Lévy–Khintchine Decomposition for Star-Bialgebras

Definitions and mathematical statement

The supplied record asks for a characterization of the $*$-bialgebras having the Lévy--Khintchine property $(\mathrm{LK})$. Its terminology requires conventions that are not fully specified in the record.

Under the usual complex, unital, algebraic interpretation, let $B$ be a unital associative complex $*$-algebra with unital $*$-homomorphisms \[ \Delta:B\longrightarrow B\otimes_{{\mathbb{C}}}B, \qquad \varepsilon:B\longrightarrow{\mathbb{C}}, \] satisfying coassociativity and the counit identities \[ (\Delta\otimes\mathrm{id})\Delta=(\mathrm{id}\otimes\Delta)\Delta, \qquad (\varepsilon\otimes\mathrm{id})\Delta=\mathrm{id} =(\mathrm{id}\otimes\varepsilon)\Delta. \] Put $K=\ker\varepsilon$. With these conventions, a generating functional is a complex-linear map $L:B\to{\mathbb{C}}$ such that \[ L(1)=0,\qquad L(a^*)=\overline{L(a)}\quad(a\in B),\qquad L(a^*a)\geq0\quad(a\in K). \] Write $\mathcal G(B)$ for this class. Under the standard algebraic convention, a generating functional $G$ is Gaussian when \[ G(abc)=0\qquad(a,b,c\in K), \] equivalently, when $G$ vanishes on the linear span $K^3$ of these products. These conventions make explicit a possible intended setting; their agreement with the cited report requires review.

The source also requires a purely non-Gaussian generating functional, but supplies no definition of this condition. Let $\mathcal P(B)\subseteq\mathcal G(B)$ denote the class intended by that source, pending verification of its definition. The requested characterization then has the form: find necessary and sufficient conditions on $B$ for \[ \begin{split} (\mathrm{LK})\qquad \forall L\in\mathcal G(B)\ \exists G\in\mathcal G(B)\ \exists N\in\mathcal P(B):\quad &L=G+N,\\ &G(K^3)=\{0\}. \end{split} \] Thus both summands must themselves be generating functionals. No uniqueness is asserted in the supplied statement. The undefined class $\mathcal P(B)$ is a substantive unresolved qualification: merely requiring that $N$ fail to be Gaussian does not supply the meaning of "purely non-Gaussian." Consequently, the display records the quantified target but is not a fully specified characterization problem until the source convention for this class and the ambient category are confirmed.

Short English statement

Characterize the $*$-bialgebras for which every generating functional is a sum of a Gaussian generating functional and a purely non-Gaussian generating functional. The supplied record does not define the latter condition; the algebraic conventions displayed above also require confirmation against the source. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004838$, title "Lévy--Khintchine Decomposition for Star-Bialgebras," exact source code catalogue entry, in the Oberwolfach Reports Open Problems collection. No proposer is identified; the supplied proposed year is 2021.
• Supplied authoritative source citation: Quantum Groups -- Algebra, Analysis and Category Theory (2022), Oberwolfach Reports, pp. 2397--2458, DOI 10.4171/owr/2021/44. The original excerpt identifies p. 2445. Source URL and supplied literature reference "Oberwolfach Report 44": https://doi.org/10.4171/owr/2021/44.
• Supplied literature reference "Levy-Khintchine decompositions": https://arxiv.org/abs/1510.03292.
• Supplied literature reference "Gaussian functionals on easy quantum groups": https://doi.org/10.1007/s10959-025-01425-w.
• The original excerpt attributes the introduction of Gaussian generating functionals in this setting to Schürmann, citing its reference [8]. The bibliographic entry for [8] is not supplied; it cannot be identified here from the record alone.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem "partially solved" and reports that property $(\mathrm{LK})$ is proved for commutative bialgebras and several quantum groups, that counterexamples are known, and that no general characterization of $*$-bialgebras with this property is available. This is supplied dated curation, not an independently verified literature assessment. The background explicitly calls for independent verification of time-sensitive status and cited claims.