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.
