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

点击翻译按钮获取中文摘要

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.