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 and Statistics, University of Glasgow, 15 University Gardens, Glasgow G12 8QW, United Kingdom
Robert Yuncken
Université Clermont Auvergne, Université Blaise Pascal, Laboratoire de Mathématiques, BP 10448, 63000 Clermont-Ferrand; and CNRS, UMR 6620, LM 63178, Aubiere, France

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

Download PDF

This article is published open access.

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 $S U_{q} (3)$ .

Cite this article

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