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. 1 The Connes-Moscovici Hopf algebra HCM The Connes-Moscovici Hopf algebras originally appeared in [5], arising from a longstanding internal problem of noncommutative geometry, the computation of the index of transversally elliptic operators on foliations. This family of Hopf algebras (one for each positive integer) was found to reduce transverse geometry to a universal geometry of affine nature, and provided the initial impetus for the development of Hopf-cyclic cohomology. The cyclic cohomology of these Hopf algebras was shown by Connes and Moscovici to serve as an organizing principle for the computation of the cocycles in their local index formula [4]. They are also closely related to the Connes-Kreimer Hopf algebras of rooted trees arising from renormalization of quantum field theories [2]. More recently these Hopf algebras have appeared in number theory, in the context of operations on spaces of modular forms and modular Hecke algebras [6] and spaces of Q-lattices [3]. They appear to play a near-ubiquitous role as symmetries in noncommutative geometry. There is also an algebraic approach to diffeomorphism groups [14], which we link to Connes and Moscovici’s work. In this paper we focus on the simplest example, the codimension one Connes-Moscovici Hopf algebra. We work with a right-handed version of this algebra, which we denote HCM. The algebras in [5] were implicitly defined over R or C, but throughout this paper we will work over an arbitrary field k of characteristic zero. Definition 1.1 We define HCM to be the Hopf algebra (over k) generated by elements X, Y , δn (n ≥ 1), with [Y,X] = X, [X, δn] = δn+1, [Y, δn] = nδn, [δm, δn] = 0 ∀ m,n ∆(X) = X ⊗ 1 + 1⊗X + Y ⊗ δ1, ∆(Y ) = Y ⊗ 1 + 1⊗ Y, ∆(δ1) = δ1 ⊗ 1 + 1⊗ δ1 Supported by an EPSRC postdoctoral fellowship †Supported during completion of the work by the Perimeter Institute, Waterloo, Ontario.