## Hopf algebras, cyclic cohomology and the transverse index theorem (1998)

Venue: | Comm. Math. Phys |

Citations: | 141 - 18 self |

### BibTeX

@ARTICLE{Connes98hopfalgebras,,

author = {A. Connes and H. Moscovici},

title = {Hopf algebras, cyclic cohomology and the transverse index theorem},

journal = {Comm. Math. Phys},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we present the solution of a longstanding internal problem of noncommutative geometry, namely the computation of the index of transversally elliptic operators on foliations.

### Citations

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Foundations of Quantum Group Theory
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Citation Context ...e get, (11) (ϕf)(s, t) = f(s + log ψ ′ (t), ψ(t)) , ψ = ϕ −1 . V. Hopf algebra H(G) associated to a matched pair of subgroups In this section we recall a basic construction of Hopf algebras ([K],[BS],=-=[M]-=-). We let G be a finite group, G1, G2 be subgroups of G such that, (1) G = G1 G2 , G1 ∩ G2 = 1 , i.e. we assume that any g ∈ G admits a unique decomposition as (2) g = k a , k ∈ G1 , a ∈ G2 . Since G1... |

488 |
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Citation Context ... of course requires to understand the Hochschild cohomology of the algebra H∗ with coefficients in the module C given by the augmentation on H∗. In order to do this we shall use the abstract version (=-=[C-E]-=- Theorem 6.1 p. 349) of the Hochschild-Serre spectral sequence [Ho-Se]. A subalgebra A1 ⊂ A0 of a augmented algebra A0 is called normal iff the right ideal J generated in A0 by the Kerε (of the augmen... |

169 | The local index formula in non commutative geometry - Connes, Moscovici - 1995 |

146 |
Connes, “Noncommutative geometry
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Citation Context ...mutative spaces and already exhibit most of the features of the general theory. The index problem for longitudinal elliptic operators is easy to formulate in the presence of a transverse measure, cf. =-=[Co]-=- [M-S], and in general it leads to the construction (cf. [C-S]) of a natural map from the geometric group to the K-theory of the leaf space, i.e. the K-theory of the associated C ∗ -algebra. This “ass... |

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algebres et geometrie differentielle
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Citation Context ...yclic cohomology of the Hopf algebra H as the Lie algebra cohomology of A, the Lie algebra of formal vector fields. The theory of characteristic classes for actions of H extends the construction (cf. =-=[Co2]-=-) of cyclic cocycles from a Lie algebra of derivations of a C ∗ algebra A, together with an invariant trace τ on A. At the purely algebraic level, given an algebra A and an action of the Hopf algebra ... |

97 | Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29 - Connes, Moscovici - 1990 |

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71 | Cyclic cohomology and the transverse fundamental class of a foliation, in Geometric Methods in Operator Algebras - CONNES - 1983 |

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Citation Context ...lgebra H∗ with coefficients in the module C given by the augmentation on H∗. In order to do this we shall use the abstract version ([C-E] Theorem 6.1 p. 349) of the Hochschild-Serre spectral sequence =-=[Ho-Se]-=-. A subalgebra A1 ⊂ A0 of a augmented algebra A0 is called normal iff the right ideal J generated in A0 by the Kerε (of the augmentation ε of A1) is also a left ideal. When this is so, one lets (112) ... |

51 |
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Citation Context ...ive spaces and already exhibit most of the features of the general theory. The index problem for longitudinal elliptic operators is easy to formulate in the presence of a transverse measure, cf. [Co] =-=[M-S]-=-, and in general it leads to the construction (cf. [C-S]) of a natural map from the geometric group to the K-theory of the leaf space, i.e. the K-theory of the associated C ∗ -algebra. This “assembly ... |

27 |
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Citation Context ...general we get, (11) (ϕf)(s, t) = f(s + log ψ ′ (t), ψ(t)) , ψ = ϕ −1 . V. Hopf algebra H(G) associated to a matched pair of subgroups In this section we recall a basic construction of Hopf algebras (=-=[K]-=-,[BS],[M]). We let G be a finite group, G1, G2 be subgroups of G such that, (1) G = G1 G2 , G1 ∩ G2 = 1 , i.e. we assume that any g ∈ G admits a unique decomposition as (2) g = k a , k ∈ G1 , a ∈ G2 .... |

16 |
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Citation Context ... ∗ (An, SO(n)) θ → HC ∗ (H, SO(n)) 47where the left hand side is the relative Lie algebra cohomology of the Lie algebra of formal vector fields. Let us recall the result of Gelfand-Fuchs (cf. [G-F], =-=[G]-=-) which allows to compute the left hand side of (19). One lets G0 = GL + (n, R) and G0 its Lie algebra, viewed as a subalgebra of G1 ⊂ An. One then views (cf. [G]) the natural projection, (20) π : An ... |

15 |
Fuks, Cohomologies of the Lie algebra of formal vector fields
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Citation Context ... (19) H ∗ (An, SO(n)) θ → HC ∗ (H, SO(n)) 47where the left hand side is the relative Lie algebra cohomology of the Lie algebra of formal vector fields. Let us recall the result of Gelfand-Fuchs (cf. =-=[G-F]-=-, [G]) which allows to compute the left hand side of (19). One lets G0 = GL + (n, R) and G0 its Lie algebra, viewed as a subalgebra of G1 ⊂ An. One then views (cf. [G]) the natural projection, (20) π ... |

7 | Sur les classes caracteristiques des feuilletages - Haefliger - 1971 |

3 |
On the linear independence of certain cohomology classes of BΓq, Studies in algebraic topology
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Citation Context ...computation of ˜ Ln by evaluating the index on the range of the assembly map, (34) µ : K∗,τ(P>⊳Γ E Γ) → K(A1), provided one makes use of the conjectured (but so far only partially verified, cf. [He], =-=[K-T]-=-) injectivity of the natural map, (35) H ∗ d (Γn, R) → H ∗ (BΓn, R) from the smooth cohomology of the Haefliger groupoid Γn to its real cohomology. One then obtains that ˜ Ln is the product of the usu... |

2 |
Heitsch : Independent variation of secondary classes, Ann. of Math. 108
- L
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Citation Context ...d the computation of ˜ Ln by evaluating the index on the range of the assembly map, (34) µ : K∗,τ(P>⊳Γ E Γ) → K(A1), provided one makes use of the conjectured (but so far only partially verified, cf. =-=[He]-=-, [K-T]) injectivity of the natural map, (35) H ∗ d (Γn, R) → H ∗ (BΓn, R) from the smooth cohomology of the Haefliger groupoid Γn to its real cohomology. One then obtains that ˜ Ln is the product of ... |