Brauer-Picard groups and pointed braided tensor categories

Tensor categories are ubiquitous in areas of mathematics involving algebraic structures. They appear, also, in other fields, such as mathematical physics (conformal field theory) and theoretical computer science (quantum computation). The study of tensor categories is, thus, a useful undertaking. Two classes of tensor categories arise naturally in this study. One consists of group-graded extensions and another of pointed tensor categories. Understanding the former involves knowledge of the Brauer-Picard group of a tensor category, while results about pointed Hopf algebras provide insights into the structure of the latter. This work consists of two main parts. In the first one we compute the Brauer-Picard group of a class of symmetric non-semisimple finite tensor categories by studying a canonical action on a vector space. In the second one we use results from the theory of Hopf algebras to prove an equivalence between the groupoid of pointed braided finite tensor categories admitting a fiber functor and a groupoid of metric quadruples