Representations and Support Theory for Bosonized Quantum Complete Intersections

Abstract

Support theories are frequently used by representation theorists when trying to understand module categories with complicated structure. We associate to an algebra A a variety where the topological structure is determined by the support of modules over A. Support comes in many flavors, from the easily described (but computationally difficult) to more abstract notions of support that are more amenable to computation. In this work, we are interested in understanding support theories on a class of algebras called bosonized quantum complete intersections (BQCIs), which are noncommutative k-algebras with some useful parallels to commutative algebra. Drawing upon prior notions of matrix factorizations and hypersurface support, we develop a notion of rank support that reduces the computation of homological support to essentially a linear algebra problem. Along the way, we develop computational tools necessary to explicitly compute support for these algebras and provide some insight into the theoretical results that power our observations.