Geometric Insights in Noncommutative Algebra

University Of California-Irvine, Irvine CA

Investigators

Abstract

Commutative algebra studies algebraic systems for which the order of multiplication is irrelevant, while noncommutative algebra allows for the possibility that the two products XY and YX may not be the same. Noncommutative algebra historically arose in the algebraic study of symmetry, in linear algebra, and perhaps most profoundly in quantum mechanics. Since the advent of algebraic geometry, commutative algebra has been revolutionized by geometric ideas. In a similar way, the development of noncommutative geometry over the past several decades has stimulated many new ideas and perspectives in noncommutative algebra, but to date it has posed more challenges than it has solved. For instance, many noncommutative algebras are expected to be well-behaved due to the geometric nature of their construction, but we lack the algebraic tools to prove that this is the case. Furthermore, noncommutative geometry largely remains a collection of loosely related frameworks that are not directly compatible with one another, which can make the field especially difficult for newcomers. This project will directly address these fundamental problems by creating new techniques to deduce good algebraic properties for geometrically constructed noncommutative algebras and providing new tools to represent geometric structures corresponding to noncommutative algebras. The project will involve research and training opportunities for graduate students. The first part of this project focuses on homological problems in noncommutative algebraic geometry. The PI and collaborators will use methods of Koszul duality and the representations of finite-dimensional algebras to attack a longstanding conjecture that Artin-Schelter (AS) regular algebras are domains. Related techniques will be used to understand when generalized AS regular algebras that are not necessarily connected, such as graded Calabi-Yau algebras, are prime rings. The second part of this project is focused on spectral problems in noncommutative geometry. The PI will use a variety of approaches to understand noncommutative discrete spaces and utilize them to construct noncommutative spectrum functors for both rings and C*-algebras. Topics to be investigated include structure sheaves for noncommutative spaces, methods to characterize dual coalgebras, discretization of C*-algebras, and a projective representation theory for rings. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.