Applications of Model Theory to Representation Theory

主要研究者打算将模型理论方法应用于李代数和非交换环的表示论，以及pointedabel范畴的K-理论。 该项目的主要目标之一是将关于有限维表示的经典结果推广到伪有限维的情况-代数的表示被称为伪有限维，如果它满足有限维表示的公理。 伪有限维表示理论已经被引入到李代数l（2，k）中，本项目致力于将这一理论推广到有限维半单李代数L以及相应的李群G（L）的情形。 该项目还将集中在结合环R上的模语言中的正本原公式的复形。我们的目的将是建立这个复杂的同源性和K-理论的自由阿贝尔范畴在R.对称性之间的关系起着至关重要的作用，在自然界中发生的物理对象的研究。 一个物体的对称性--比如说，一个水晶或一只蝴蝶--是使物体回到自身的刚性运动。这些对称性形成了一个群结构，称为群。这样，我们就可以把每一个物理对象与它的对称群联系起来。 在表示论中，数学家认为群G是一个抽象的对象，并试图对所有G是对称群的对象进行分类。 传统上，研究集中在对象是有限维向量空间的情况下。在这个项目中，数理逻辑的方法被应用到包括称为伪有限维的无限维情况。 这些都是数学语言在使用中表现力有限无法与有限维区分开来的情况。已经证明，许多这样的物体存在，但要解开其中任何一个特定物体背后的秘密仍然是一个令人困惑的问题。

The principal investigator intends to apply model-theoreticmethods to the representation theory of Lie algebras andnoncommutative rings, as well as to the K-theory of a pointedabelian category. One of the main goals of the project will beto generalize classical results about finite-dimensionalrepresentations to the pseudo-finite dimensional case - arepresentation of an algebra is called pseudo-finite dimensionalif it satisfies the axioms for a finite-dimensionalrepresentation. A theory of pseudo-finite dimensionalrepresentations has already been introduced for the Lie algebrasl (2,k), and the project is devoted to generalizing this theoryfor the case of a finite-dimensional semisimple Lie algebra L ,as well as the corresponding Lie group G(L). The project willalso focus on the complex of positive-primitive formulae in thelanguage of modules over an associative ring R. The aim will beto establish a relationship between the homology of this complexand the K-theory of the free abelian category over R.Symmetry plays an essential role in the study of physical objectsthat occur in Nature. The symmetries of an object - think, forexample, of a crystal or a butterfly - are the rigid motions thatbring the object back onto itself. These symmetries form analgebraic structure, called a group. In this way, we canassociate to every physical object its group of symmetries. Inrepresentation theory, the mathematician considers a group G asan abstract object and attempts to classify all the objects ofwhich G is the symmetry group. Traditionally, research hasfocused on the case where the objects are finite-dimensionalvector spaces. In this project, the methods of mathematical logicare applied to include the infinite-dimensional cases calledpseudo-finite dimensional. These are the situations that thelimited power of expression of the mathematical language in usecannot distinguish from the finite-dimensional. It has been shownthat many such objects exist, but to unlock the secrets behindany particular one of them remains a baffling question.