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

D-大域的模型理论以及与表示理论的联系。

基本信息

批准号：
EP/V03619X/1
负责人：
Omar Leon Sanchez
金额：
$ 44.73万
依托单位：
University of Manchester
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV03619X%2F1
关键词：
Model theory large fields connections

项目摘要

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.

这个项目的目的是建立模型理论和代数表示理论之间的进一步联系。这是由PI（与Bell，Launois和Moosa一起）五年前发起的一系列有希望的研究所推动的，该研究利用模型理论的几何稳定性机制为Dixmier-Moeglin等价提供了一种新方法-一种对诺特代数的不可约表示进行分类的程序。更具体地说，本项目旨在进一步探索和统一具有一般算子的驯服场的模型理论。我们研究了具有一般加法算子（记为D）的大场的模型论性质，这些算子服从一定的乘法和交换规则。我们的结果自然导致D-大场的概念，模拟大场的D-运营商设置，我们探讨他们的作用，在D-域算术和逆D-伽罗瓦问题。这些发展，然后部署在诺特代数的表示理论。也就是说，我们使用模型理论的机器来表征原始的理想，其中大致分类不可约表示，在纯粹的拓扑和代数方面的广泛的一类noetherian霍普夫代数。模型理论代数（或者更确切地说，模型理论的领域与运营商）研究特别是代数，也多次分析，结构的环配备了交换导子。经典的例子是光滑函数的环和亚纯函数的域（在多个变量中），配备了通常的微分算子。大多数的微分场论可以与经典的代数场论并行地研究。例如，有代数闭域、真实的闭域和p-代数闭域的微分类似物。此外，在多项式方程伽罗瓦理论的精神下，线性微分方程的微分伽罗瓦理论也被发展出来并用于泛函超越问题。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但非常基本的问题驱动的。特别是，一个基本的问题是分类一个给定的诺特代数的不可约表示（这往往是相当困难的）。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，Dixlane和Moeglin证明了本原理想可以被描述为纯代数和拓扑的。广义地说，这个项目有两个主要的指导思想：（1）导子是由对偶数（局部有限代数的一个特例）导出的简单的加法算子，我们的目标是统一所有算子的模型理论和伽罗瓦理论，这些算子具有由任意局部有限代数（在特定的大域类上）导出的乘法和交换规则。这包括哈塞-施密特导子的重要情况（例如代数上和真实的闭域上）。(2)利用上面的模型论结果（特别是几何稳定性工具）来处理诺特代数的不可约表示的分类。更确切地说，揭示了贝尔-梁猜想，说明所有有限Gelfand-Kirillov维数的N-生成诺特Hopf代数满足Dixmier-Moeglin等价。我们的目标是证明这种等价性的一个广泛的家庭的重要情况下（迭代Hopf-Ore扩展），其泊松版本的充分的一般性。