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

在20世纪80年代，模的模型理论在有限维代数的表示理论中的意外应用被发现，从那时起，这些领域之间存在着进一步的，有时是深入的相互作用。模型理论使用数理逻辑的思想和结果来研究关于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，通常会给数学的其他部分带来新的见解。模型理论几乎总是大量使用数理逻辑的紧致性定理，为此，一个人需要在一个有空间进行无限构造的环境中工作。在有限维代数的表示理论的特定背景下，其中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一些无限维表示上，即使我们最终的应用回到有限维表示的背景下。这一特定的项目将加深模型理论和表现理论之间的互动。这个项目的基本问题是“一个特定的表示集合有多复杂？”；已经研究了回答这个问题的各种方法，主要目的是证明最标准的代数答案--根据一个集合在另一个集合中的某些嵌入--与模型理论的答案很好地吻合。后者是关于解释的概念，它本质上是从一种语言(与一组表示相关联)到另一种语言的翻译。这已经被证明相当于一种特别好的嵌入，但还不知道如何缩小这种嵌入与嵌入之间的差距，后者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，该项目的目标是对现有的相当广泛的复杂类别的代数分类进行实质性的改进，将其分为TAME和WARD(进一步改进TAME)。该项目将结合非常一般的方法，其中一些方法受到代数几何和抽象范畴理论的启发，与非常具体的研究特定代数的表示，其中完全明确的描述是目的。它将借鉴两门成熟的学科：模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性函子范畴理论的技巧。鉴于所需投入广泛，并考虑到该项目目标的数量和性质，两个PDRA将与私营部门合作，共同组成研究小组。

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.