This dissertation aims to contribute to the cyclic cohomology theory of Hopf algebras as defined by Connes and Moscovici. To date, the most important Hopf algebras whose periodic cyclic cohomology group is computed are the members of the family of Connes-Moscovici Hopf algebras of transverse geometry. We will focus our attention to 𝓗(1), the first member of this family of Hopf algebras. While the periodic theory of 𝓗(1) is well understood and can be obtained from the Gelfand-Fuks cohomology of Lie algebras of vector fields, the non-periodic theory turns out to be much more complicated. One of the main results of this dissertation is the computation of the first cyclic cohomology group of 𝓗(1). The dissertation also contains calculations pertaining to the second cyclic cohomology group of 𝓗(1), but these are only partial results.
The next main result is the computation of the periodic cyclic cohomology group of some Hopf algebras appearing as (continuous) duals to some infinite dimensional pro-nilpotent Lie algebras. Among these Hopf algebras is 𝓗δ, the maximal commutative sub-algebra of 𝓗(1).
The Connes-Moscovici cyclic theory of Hopf algebras leaves room for incorporating coefficients as discussed by Khalkhali and his collaborators. Chapter 5 of the present dissertation explains the connection of this theory and the theory of bicovariant bimodules.