Representation Theory of Finite Dimensional Algebras

研究者的主题是探索野生有限维代数的表示理论，通过分析特定类别的表示和它们之间的同态与一系列的方法，从几何通过组合到同调。 这些类大致可以分为以下几类：（1）具有固定的基本层序列的模类，更一般地说，具有固定的顶点和固定的向量空间维数的模类。(2)模的类，它们在全范畴内形成反变有限或协变有限的子范畴。(3)一类自同态环上具有有限性条件的无限直和类（主要是一般模和广义生成模）. 一个相关的调查旨在几何的有限复合物的模和衍生的范畴作为一个替代设置的研究地图之间的表示。这些线之间有方法论上的相互联系，以及在获得更好地理解具有野生表示类型的代数的总体策略方面的强联系。从更广泛的角度来看，这个项目应该在以下框架中被看到：我们的大多数宇宙物理模型都被放置在有限维向量空间中。通常，这样的空间带有额外的结构，例如Lie或结合代数结构。结合有限维代数是研究者的研究对象。理解它们的一种方法是探索它们的表征，可以认为它们是一组正方形矩阵中的“线性照片”，粗粒度和细粒度的照片都包含有价值的信息。所考虑的代数的表示类型被称为“驯服”，如果任何固定矩阵大小的表示可以被细分为许多“可管理的”系列（这些系列本身可能是无限的），否则它被称为“野生"。野生情况下是一个主要遇到的自然，这是调查人员目前正在处理的情况。她的想法是彻底理解野生型代数表示的完整类的重要子类，从而建立一个相互关联的马赛克分析领域。人们的注意力集中在选择与相邻领域（如代数或量子群理论）的需求以及与已经建立的洞察力相联系的最佳线路上。

The investigator's main theme is to explore the representation theoryof wild finite dimensional algebras by analyzing specific classes ofrepresentations and the homomorphisms among them with an array ofmethods, ranging from geometric through combinatorial tohomological. Roughly, these classes can be grouped under thefollowing headings: (1) Classes of modules having fixed sequence ofradical layers and, more generally, classes of modules with fixed topand fixed vector space dimension. (2) Classes of modules formingcontravariantly finite or covariantly finite subcategories within thefull category of finitely generated modules. (3) Classes of infinitedirect sums (mainly of generic and finitely generated modules)enjoying finiteness conditions over their endomorphism rings. Arelated investigation aims at the geometry of finite complexes ofmodules and derived categories as an alternate setting for the studyof maps among representations. There are methodologicalinterconnections among these lines, as well as strong links in termsof the overall strategy of gaining a better understanding of algebrasthat have wild representation type. Under a wider angle, the project should be seen in the followingframework: The majority of our physical models for the universe areplaced within finite dimensional vector spaces. Frequently, suchspaces carry additional structure, such as a Lie or associativealgebra structure, for example. Associative finite dimensionalalgebras are the objects of the investigator's research. One approachtowards understanding them is to explore their representations, whichcan be thought of as 'linearized photographs' within sets of squarematrices, both coarse-grained and fine-grained pictures containingvaluable information. The representation type of the consideredalgebra is called 'tame' if the representations of any fixed matrixsize can be subdivided into finitely many 'manageable' series (theseseries themselves may be infinite), otherwise it is called 'wild'.The wild case being the one predominantly encountered in mathematicalnature, this is the case which the investigator is presentlytackling. Her idea is to thoroughly understand important subclassesof the full class of representations of algebras of wild type, and tothus establish an interlinked mosaic of analyzed territory. Attentionis being paid to choosing lines that optimally connect with the needsof adjacent fields (such as the theory of algebraic or quantum groups)and with the body of insights already established.