C*-algèbres de Hopf et théorie de Kasparov équivariante

Baaj, Saad; Skandalis, Georges

doi:10.1007/bf00538428

Cited by 132 publications

(256 citation statements)

References 16 publications

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“…The same holds for the equivariant Kasparov theory KK S with respect to any C * -bialgebra S; this theory was defined by Baaj and Skandalis in [2].…”

Section: What Is a Triangulated Category?mentioning

confidence: 82%

“…This requires a quantum group analogue of the ideals VC F in KK G of Example 2.7. If G is a locally compact quantum group, then Saad Baaj and Georges Skandalis construct a G-equivariant Kasparov category KK G in [2]. There is a forgetful functor Res H G : KK G → KK H for each closed quantum subgroup H ⊆ G. Therefore, a family F of closed quantum subgroups yields an ideal VC F in KK G as in Example 2.7.…”

Section: Introductionmentioning

confidence: 99%

“…Any crossed product G ⋉ A carries a canonical coaction of G, that is, a coaction of the discrete quantum group C * (G). Baaj-Skandalis duality asserts that this yields an equivalence of categories KK G ∼ = KK C * (G) (see [2]). We get back the crossed product functor KK G → KK by composing this equivalence with the restriction functor KK C * (G) → KK for the trivial quantum subgroup.…”

Section: Introductionmentioning

confidence: 99%

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Homological algebra in bivariant K-theory and other triangulated categories. I

Meyer¹,

Nest²

2010

Triangulated Categories

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Abstract. Bivariant (equivariant) K-theory is the standard setting for noncommutative topology. We may carry over various techniques from homotopy theory and homological algebra to this setting. Here we do this for some basic notions from homological algebra: phantom maps, exact chain complexes, projective resolutions, and derived functors. We introduce these notions and apply them to examples from bivariant K-theory.An important observation of Beligiannis is that we can approximate our category by an Abelian category in a canonical way, such that our homological concepts reduce to the corresponding ones in this Abelian category. We compute this Abelian approximation in several interesting examples, where it turns out to be very concrete and tractable.The derived functors comprise the second page of a spectral sequence that, in favourable cases, converges towards Kasparov groups and other interesting objects. This mechanism is the common basis for many different spectral sequences. Here we only discuss the very simple 1-dimensional case, where the spectral sequences reduce to short exact sequences.

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“…The same holds for the equivariant Kasparov theory KK S with respect to any C * -bialgebra S; this theory was defined by Baaj and Skandalis in [2].…”

Section: What Is a Triangulated Category?mentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Homological algebra in bivariant K-theory and other triangulated categories. I

Meyer¹,

Nest²

2010

Triangulated Categories

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“…We also introduce a dual theoryKK G for G-graded algebras and establish a duality result similar to that proved by Baaj and Skandalis in [1].…”

mentioning

confidence: 65%

Equivariant algebraic kk-theory and adjointness theorems

Ellis

2014

Journal of Algebra

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“…Note that KÂ 1 (B) as defined in [BS1] is identical to K 1 of the Banach category of α-equivariant f.g. right Hilbert B-modules [K]. To get the pairing with cyclic cohomology, one needs a continuous version of equivariant cyclic cohomology, but we shall not discuss this any further here.…”

Section: Julg's Theoremmentioning

confidence: 99%

Hopf Algebra Equivariant Cyclic Cohomology, K-theory and Index Formulas

Neshveyev¹,

Tuset²

2004

K-Theory

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For an algebra B with an action of a Hopf algebra H we establish the pairing between equivariant cyclic cohomology and equivariant K-theory for B. We then extend this formalism to compact quantum group actions and show that equivariant cyclic cohomology is a target space for the equivariant Chern character of equivariant summable Fredholm modules. We prove an analogue of Julg's theorem relating equivariant K-theory to ordinary K-theory of the C * -algebra crossed product, and characterize equivariant vector bundles on quantum homogeneous spaces.

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C*-algèbres de Hopf et théorie de Kasparov équivariante

Cited by 132 publications

References 16 publications

Homological algebra in bivariant K-theory and other triangulated categories. I

Homological algebra in bivariant K-theory and other triangulated categories. I

Equivariant algebraic kk-theory and adjointness theorems

Hopf Algebra Equivariant Cyclic Cohomology, K-theory and Index Formulas

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