Bicrossproduct approach to the Connes–Moscovici Hopf algebra

We give a rigorous proof that the (codimension one) Connes–Moscovici Hopf algebra HCM is isomorphic to a bicrossproduct Hopf algebra linked to a group factorisation of the diffeomorphism group Diff+(R). We construct a second bicrossproduct UCM equipped with a nondegenerate dual pairing with HCM. We give a natural quotient Hopf algebra kλ[Heis] of HCM and Hopf subalgebra Uλ(heis) of UCM which again are in duality. All these Hopf algebras arise as deformations of commutative or cocommutative Hopf algebras that we describe in each case. Finally we develop the noncommutative differential geometry of kλ[Heis] by studying first order differential calculi of small dimension.