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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.

该项目的目的是在模型理论和代数表示论之间建立进一步的联系。这是由 PI（与 Bell、Launois 和 Moosa 一起）五年前发起的一项有前景的研究推动的，该研究利用模型理论中的几何稳定性机制，为 Dixmier-Moeglin 等价提供了一种新方法——一种对诺特代数的不可约表示进行分类的程序。更详细地说，该项目旨在进一步探索并将驯服场的模型理论与通用算子统一起来。我们研究配备了遵守某些乘法和交换规则的通用加法算子（用 D 表示）的大域的模型理论属性。我们的结果自然引出了 D-大域的概念，即 D-算子设置中大域的类比，并且我们探讨了它们在 D-域算术和逆 D-伽罗瓦问题中的作用。这些发展随后被运用到诺特代数的表示论中。也就是说，我们使用模型理论机制来描述原始理想，这些理想以纯拓扑和代数术语对一类诺特霍普夫代数的不可约表示进行粗略分类。模型理论代数（或者更确切地说，具有算子的域模型理论）特别研究具有通勤推导的环的代数结构，并且很多时候也是解析结构。经典的例子是光滑函数环和亚纯函数域（在多个变量中），配备了常用的微分算子。大多数微分场论可以与其经典代数对应物并行地探索。例如，存在代数闭域、实闭域和 p 进闭域的微分类似物。此外，本着多项式方程伽罗瓦理论的精神，一种美丽的线性微分方程微分伽罗瓦理论已经被开发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又非常基本的问题推动的。特别是，基本问题之一是对给定诺特代数的不可约表示进行分类（这通常相当困难）。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，Dixmier 和 Moeglin 证明了原始理想可以用纯代数和拓扑来表征。这些特征引发了人们对当今所谓的 Dixmier-Moeglin 等价的兴趣。从广义上讲，该项目以两个广泛的愿景为指导：（1）导数只是由对偶数（局部有限代数的特殊情况）导出的加法算子，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（在特定的情况下）导出的乘法和交换规则统一起来。大领域的类别）。这包括哈斯-施密特推导的重要情况（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）来解决诺特代数不可约表示的分类问题。更准确地说，阐明贝尔-梁猜想，指出有限 Gelfand-Kirillov 维数的所有有限生成的诺特霍普夫代数满足 Dixmier-Moeglin 等价。我们的目标是证明这一等价性对于一系列重要的情况（迭代的 Hopf-Ore 扩展）及其具有完全普遍性的泊松版本。