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Let $\mathbb{G}$ be a compact quantum group, $B$ be a $C^*$-algebra together with a right action $$\beta: B \to B\otimes C(\mathbb{G})$$ which is a non-degenerate $*$-homomorphism satisfying $(\beta \otimes \iota)\beta = (\iota \otimes \Delta)\beta$ and the Podles density condition. A right $B$-Hilbert module $X$ is called $\mathbb{G}$-equivariant if it is equipped with a coaction $$\delta: X \to X\otimes C(\mathbb{G})$$ such that

  • $\beta(\langle x,y\rangle_X) = \langle\delta(x), \delta(y)\rangle_{X\otimes C(\mathbb{G})}$
  • $(\delta \otimes \iota)\delta = (\iota \otimes \Delta)\delta$
  • $[\delta(X)(1\otimes C(\mathbb{G}))]= X\otimes C(\mathbb{G})$.

Example: Let $U\in B(H_U)\otimes C(\mathbb{G})$ be a finite-dimensional representation of $\mathbb{G}$. Then the right $B$-Hilbert module $H_U\otimes B$ becomes $\mathbb{G}$-equivariant for $$\delta(\xi\otimes b) = U_{13} (\xi \otimes \beta(b)).$$ We denote this $\mathbb{G}$-equivariant module by $B\times U.$


I'm looking for references/proofs of the following facts:

(1) If $X$ is finitely generated as a right Hilbert module and $\mathbb{G}$-equivariant, then $X$ embeds equivariantly in a finite direct sum $\bigoplus_{i=1}^n (B\times U_i)$ where the $U_i$ are irreducible representations of $\mathbb{G}$.

(2) If $\beta$ is a homogeneous action, i.e. $B$ is unital and $\{b\in B: \alpha(b) = b\otimes 1\}= \mathbb{C}1_B$, then every $\mathbb{G}$-equivariant right Hilbert $B$-module decomposes as a direct sum of equivariant irreducible submodules.


Thanks in advance for your help.

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1 Answer 1

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Let's assume that $G$ is a reduced compact quantum group, that is, the Haar state on $C(G)$ is faithful.

(1): A direct reference is [1, Lemma 4.2]. You can also get this from a careful study of [2, Théorème 3.2].

(2): This (for countably generated modules) follows from [2, Théorème 3.2] and [3, Proposition 4.6].

  1. Neshveyev, Sergey; Tuset, Lars, Hopf algebra equivariant cyclic cohomology, (K)-theory and index formulas, (K)-Theory 31, No. 4, 357-378 (2004). ZBL1067.19002.
  2. Roland Vergnioux, KK-théorie équivariante et opératour de Julg-Valette pour les groupes quantiques, Ph.D. thesis, Universit ́e Paris Diderot-Paris 7, 2002.
  3. De Commer, Kenny; Yamashita, Makoto, Tannaka-Krein duality for compact quantum homogeneous spaces. I: General theory, Theory Appl. Categ. 28, 1099-1138 (2013). ZBL1337.46045.
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  • $\begingroup$ Thanks for your answer! It appears that the first article you link to is behind a paywall. Is there some other way to access this article? The Arxiv version arxiv.org/pdf/math/0304001.pdf does not have a Lemma 4.2. $\endgroup$
    – J. De Ro
    Commented Nov 21, 2022 at 21:47
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    $\begingroup$ The section numbering is off by one, so it’s Lemma 3.2 in the arXiv version. $\endgroup$ Commented Nov 22, 2022 at 2:36

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