Morita Classification of Quantum SU-Two Projective Representations

Definitions and mathematical statement

For a compact quantum group $\mathbb G$, write $A=L^\infty(\mathbb G)$ and let $\Delta:A\to A\,\bar\otimes\,A$ denote its coproduct, where $\bar\otimes$ is the spatial von Neumann algebra tensor product. To express the questions, use the right-coaction convention: a coaction on a von Neumann algebra $M$ is a normal injective unital $*$-homomorphism $\alpha:M\to M\,\bar\otimes\,A$ satisfying \[ (\alpha\otimes\mathrm{id})\alpha=(\mathrm{id}\otimes\Delta)\alpha. \] It is ergodic if \[ M^\alpha:=\{x\in M:\alpha(x)=x\otimes1\}={\mathbb{C}}1. \] In this convention a coideal von Neumann subalgebra is a unital von Neumann subalgebra $N\subseteq A$ such that $\Delta(N)\subseteq N\,\bar\otimes\,A$, equipped with the restricted coproduct. The supplied record does not specify the coaction side; this convention makes the notation explicit and does not reconstruct the report's conventions.

Two coactions of the same quantum group are equivariantly $W^*$-Morita equivalent if they can be realized as the two diagonal corners of a linking von Neumann algebra carrying a coaction: the complementary corner projections are fixed by the coaction, each has central support $1$, and the corner identifications intertwine the coactions. Denote this relation by $\sim_{\mathbb G}$. A type-I von Neumann algebra is one in which every nonzero central summand contains a nonzero abelian projection. A type-I factor is isomorphic to $B(H)$ for a nonzero Hilbert space $H$; the supplied record imposes no dimension or separability restriction on $H$.

Question in the source's parameter notation. Write $A_F=L^\infty(FO_F)$. For every ergodic coaction $(M_F,\alpha_F)$ covered by the source's parameter conventions, is there a parameter $G\in F_q$ and a coideal von Neumann subalgebra $N\subseteq A_G$ such that \[ (M_G,\alpha_G)\sim_{FO_G}(N,\Delta_G|_N)? \] Here $(M_G,\alpha_G)$ must be the coaction associated with the given $(M_F,\alpha_F)$ by the source's intended construction. It is not an independently chosen coaction. The supplied record does not define $F_q$, the admissible parameters $F$, the quantum groups $FO_F$, or this association. Consequently this question remains a conditional formulation in the source's notation, rather than a fully specified assertion. The cleaned statement additionally asks, if the universal assertion fails, for a characterization of the coactions having such a coideal representative; that follow-up is not present in the supplied original excerpt.

Question for quantum $SU_q(2)$. Put $A_q=L^\infty(SU_q(2))$. For every ergodic coaction $\alpha:M\to M\,\bar\otimes\,A_q$ on a type-I von Neumann algebra $M$, does there exist a coideal von Neumann subalgebra $N\subseteq A_q$ such that \[ (M,\alpha)\sim_{SU_q(2)}(N,\Delta_q|_N)? \] The admissible values of $q$ and the convention for $SU_q(2)$ are not supplied, so the quantifier over $q$ cannot be completed from this record.

Scope qualification. The original excerpt motivates the first question by classification of projective unitary representations of quantum $SU(2)$, described there as coactions of $L^\infty(SU_q(2))$ on type-I factors $B(H)$. It attributes explicit knowledge of the von Neumann algebra structure of coideals to [Tom08] and the projective-representation interpretation to [DC11]. The cleaned question instead explicitly assumes ergodicity and permits any type-I von Neumann algebra. These targets are not identified here: the supplied excerpt does not establish their equivalence or supply the missing parameter-transport construction. Its assertion that the first question would yield the stated classification is retained as source motivation, not proved as part of this draft.

Short English statement

The questions ask whether the specified ergodic quantum-group coactions have representatives, up to equivariant $W^*$-Morita equivalence, that arise from coideal von Neumann subalgebras. The general $FO_F$ question lacks definitions of its parameter family and of the passage from $M_F$ to $M_G$. The $SU_q(2)$ question lacks a range for $q$, and its type-I-algebra formulation differs from the original excerpt's projective-representation motivation on factors $B(H)$. These qualifications require mathematical review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30004840$, source problem code catalogue entry, title "Morita Classification of Quantum SU-Two Projective Representations." The supplied record includes both an original excerpt and a cleaned formulation.
• Supplied primary-source citation: Quantum Groups -- Algebra, Analysis and Category Theory (2022), Oberwolfach Reports, pp. 2397--2458, DOI $10.4171/owr/2021/44$, https://doi.org/10.4171/owr/2021/44. The supplied excerpt carries page number 2451. The registry gives proposed year 2021 and names no proposer; its report citation gives 2022.
• Supplied literature source, "Oberwolfach Report 44": https://doi.org/10.4171/owr/2021/44.
• Supplied literature source, "Projective representations for compact quantum groups": https://arxiv.org/abs/1006.2278.
• Supplied literature source, "Projective representation theory and Baum--Connes": https://arxiv.org/abs/2112.04365.
• The original excerpt cites [Tom08] for explicit coideal von Neumann algebra structures and [DC11] for the interpretation of projective unitary representations as coactions on $B(H)$. Full bibliographic entries and a verified correspondence with the supplied links are absent.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem "partially solved" and states that projective representations are characterized through actions on type-I factors and cocycle regularity, with explicit $SU_q(2)$ examples, while a complete classification of all such coactions remains unavailable. This is supplied dated curation derived from the OWR collection, not independently verified mathematical or current-status information. The original excerpt describes the classification problem as open at its source date.
• The registry labels the statement corrected_verified and reports that the linked report was inspected and both bullet-point questions recovered with their hypotheses. This is a supplied verification claim; no source visit or independent verification is asserted in this draft.