DOCUMENTA MATHEMATICA, Vol. 20 (2015), 433-490

Christian Voigt and Robert Yuncken

Equivariant Fredholm Modules for the Full Quantum Flag Manifold of SU_q(3)

We introduce $C^*$-algebras associated to the foliation structure of a quantum flag manifold. We use these to construct $\{SL}_q(3,\{C})$-equivariant Fredholm modules for the full quantum flag manifold $X_q = \{SU}_q(3)/T$ of $\{SU}_q(3)$, 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 $\{SU}_q(3)$.

2010 Mathematics Subject Classification: Primary 20G42; Secondary 46L80, 19K35.

Keywords and Phrases: Noncommutative geometry, quantum groups, quantum flag manifolds, Poincaré duality, Bernstein-Gelfand-Gelfand complex, Kasparov theory, Baum-Connes Conjecture.

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