The Lévy–Khintchine Property for Quantum Groups

Definitions and mathematical statement

Under the usual complex algebraic convention, let $A$ be a unital $*$-bialgebra over ${\mathbb{C}}$, with coproduct $\Delta:A\to A\otimes A$ and counit $\varepsilon:A\to{\mathbb{C}}$. Here the tensor product is algebraic, the coproduct is coassociative, and $\Delta$ and $\varepsilon$ are unital $*$-homomorphisms satisfying the counit identities. No topology or analytic completion is specified in the supplied record. Put $I=\ker\varepsilon$.

A generating functional is a complex-linear map $L:A\to{\mathbb{C}}$ satisfying \[ L(1)=0,\qquad L(a^*)=\overline{L(a)}\quad(a\in A),\qquad L(x^*x)\geq0\quad(x\in I). \] Write $\mathcal G(A)$ for the set of these functionals. In the usual algebraic terminology, a Gaussian generating functional $G$ satisfies \[ G(abc)=0\qquad(a,b,c\in I). \] These definitions use the standard complex algebraic convention; the supplied record does not explicitly specify the scalar field or an analytic setting.

Requested classification, with an unresolved definition. Determine which unital $*$-bialgebras have the Lévy--Khintchine property: every generating functional admits a Lévy--Khintchine-type decomposition. The supplied record does not define the admissible components or conditions of this decomposition. To display the missing information explicitly, let $\mathcal D(A)\subseteq\mathcal G(A)\times\mathcal G(A)$ denote the pairs admissible under the intended decomposition convention. The requested property would then be \[ \forall L\in\mathcal G(A)\quad\exists (G,J)\in\mathcal D(A)\quad\forall a\in A,\qquad L(a)=G(a)+J(a). \] A Gaussian-plus-jump formulation would require $G$ to be Gaussian and would need a precise condition on $J$, together with any required compatibility or maximality condition. Those conditions are not supplied. Merely allowing arbitrary generating functionals as the second component would make existence trivial by taking $G=0$ and $J=L$. Thus $\mathcal D(A)$ remains an explicit unresolved part of the intended statement, and the display is a specification of the classification target rather than a complete definition of the Lévy--Khintchine property.

The original wording additionally singles out $SU_q(d)$ for integers $d\geq3$ as unresolved examples. It specifies neither the allowed values of $q$ nor the associated function or coordinate $*$-bialgebra. The supplied later literature assessment instead reports the property for all $SU_q$ and $U_q$ quantum groups, without parameter or algebra conventions. Preserve the former as a historical target and the latter as supplied dated curation; their precise scope and relationship require review.

Short English statement

Classify the unital $*$-bialgebras for which every generating functional has the intended Lévy--Khintchine decomposition. The supplied record leaves that decomposition undefined, so its admissible components must be clarified before the classification has a fully precise target. The original $SU_q(d)$, $d\geq3$, examples also lack parameter and algebra conventions. The supplied partially solved label and later claims about quantum groups are dated curation, not independently verified conclusions.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30003805$, title The Lévy--Khintchine Property for Quantum Groups, source problem code catalogue entry, in Oberwolfach Reports Open Problems. The record gives proposed year 2018 and names no individual proposer.
• Supplied report attribution: Interactions between Operator Space Theory and Quantum Probability with Applications to Quantum Information (2019), Oberwolfach Reports, pp. 1293--1379, DOI 10.4171/owr/2018/22. https://doi.org/10.4171/owr/2018/22. The background describes this report as documenting workshop open-problem discussions. No exact page or passage identifying this problem is supplied.
• Supplied literature source, labelled Hunt formula for SUq and Uq: https://eprints.lancs.ac.uk/id/eprint/222139.
• Supplied literature source, labelled Generating functional decompositions: https://arxiv.org/abs/1510.03292.
• Supplied literature source, labelled Oberwolfach Report 22: https://doi.org/10.4171/owr/2018/22.
• Literature assessment dated exactly 2026-08-21: the supplied status is partially solved; the supplied assessment reports the LK property for broad classes and all $SU_q$ and $U_q$ quantum groups, while reporting counterexamples and an outstanding general classification. This is supplied dated curation derived from the OWR record, not independently verified here. Its time-sensitive status and cited claims require independent verification.