Monoidal Triangular Categories: Representation Theory, Cohomology, and Geometry

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Representation theory has emerged as a central area of modern mathematics with connections to combinatorics, algebraic geometry, topology and number theory. In addition, it has many applications to chemistry and physics. Representations are mappings of complicated algebraic objects (such as groups, rings, Lie algebras, and Lie superalgebras) to an array of numbers (matrices). These realizations by matrices encode important data that can yield deep insights into these complicated algebraic objects. In recent years, a useful approach has been to understand the entire collection of representations of an object. Representations for a certain algebraic object often form a tensor triangulated category. Techniques from homological algebra can be used to build bridges between tensor triangulated categories and geometric objects. Uncovering this hidden geometry often leads to new insights about the algebraic object and its representations. This project includes research and training opportunities for graduate students and postdoctoral fellows in algebra and representation theory. In this project the PI will develop new methods to understand tensor structures in monoidal tensor categories. This will entail the development of monoidal triangular geometry and explicit computations of Balmer spectra. The PI will also introduce new geometric and topological methods to provide concrete calculations of cohomology for algebraic/finite groups, Lie superalgebras, quantum groups, and Frobenius kernels. The development of the cohomology theory was an important tool to resolving 30 year old problems that deal with tilting modules in connection with filtrations of representations of algebraic groups. The PI will study representations of Lie superalgebras via the application of super geometry to construct representations and to compute higher sheaf cohomology. The PI will develop a new Lie theory that entails the use of detecting and parabolic subalgebras, in addition to, a nilpotent cone to formulate a geometric setting in order to compute characters for irreducible representations.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.