Duality involutions, representations, and geometry

Abstract

In this thesis we give an exposition of the theory of duality involutions, and within this context we present the results of two different research projects.
Loosely speaking, a duality involution on a category C is a self-adjoint contravariant endofunctor of C. A prototypical example of such is the usual notion of duality for the finite dimensional vector spaces. We also consider duality involutions for bicategories, as defined by Shulman.
The first project concerns classification problems in symplectic linear algebra. In this part, we discuss results regarding the symplectic group in its Lie algera, as well as work on systems of subspaces in symplectic vector spaces. In the language of duality involutions, symplectic structures are encoded as fixed point structures.
The second project is about the Morita bicategory of finite-dimensional k-algebras and bimodules, and the representation pseudofunctor which sends an algebra to its category of representations. We show that this representation pseudofunctor is equivariant in a natural manner with respect to duality involutions which we define on its source and target.