schmidt derivations (on algebraically and real closed fields, for instance). (2) Exploit the above model-theoretic results (in particular, the geometric-stability tools) to tackle the classification of irreducible representations of noetherian algebras. More precisely, shed a light in the Bell-Leung conjecture stating the all finitely generated noetherian Hopf algebras of finite Gelfand-Kirillov dimension satisfy the Dixmier-Moeglin equivalence. We aim to prove this equivalence for a wide family of important cases (iterated Hopf-Ore extensions), and its Poisson-version in full generality.