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Let $\mathbb{G}$ be a discrete quantum group (in the sense of Vaes-Kustermans) with function algebra $(\ell^\infty(\mathbb{G}), \Delta)$. A (right) action of $\mathbb{G}$ on a von Neumann algebra $M$ is an injective, unital, normal $*$-homomorphism $\alpha: M \to M \otimes \ell^\infty(\mathbb{G})$ such that $$(\iota \otimes \Delta)\alpha = (\alpha \otimes \iota)\alpha.$$

Are there any interesting examples of (genuine) discrete quantum groups (i.e. not classical discrete groups) acting on commutative von Neumann algebras?

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The dual coideals for Podleś sphere algebras are commutative coideal subalgebras of $\ell^\infty \widehat{\mathit{SU}}_q(2)$. They can be thought as ‘quantized’ $L(T)$ for the 1-dimensional toruses $T < \mathit{SU}(2)$ sitting as coisotropic subgroups.

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  • $\begingroup$ Thanks! This actually was the only non-trivial example I knew. Do you know of any others? $\endgroup$
    – J. De Ro
    Commented Aug 2, 2023 at 6:17
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    $\begingroup$ A classical analogue of this is $L(T) < L(G)$ for any inclusion of compact groups $T < G$, where $T$ is commutative. In another direction, some graphs $\Gamma$ have finite quantum symmetry, meaning the free autmorphism group $\mathrm{Aut}^+(\Gamma)$ is represented by a finite dimensional noncommutative C$^*$-algebra. Then the defining action of $\mathrm{Aut}^+(\Gamma)$ on $C(V_\Gamma)$ is an example. $\endgroup$ Commented Aug 2, 2023 at 6:54

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