Free Resolutions and Representation Theory

本研究计画涉及代数的两个分支：交换代数与箭图的表示。交换代数研究由多项式方程定义的集合。提案的这一部分涉及研究定义某些几何特征集的多项式方程。 一个向量的表示是一种将向量数据与某个有向图的顶点相关联的方法。图的边可以看作是这些数据之间的关系。 抽象代数允许系统地研究这样的对象。 这项研究的结果可能会导致更好的算法处理线性代数问题，这在其他数学领域以及在其他科学中有许多应用。 研究人员的一些已发表的研究导致了这样的算法。PI将邀请研究生参与他的研究。这个项目由几个相互关联的部分组成。第一部分是研究有限自由分解的一般环的结构。研究人员建议发展他的发现之间的联系通用环的决议长度为3和卡茨穆迪李代数有关的图T（p，q，r）。特别地，一般环是诺特环当且仅当T（p，q，r）是Dynkin图。 PI建议努力明确描述通用环的生成元，当它们是诺特环时。他还建议继续推广这种方法，以完善复杂的长度3，决议Gorenstein理想的余维4和决议的更高的长度。在一个相关的项目中，PI建议研究Buchsbaum-Rim链接的作用，它反映了图T（p，q，r）的对称性。第二部分是关于局部上同调的计算。PI计划发展他的计算局部上同调的多项式环一般矩阵支持的行列式理想。他计划找到类似的结果Pfweans的反对称矩阵和未成年人的对称矩阵。然后，他计划开发的技术，用于能够找到自由决议的不可约等变理想在所有三种情况下。 第三部分涉及非交换代数。研究者建议进一步研究具有潜力的颤抖。他计划追求猜想的大卫贝伦斯坦关于全球层面的雅可比代数的一个潜在的。他还计划研究一个雅可比代数和相应的完备雅可比代数的性质之间的关系。PI建议研究I型Vinberg表示中轨道闭包的非交换解决方案，特别是对于反对称矩阵和对称矩阵的子矩阵的Pfrons。最后，他建议研究几何的组成部分，代表空间的颤抖的关系。最有趣的问题是试图决定一个给定的代数的表示类型到其表示空间的几何。在这种情况下，PI对这些组分的正态性以及它们的MF和DO性质感兴趣。

This research project is related to two branches of algebra: commutative algebra and representations of quivers. Commutative algebra studies sets defined by polynomial equations. This part of the proposal involves studying polynomial equations defining certain sets characterized geometrically. A representation of a quiver is a way to associate vector data to the vertices of some oriented graph. The edges of a graph can be viewed as relations between these data. Abstract algebra allows the study of such objects systematically. The results of this research might lead to better algorithms dealing with linear algebra problems, which have many applications in other areas of mathematics as well as in other sciences. Some of the published research of the investigator led to such algorithms. The PI will involve graduate students in his research.This project consists of several interrelated parts. The first part is to study the structure of generic rings for finite free resolutions. The investigator proposes to develop his discovery of a link between the generic ring for resolutions of length 3 and Kac-Moody Lie algebras related to graphs T(p,q,r). In particular the generic ring is Noetherian if and only if T(p,q,r) is a Dynkin diagram. The PI proposes to work towards an explicit description of the generators of the generic ring in the case when they are Noetherian. He also proposes to continue to generalize this approach to perfect complexes of length 3, for resolutions of Gorenstein ideals of codimension 4 and for resolutions of higher length. In a related project the PI proposes to study the role of Buchsbaum-Rim linkage which reflects the symmetry of the graphs T(p,q,r). The second part is related to calculating local cohomology. The PI plans to develop his calculation of local cohomology of the polynomial ring on generic matrix supported in the determinantal ideal. He plans to find similar results for Pfaffians of skew symmetric matrices and minors of symmetric matrices. He then plans to develop the techniques used to be able to find the free resolutions of irreducible equivariant ideals in all three cases. The third part involves noncommutative algebra. The investigator proposes to further study quivers with potential. He plans to pursue the conjecture of David Berenstein concerning global dimension of the Jacobian algebra of a quiver with potential. He also plans to study the relation between properties of a Jacobian algebra and the corresponding completed Jacobian algebra. The PI proposes to study the noncommutative resolutions of orbit closures in Vinberg representations of type I, in particular for Pfaffians of skew symmetric matrices and minors of symmetric matrices. Finally, he proposes to study the geometry of the components of representation spaces for quivers with relations. The most interesting problem is trying to decide the representation type of a given algebra to the geometry of its representation spaces. In this context the PI is interested in normality of these components as well as in their MF and DO properties.