Rigidity of action of compact quantum groups I

Details Rigidity of action of compact quantum groups I Rigidity of action of compact quantum groups I

2011-06-25

arxiv.org/abs/1106.5107v3

Mathematics

Quantum Algebra

revised version: some mistakes corrected, which also adds the restriction to types A,D,E of Lie algebras

Scientific paper

Suppose that a compact quantum group $\clq$ acts faithfully on the commutative $C^*$ algebra of continuous functions on $G/T$, where $G$ is a compact, semisimple, centre-less, connected Lie group of simply-laced type, i.e. A,D or E types, with a maximal torus $T$, Lie algebra $\clg$ and a regular element $H$, and assume also that the action is linear in the sense that it leaves invariant the space ${f_\lambda, \lambda \in \clg^\prime}$, $f_\lambda(g):=\lambda({\rm Ad}_g(H))$. It is proved that $\clq$ must be commutative, i.e. of the form $C(K)$ for some compact group $K$. As a corollary, it is also shown that any compact quantum group having a faithful action on the noncommutative manifold obtained by Rieffel deformation of $G/T$ satisfying a similar `linarity' condition must be a Rieffel-Wang type deformation of some classical group.

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