Equivariant Fredholm modules for the full quantum flag manifold of $SU_q(3)$

Robert Yuncken (1) , Christian Voigt (2)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 School of Mathematics and Statistics

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 Poincar\'e duality in equivariant KK-theory. Moreover, we show that the Baum-Connes conjecture with trivial coefficients holds for the discrete quantum group dual to $SU_q(3)$.

Domaines: K-théorie et homologie [math.KT]; Algèbres d'opérateurs [math.OA]; Algèbres quantiques [math.QA]

Robert Yuncken, Christian Voigt. Equivariant Fredholm modules for the full quantum flag manifold of $SU_q(3)$. Documenta Mathematica, 2015, 20, pp.58. ⟨hal-01271863⟩