Deformations and Representation Theory

变形与表示理论

• 批准号: 0139737
• 负责人: Frauke Bleher
• 金额: $ 5.53万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0139737&HistoricalAwards=false
• 关键词: Deformations; Representation; Theory

The investigator applies tools from group representationtheory to study deformations of representations of profinitegroups and of complexes of modules for such groups. Theinvestigator's main theme is to examine problems andapplications related to generalizations of the deformationtheory of group representations developed by B. Mazur. Thegeneralizations are extensions of this theory to (1)deformations of objects in derived categories, and (2)deformations of representations whose images lie in variouslinear algebraic groups, such as orthogonal groups. Theinvestigator studies applications of these theories to certainembedding problems. The second theme is to determine for finitegroups the ring structure of the universal deformation ring ofan irreducible modular representation, using only the localstructure of the finite group. The investigator also works onone other project which concerns moduli spaces of representationsof finite dimensional algebras. The goal is to use the richalgebro-geometric structure of moduli spaces to obtain a betterunderstanding of the geometric behavior of algebra automorphisms.Groups are abstract mathematical objects by which one may encodeand study symmetry, for example in chemical molecules, crystals, networks,or abstract mathematical structures. Representations of groups providea way to extract information about the internal structures of a group.Roughly speaking, representations can be thought of as "linearizedsnapshots" of the group which are given by explicitly describedmatrices. In this project, the investigator studies deformations ofrepresentations. The deformations of a given representation form afamily of representations which are associated to this representationin a certain way. In case there is a single deformation which can beused to describe all deformations, one talks about a universaldeformation. Universal deformations provide universal constructions whichcan be used to solve certain problems all at once, which otherwise wouldhave to be solved in a case-by-case fashion. It is the goal of thisproject to study certain generalizations of deformations ofrepresentations which in turn should lead to more powerful applicationsin different areas.

研究者应用群表示理论的工具来研究有限群的表示以及此类群的模复合体的表示变形。研究者的主题是研究与 B. Mazur 提出的群表示变形理论的推广相关的问题和应用。这些概括是该理论对（1）派生类别中的对象的变形，以及（2）其图像位于各种线性代数群（例如正交群）中的表示的变形的扩展。研究者研究这些理论在某些嵌入问题上的应用。第二个主题是仅使用有限群的局部结构来确定有限群的不可约模表示的通用变形环的环结构。研究人员还从事另一个项目，该项目涉及有限维代数表示的模空间。目标是利用模空间丰富的代数几何结构来更好地理解代数自同构的几何行为。群是抽象的数学对象，人们可以通过它来编码和研究对称性，例如在化学分子、晶体、网络或抽象中数学结构。群的表示提供了一种提取有关群内部结构的信息的方法。粗略地说，表示可以被认为是由明确描述的矩阵给出的群的“线性化快照”。在这个项目中，研究者研究表征的变形。给定表示的变形形成一系列表示，这些表示以某种方式与该表示相关联。如果有一个单一的变形可以用来描述所有的变形，我们就称之为通用变形。通用变形提供了通用结构，可用于一次性解决某些问题，否则这些问题必须以具体情况具体分析的方式解决。该项目的目标是研究表示变形的某些概括，这反过来又会在不同领域带来更强大的应用。