Bicrossproduct approach to the Connes–Moscovici Hopf algebra

Abstract

We give a rigorous proof that the (codimension one) Connes–Moscovici Hopf algebra H CM is isomorphic to a bicrossproduct Hopf algebra linked to a group factorisation of the diffeomorphism group Diff + ( R ) . We construct a second bicrossproduct U CM equipped with a nondegenerate dual pairing with H CM . We give a natural quotient Hopf algebra k λ [ Heis ] of H CM and Hopf subalgebra U λ ( heis ) of U CM 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 k λ [ Heis ] by studying first order differential calculi of small dimension.