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Interpretation functors and infinite-dimensional representations of finite-dimensional algebras

解释函子和有限维代数的无限维表示

基本信息

批准号：
EP/K022490/1
负责人：
Mike Prest
金额：
$ 76.3万
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：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FK022490%2F1
关键词：
Interpretation functors infinite dimensional representations

项目摘要

In the 1980s unexpected applications of the model theory of modules to the representation theory of finite-dimensional algebras were discovered and since then there has been further, sometimes deep, interaction between these areas. Model theory uses ideas and results from mathematical logic to investigate general questions about mathematical structure and also to obtain new results in other parts of mathematics. It provides a particular perspective which often gives new insights into other parts of mathematics. Almost always model theory makes heavy use of the Compactness Theorem of mathematical logic and, for that, one needs to be working in a context within which there is room to make infinitary constructions. In the specific context of the representation theory of finite-dimensional algebras, where interest is typically focussed on finite-dimensional representations, that means that we have to extend our interest to at least some of the infinite-dimensional representations, even if our eventual applications are back in the context of the finite-dimensional ones. This particular project will deep the interaction of model theory and representation theory. The question underlying the project is "How complex is a particular collection of representations?"; various ways of answering this question have been investigated already and the principal aim is to show that the most standard algebraic answer - which is given in terms of certain embeddings of one collection in another - fits well with the model-theoretic one. The latter is in terms of the notion of interpretation, which is essentially a translation from one language (associated to a collection of representations) to another. That has already been shown to be equivalent to a particularly nice kind of embedding but it is not known how to close the gap between that and kind of embedding which is the standard algebraic answer to the above question. Closing that gap is one of the aims of the project. Going beyond that, the project has as an aim a substantial refinement of the existing rather broad algebraic classification of complexity classes into tame and wild (with further refinements of tame).The project will combine very general methods, some being inspired by algebraic geometry and abstract category theory, with very specific investigations of the representations of particular algebras where entirely explicit descriptions are the aim. It will draw on two well-developed subjects; the model theory of modules and the representation theory of finite-dimensional algebras, and will use techniques from homological algebra and additive functor category theory. In view of that breadth of necessary input as well as on account of the number and nature of the aims of the project, two PDRAs, working together with the PI, all sharing their expertise, will form the research team.

在20世纪80年代意想不到的应用模型理论的模块表示理论的有限维代数被发现，从那时起，有进一步的，有时深，相互作用之间的这些领域。模型论使用数理逻辑的思想和结果来研究数学结构的一般问题，并在数学的其他部分获得新的结果。它提供了一个特殊的视角，经常给数学的其他部分带来新的见解。模型论几乎总是大量使用数理逻辑的紧性定理，为此，人们需要在一个有空间进行无限构造的上下文中工作。在有限维代数表示论的特定背景下，人们的兴趣通常集中在有限维表示上，这意味着我们必须将我们的兴趣扩展到至少某些无限维表示，即使我们最终的应用又回到了有限维表示的背景下。这个特别的项目将加深模型理论和表示理论的相互作用。该项目的基本问题是“一个特定的表征集合有多复杂？";回答这个问题的各种方法已经被研究过了，主要目的是表明最标准的代数答案--它是根据一个集合在另一个集合中的某些嵌入给出的--与模型理论的答案很好地吻合。后者是在解释的概念方面，它本质上是从一种语言（与一系列表征相关联）到另一种语言的翻译。这已经被证明等价于一种特别好的嵌入，但不知道如何缩小这与嵌入类型之间的差距，这是上述问题的标准代数答案。缩小这一差距是该项目的目标之一。除此之外，该项目的一个目标是对现有的相当广泛的复杂性代数分类进行实质性的改进，将复杂性分类分为驯服和野性（进一步改进驯服）。该项目将结合联合收割机非常普遍的方法，其中一些方法受到代数几何和抽象范畴理论的启发，并对特定代数的表示进行非常具体的研究，其中完全明确的描述是目标。它将借鉴两个发展良好的主题;模块的模型理论和有限维代数的表示理论，并将使用同调代数和加法函子范畴理论的技术。鉴于所需投入的范围之广以及项目目标的数量和性质，将由两名PDRA与PI合作，分享其专业知识，组成研究小组。