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Hopf algebras, cyclic cohomology and the transverse index theorem. (English) Zbl 0940.58005

The authors give the solution to a longstanding internal problem in noncommutative geometry, i.e., the computation of the index of transversally elliptic operators on foliations. The computation of the local index formula is obtained for these operators by using a specific Hopf algebra \(\mathcal{H}(n)\) associated with the foliation.
In the previous paper [Geom. Funct. Anal. 5, No. 2, 174-243 (1995; Zbl 0960.46048)], the authors gave the spectral triple \((\mathcal{A}, \mathcal{H}, D)\) for foliation and showed that it satisfies the hypothesis of a general abstract index theorem. A local formula for the cyclic cocycle \(\text{ch}_*(D)\), the Chern character, was also given in the residue forms. The general index formula reduces to the local form of the Atiyah-Singer index theorem when \(D\) is say a Dirac operator on a manifold, but the actual explicit computation of all the terms of components \((\varphi)\) involved in \(\text{ch}_*(D)\) is a rather formidable task even, for instance, in a codimension one foliation case. Hence a new organizing principle for the computation is needed for general codimension \(n\).
In this paper, the calculation of the local formula of \(\text{ch}_*(D)\) above is explicitly settled by virtue of the cyclic cohomology theory of the Hopf algebra \(\mathcal{H}(n)\) in the following way.
In the first part, the specific Hopf algebra \(\mathcal{H}(n)\) is introduced which acts on the \(C^*\)-algebra of the transverse frame bundle of codimension \(n\) foliation \((V, F)\), and the cyclic cohomology theory of \(\mathcal{H}(n)\) is developed.
An isomorphism \(\theta\) is constructed between the Lie algebra cohomology of \(\mathcal{A}\) and the cyclic cohomology of the Hopf algebra \(\mathcal{H}(n)\) (Theorem 11 in Section 7). The components \((\varphi)\) of the Chern character give an element of the Hochschild cohomology \(HC^*(\mathcal{A}_1)\) of the subalgebra \(\mathcal{A}_1=\mathcal{A}^{SO(n)}\). In Section 9, a morphism is introduced between the cohomologies \[ HC^*(\mathcal{H}(n), SO(n)) \to HC^*(\mathcal{A}_1) \] called the characteristic map and it is shown that \((\varphi)\) is in the image of the characteristic map (Proposition 3 in Section 9).
By \(\theta\), one gets an isomorphism \[ \theta: \mathcal{H}^*(\mathcal{A}_n, SO(n)) \to HC^*(\mathcal{H}(n), SO(n)) \] where \(\mathcal{H}^*(\mathcal{A}_n, SO(n))\) is the relative Lie algebra cohomology of the Lie algebra of formal vector fields. The Gelfand-Fuchs cohomology theory gives an isomorphism \[ \widetilde {\varphi} : H^*(WSO(n)) \to H^*(\mathcal{A}_n, SO(n)) \] where \(WSO(n)\) is a subcomplex of the elements of \(W_n\) which are basic relative to the action of \(SO(n)\), and \(W_n\) is a certain quotient of the Weil complex. Combining the results above, the authors give the result of the paper (Theorem 5 in Section 9); namely, there exists a universal polynomial \(L_n \in H^*(WSO(n))\) such that \[ \text{ch}_*(Q)=\underline{\theta}(L_n), \] where \(Q=D|D|\) and \(\underline{\theta}=\theta \circ \widetilde {\varphi}\).

MSC:

58B34 Noncommutative geometry (à la Connes)
58J20 Index theory and related fixed-point theorems on manifolds
16W30 Hopf algebras (associative rings and algebras) (MSC2000)

Citations:

Zbl 0960.46048
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