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PI（与Bell， Launois和Moosa一起）在五年前发起的一项有前途的研究推动的，该研究利用模型理论中的几何稳定性机制为Dixmier-Moeglin等价提供了一种新的方法-一种对noether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驯服域与泛型算子的模型理论。研究了具有一般加性算子（记为D）的大域的模型论性质，这些加性算子服从一定的乘法和交换规则。我们的结果自然导致了d -大场的概念，d -算子设置中的大场的模拟，我们探索了它们在d -场算术和逆d -伽罗瓦问题中的作用。这些发展随后被应用于诺埃尔代数的表示理论。也就是说，我们使用模型论机制来描述原始理想，这些原始理想大致分类不可约的表示，在纯拓扑和代数术语中，用于广泛的noether Hopf代数。模型理论代数（或者更确切地说，带算子场的模型理论）研究的是具有交换导数的环的代数结构，以及很多时候的解析结构。经典的例子是光滑函数的环和亚纯函数的域（在几个变量中），配备了通常的微分算子。微分场论的大部分内容可以与经典代数理论并行探讨。例如，存在代数闭域、实闭域和p基闭域的微分类似物。此外，在多项式方程的伽罗瓦理论的精神下，一个优美的线性微分方程的微分伽罗瓦理论被发展并用于泛函超越问题。另一方面，表示理论是纯数学中最具影响力的领域之一。中国的发展是由具有挑战性但又非常基本的问题推动的。特别是，其中一个基本问题是对给定的诺etherian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标准方法是研究不可约表示的核——即所谓的原始理想。对于有限维复李代数的包络代数，Dixmier和Moeglin证明了原始理想可以用纯代数和纯拓扑来表征。这些特征引发了人们对现在所知的迪克米尔-莫格林等价的兴趣。从广义上讲，本项目的指导思想是：(1)推导是由对偶数（局部有限代数的特殊情况）推导出的简单的加性算子，我们的目标是统一所有算子的模型论和伽罗瓦理论，这些算子具有从任何局部有限代数（在特定类的大域上）推导出的乘法和交换规则。这包括Hasse-Schmidt推导的重要情况（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）来解决noether代数的不可约表示的分类问题。更准确地说，阐明了Bell-Leung猜想，说明所有有限生成的有限Gelfand-Kirillov维的noetherian Hopf代数满足Dixmier-Moeglin等价。我们的目标是证明这个等价的一个广泛的家族的重要情况（迭代的Hopf-Ore扩展），并在完全一般的泊松版本。

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.