Free Decompositions of Compact Quantum Groups

Definitions and mathematical statement

Let $(G,u)$ be a compact matrix quantum group, with distinguished finite-dimensional corepresentation $u$ on a complex vector space $H$. Write $C(G)$ for its function algebra and $1$ for its unit. For each integer $n\geq 1$, define \[ V_n=\operatorname{Fix}(u^{\otimes n})=\{\xi\in H^{\otimes n}:u^{\otimes n}(\xi\otimes 1)=\xi\otimes 1\}. \] The fusion semiring $R^+(G)$ is the semiring of equivalence classes of finite-dimensional unitary corepresentations of $G$, with addition induced by direct sum and multiplication induced by tensor product; its distinguished additive basis consists of irreducible corepresentation classes.

Source problem. Characterize the pairs $(G,u)$ for which the sequence $(V_n)_{n\geq 1}$ admits a free decomposition. Determine whether, for every compact matrix quantum group with its distinguished corepresentation $(G,u)$, \[ (V_n)_{n\geq 1}\text{ admits a free decomposition} \quad\Longleftrightarrow\quad G\text{ has a free fusion semiring}. \] Develop a generalization of free decomposition that permits possibly negative free cumulants.

Unresolved qualifications. The supplied record does not define free decomposition or the additional structure required for a fusion semiring to be free. Thus the displayed equivalence records the intended question, with both specialized predicates awaiting their source definitions. In particular, the left side involves the distinguished corepresentation $u$, whereas the right side is stated as a property of $G$.

For clarity, numerical free cumulants $(r_n)_{n\geq 1}$ of a scalar moment sequence $(m_n)_{n\geq 0}$, with $m_0=1$, are determined recursively by \[ m_n=\sum_{\pi\in\operatorname{NC}(n)}\prod_{B\in\pi}r_{|B|},\qquad n\geq 1, \] where $\operatorname{NC}(n)$ denotes the partitions of $\{1,\ldots,n\}$ having no distinct blocks containing respectively $a,c$ and $b,d$ with $a<b<c<d$. One numerical sequence associated with the fixed spaces is $m_n=\dim V_n$. The supplied record does not specify whether this is the intended moment sequence or what generalized decomposition must realize when a cumulant is negative. This numerical interpretation is therefore conditional, and no precise existence assertion for the proposed generalization is supplied.

Short English statement

Characterize when the tensor-power fixed spaces of a compact matrix quantum group admit the source's intended free decomposition, ask whether this is equivalent to having a free fusion semiring, and seek an extension allowing negative free cumulants. The supplied record omits definitions needed to make these targets fully precise.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004261$, title Free Decompositions of Compact Quantum Groups, source problem code catalogue entry, collection Oberwolfach Reports Open Problems. No individual proposer is identified; the supplied proposed year is 2019.
• Supplied report citation: Mini-Workshop: Operator Algebraic Quantum Groups (2020), Oberwolfach Reports, pp. 2821--2867, DOI 10.4171/owr/2019/45. https://doi.org/10.4171/owr/2019/45. The supplied report title's year and the proposed year differ; this draft preserves both.
• Supplied literature source, label OWR 2019 45: https://ems.press/content/serial-article-files/46823.
• Supplied literature source, attribution and label Freslon, Free fusion: https://www.imo.universite-paris-saclay.fr/~amaury.freslon/Documents/Free_fusion.pdf.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that free decompositions and an extension involving negative free cumulants remain open, and that existing fusion-semiring results do not settle the proposed equivalence or extension. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry supplies the statement label corrected_verified and reports that its statement was checked against the official Oberwolfach report. These are supplied verification claims; no source was independently visited or checked in preparing this draft.