On weak quantum symmetry and Frobenius-Perron theory

Abstract

Chapter 1 describes the history and notion of a fusion category, as well as some natural settings where fusion categories appear. We discuss the fact that every fusion category is equivalent to the representation category of a weak Hopf algebra, and the importance of an invariant called the Frobenius-Perron (FP) dimension. Chapter 2 explores algebraic structures in corepresentation categories over weak bialgebras, an investigation which was inspired by studying the symmetries of path algebras of quivers. We prove that for a weak bialgebra H, the formulaic notions of H-comodule algebra, H-comodule coalgebra, and H-comodule Frobenius algebra are equivalent via an isomorphism of categories to the categorical notions of algebra, coalgebra, and Frobenius algebra in the monoidal category of right H-comodules. We also provide a monoidal functor which constructs "weak quantum symmetries'' from "quantum symmetries.'' Chapter 3 takes a slightly different perspective, and seeks to generalize the categories where we can apply the notion of FP dimension. We generalize the definition of the FP dimension of an object in a fusion category to any k-linear category with a chosen endofunctor. In particular, we define the FP dimension for abelian and derived categories, as well as provide examples and applications. This work has particular significance because it defines a new invariant for derived categories, and not many derived invariants are known. Chapter 4 takes this one step further by calculating the Frobenius-Perron dimension for the abelian categories associated to a special generalization of A-D-E quiver algebras, and introducing a family of a abelian categories which produce arbitrarily large FP dimension. Along the way, we introduce a result that can be applied to calculate the FP dimension of a radical square zero bound quiver algebra.