Infinite iterated crossed products of Hopf Algebras, Drinfeld doubles and planar algebras[HBNI Th90]

Abstract

This thesis deals with the mathematical objects known as planar algebras and their connection with Hopf algebras and their Drinfeld doubles. The motivation for this thesis comes from a series of talks delivered by Prof. Masaki Izumi at IMSc., Chennai, during one of which he asserted that for a Kac algebra subfactor, a related subfactor to its asymptotic inclusion comes from an outer action of its Drinfeld double. This is a folklore result in subfactor theory and in the process of trying to prove this, we noticed a purely algebraic result which also seemed quite interesting and this is one of the main results in the thesis. Given a finite dimensional Hopf algebra H over any field, we associate to it a very natural inclusion A ⊆ B of infinite iterated crossed product algebras.

The thesis is divided into four chapters. Chapters 1 and 2 are devoted to a discussion of preliminary notions, namely, Hopf algebras and planar algebras, while the main content of the thesis is contained in Chapters 3 and 4.