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Cyclic cohomology and Hopf algebra symmetry. (English) Zbl 0970.58005

Dito, Giuseppe (ed.) et al., Conférence Moshé Flato 1999: Quantization, deformation, and symmetries, Dijon, France, September 5-8, 1999. Volume I. Dordrecht: Kluwer Academic Publishers. Math. Phys. Stud. 21, 121-147 (2000).
The authors say that, in noncommutative geometry, cyclic cohomology and Hopf algebras are playing similar roles to that of de Rham cohomology and the Lie group/algebra actions in classical geometry. In this paper, following the authors’ papers [Geom. Funct. Anal. 5, No. 2, 174–243 (1995; Zbl 0960.46048) and Commun. Math. Phys. 198, No. 1, 199–246 (1998; Zbl 0940.58005)], this argument is explained, mostly taking the authors’ proof of the transverse index theorem as the example.
The outline of the paper is as follows: In section 1, definition and properties of cyclic cohomology are reviewed including the theory of the Chern character. a cyclic theory for Hopf algebras is presented in section 2, by using the notion of modular pair in involution, which is defined as follows: Let \({\mathcal H}\) be a Hopf algebra with the counit \(\varepsilon\) and coproduct \(\Delta\). If a pair \((\delta,\sigma)\), \(\delta\) is an element of \({\mathcal H}^*\) such that \(\delta(ab)= \delta(a)\delta(b)\) and \(\sigma\) is an element of \({\mathcal H}\) such that \(\Delta(\sigma)= \sigma\otimes\sigma\), \(\varepsilon(\sigma)= 1\), satisfies \(\delta(\sigma)= 1\), then this pair is called a modular pair. Let \(S\) be the \(\delta\)-twisted antipode. The pair \((\delta,\sigma)\) is called a modular pair in involution if \(((\sigma)^{-1}\circ\widetilde S)^2= 1\). In section 3, an outline of the proof of the transverse index theorem is reviewed. Let \((V,{\mathcal F})\) be a foliation. Treatise of this foliation by noncommutative geometry begins to construct a corresponding spectral triple \(({\mathcal A},{\mathcal H},D)\), which is constructed assuming \({\mathcal F}\) is transversely oriented.
The index of \(D\) in this spectral triple is expressed by using the Chern character taking values in the cyclic cohomology of a suitable space. To compute this cyclic cohomology, the authors introduce a Hopf algebra \({\mathcal H}(n)\) with its modular pair in involution. It is remarked that the cyclic cohomology of \({\mathcal H}(n)\) with respect to this pair is related to the Gelfand-Fuks cohomology. In practice, \(D\) is replaced by the hypoelliptic signature operator \(Q= D|D|^{-1}\), which belongs to the \(K\)-homology class of \(D\). Its explicit form is also given. Besides the transverse index theorem, some problems in noncommutative geometry such as to give correct generalizations of the notion of Riemannian curvature in noncommutative geometry are also stated in this section. In the final section 4 to illustrate the naturality of the existence of a modular pair in involution, it is shown that quantum groups and their duals do become equipped with an intrinsic modular pair, according to the authors’ paper [Lett. Math. Phys. 48, No. 1, 97–108 (1999; Zbl 0941.16024)].
For the entire collection see [Zbl 0947.00049].

MSC:

58B34 Noncommutative geometry (à la Connes)
46L87 Noncommutative differential geometry
58B32 Geometry of quantum groups
81R60 Noncommutative geometry in quantum theory
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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