Free Resolutions and Representation Theory

本研究项目涉及到代数的两个分支：交换代数和箭图的表示。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刻画的集合的多项式方程。箭图的表示是将向量数据与某个有向图的顶点相关联的一种方式。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来处理线性代数问题，这在其他数学领域以及其他科学中都有很多应用。这位研究人员发表的一些研究导致了这样的算法。PI将让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。第一部分研究有限自由分解的一般环的结构。这位研究者建议进一步发展他的发现，即长度为3的分解的通用环与与图T(p，q，r)相关的Kac-Moody李代数之间的联系。特别地，一般环是Noether环当且仅当T(p，q，r)是动态图。PI建议在一般环的生成元是Noether的情况下，对它们进行明确的描述。他还建议继续将这种方法推广到长度为3的完美复形、余维为4的Gorenstein理想的解和长度更高的解。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计划发展行列式理想支持的一般矩阵上多项式环的局部上同调的计算。他计划为斜对称矩阵的Pfaffian和对称矩阵的子式找到类似的结果。然后，他计划开发能够在所有三种情况下找到不可约等变理想的自由解的技术。第三部分涉及非对易代数。这位研究人员建议进一步研究具有潜力的颤动。他计划继续大卫·贝伦斯坦关于有势箭图的雅可比代数的整体维度的猜想。他还计划研究雅可比代数的性质与相应的完备雅可比代数之间的关系。PI建议研究第I类Vinberg表示中轨道闭包的非对易分解，特别是对于斜对称矩阵的Pfaffian和对称矩阵的子式。最后，他建议研究带关系箭图的表示空间分支的几何。最有趣的问题是试图确定给定代数的表示空间的几何表示类型。在这种情况下，PI对这些组件的正规性以及它们的Mf和DO属性感兴趣。

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.