Equivariant Fredholm modules for the full quantum flag manifold of $\mathrm{SU}_q(3)$

Christian Voigt; Robert Yuncken

doi:10.4171/dm/495

JournalsdmVol. 20pp. 433–490

Equivariant Fredholm modules for the full quantum flag manifold of $SU_{q} (3)$

Christian Voigt
School of Mathematics (1) Université Clermont Auvergne and Statistics Université Blaise Pascal University of Glasgow Laboratoire de Mathématiques 15 University Gardens BP 10448 Glasgow G12 8QW F-63000 Clermont-Ferrand United Kingdom France (2) CNRS, UMR 6620, LM F-63178 Aubiere France Robert.Yuncken
Robert Yuncken
School of Mathematics (1) Université Clermont Auvergne and Statistics Université Blaise Pascal University of Glasgow Laboratoire de Mathématiques 15 University Gardens BP 10448 Glasgow G12 8QW F-63000 Clermont-Ferrand United Kingdom France (2) CNRS, UMR 6620, LM F-63178 Aubiere France Robert.Yuncken

$Equivariant Fredholm modules for the full quantum flag manifold of $\mathrm{SU}_q(3)$ cover$

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Abstract

We introduce $C^{*}$ -algebras associated to the foliation structure of a quantum flag manifold. We use these to construct $S L_{q} (3, C)$ -equivariant Fredholm modules for the full quantum flag manifold $X_{q} = S U_{q} (3) / T$ of $S U_{q} (3)$ , based on an analytical version of the Bernstein-Gelfand-Gelfand complex. As a consequence we deduce that the flag manifold $X_{q}$ satisfies Poincaré duality in equivariant $KK$ -theory. Moreover, we show that the Baum-Connes conjecture with trivial coefficients holds for the discrete quantum group dual to $S U_{q} (3)$ .

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Christian Voigt, Robert Yuncken, Equivariant Fredholm modules for the full quantum flag manifold of $SU_{q} (3)$ . Doc. Math. 20 (2015), pp. 433–490

DOI 10.4171/DM/495