Hypoelliptic operators, Hopf algebras and cyclic cohomology (Q2708720)

This is a survey paper on the author's computation (with H. Moscovici) of the index of transversally elliptic operator on foliations [\textit{A. Connes} and \textit{H. Moscovici}, Commun. Math. Phys. 198, No. 1, 199-246 (1998; Zbl 0940.58005)]. The local index formula for transverse elliptic operators on foliations has been given in [\textit{A. Connes} and \textit{H. Moscovici}, Geom. Funct. Anal. 5, No. 2, 174-243 (1995; Zbl 0960.46048)]. The remainding problem is the explicit computation of all the terms involved in the cocycle \(\text{ch}^*(D)\). The author says, even in the codimension 1 case, this computation takes around one hundred pages without using some bypass. The author overcomes this difficulty by introducing a specific Hopf algebra \({\mathcal H}(n)\) for each natural number \(n\) (corresponding to the codimension of the foliation) such that the index computation takes place the cyclic cohomology of \({\mathcal H}(n)\), and compute this cyclic cohomology explicitly as Gelfand Fuchs cohomology. The author also emphasizes that the used techniques of noncommutative geometry allows to consider general foliated manifolds without assumptions such as existence of a holonomy invariant transverse metric as in Riemannian foliation. So there is no restriction on the holonomy pseudogroup of the foliation.NEWLINENEWLINENEWLINEThe computation is done by using techniques of noncommutative geometry. In section I, infinitesimal calculus in noncommutative geometry (spectre triple \(({\mathcal A},{\mathcal H},D))\) is reviewed. The author does not use the Schatten ideal in the definition of noncommutative (quantum) object of classically infinitesimal of order \(\alpha\). It is defined as a compact operator with eigenvalues \(\mu_n\) satisfying \(\mu_n=O(n^{-\alpha}) \), \(n\to\infty\). Noncommutative version of derivation is the commutator \([F,f]\), where \(F\) is the involution of \({\mathcal H}\) and the integral is replaced by the Dixmier trace. Definition and meanings of the Dixmier trace are precisely exposed. In section II, under the regularity hypothesis on \(({\mathcal A},{\mathcal H},D)\), local index theorem is presented. Then taking \({\mathcal A}\) as the crossed product of a manifold by a group of diffeomorphisms, the transverse fundamental class is constructed. It is shown for any \(\mathcal A\) constructed from a smooth manifold \(M\), there exists a Morita equivalent \({\mathcal A}'\) which is constructed from a flat manifold \(M'\). This result provide a noncommutative counterpart of the curvature. Then the fundamental class is constructed as the hypoelliptic signature operator. In section IV, the Hopf algebra \({\mathcal H}(n)\) is defined. The relation of \({\mathcal H}(1)\) and the standard calculus of Taylor expansion is also explained. To compute cyclic cohomology of \({\mathcal H}(n)\), the periodic cyclic cohomology of \(HC^*({\mathcal H})\) for \({\mathcal H}={\mathcal U}(G)\), the envelopping algebra of a Lie algebra \(G\), or \({\mathcal H}={\mathcal U}(G)_*\), the dual of \({\mathcal U}(G)\), are treated in section V. By using these results, it is shown the cyclic cohomology of \({\mathcal H}(n)\) can be interpreted as the Gelfand-Fuchs cohomology of \(\mathbb{R}^n\) (expositions are given assuming \(n=1)\). After these preliminaries, the index theorem (existence of the universal polynomial \(L_n\in H^*(WSO_n)\), \(WSO_n\) is the Weil algebra of \(SO_n\), which gives the Chern character of the fundamental class \(Q)\) is shown in section VI. Remarks on the explicit form of \(L_n\) are also given.NEWLINENEWLINEFor the entire collection see [Zbl 0949.00018].