Free Resolutions and Representation Theory

University Of Connecticut, Storrs CT

Investigators

Abstract

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.