Gaussian Functionals on q-Deformed Compact Lie Groups

Definitions and mathematical statement

For terminology, let $A=\operatorname{Pol}(G_q)$ be the Hopf $*$-algebra of representative functions of a compact quantum group $G_q$, with counit $\varepsilon:A\to{\mathbb{C}}$, and put $K=\ker\varepsilon$. A generating functional is a linear map $L:A\to{\mathbb{C}}$ satisfying \[ L(1)=0,\qquad L(a^*)=\overline{L(a)}\quad(a\in A),\qquad L(a^*a)\geq0\quad(a\in K). \] A generating functional is called Gaussian if \[ L(abc)=0\qquad(a,b,c\in K). \] These standard definitions provide terminology; they do not determine the particular decomposition method intended by the source.

Source conjecture. Does the Lévy--Khintchine decomposition method described in the cited report for the quantum group denoted there by $U_q(n)$ extend to the $q$-deformations $G_q$ of other simple compact Lie groups $G$?

The intended assertion concerns extending that particular method across other simple compact Lie groups. The supplied record does not specify the admissible values of $q$, the precise construction of $G_q$, any restrictions on the global form of $G$, the class of functionals covered, or the exact decomposition and its existence or uniqueness requirements. Consequently, a fully quantified decomposition conjecture cannot be recovered from this record alone. In particular, no particular decomposition formula or universal range of $q$ is asserted here.

The original extraction is spliced with auxiliary material. It includes an assertion involving the undefined abbreviation "GFs", a formula resembling \[ \psi(a)=\langle v,[\pi(a)-\varepsilon(a)I]v\rangle, \] and a statement that each $\eta_j$ is determined by a value at $u_{jj}$, followed by a limit involving $\pi(a)-\varepsilon(a)I$, $(I-p\pi(u_{jj}))^{-1}$, and $p\to1$. The extraction alternates between $\eta(u_{jj})\in H_j$ and $\eta_j(u_{jj})$, and supplies no definitions or hypotheses for these expressions. These fragments do not specify additional claims of the present conjecture; their intended scope and relation to the method require the original context.

Short English statement

The question asks whether the report's Lévy--Khintchine decomposition method for $U_q(n)$ extends to quantum deformations of other simple compact Lie groups. The parameter range, functional class, and exact decomposition are unspecified. The supplied dated assessment labels the problem partially solved; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30004839$, source problem code catalogue entry, title "Gaussian Functionals on q-Deformed Compact Lie Groups". The supplied record names no proposer and records proposed year $2021$.
• Supplied primary citation: Quantum Groups -- Algebra, Analysis and Category Theory (2022), Oberwolfach Reports, pp. 2397--2458, DOI 10.4171/owr/2021/44. Source URL and literature-source label "Oberwolfach Report 44": https://doi.org/10.4171/owr/2021/44. The supplied background describes the reports as documenting workshops and their open-problem discussions.
• Supplied literature source, "Hunt formula for SUq and Uq": https://eprints.lancs.ac.uk/id/eprint/222139/. No authors or further bibliographic details are supplied.
• Supplied literature source, "Connectedness and Gaussian parts": https://arxiv.org/abs/2203.08030. No authors or further bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem "partially solved" and states that Lévy--Khintchine decompositions are established for $SU_q$ of arbitrary finite rank and several related quantum groups, while extension to $q$-deformations of all simple compact Lie groups is not established. This is supplied dated curation derived from the OWR record, not an independently verified assessment; its time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified and records that the linked report was inspected and its concluding conjecture isolated from a spliced formula and acknowledgements. This is a supplied verification claim, not a claim that this draft's preparer visited or checked the report.
• The spliced source acknowledgements attribute partial support to an unnamed author from the Polish National Science Center, SONATA grant 2016/21/D/ST1 /03010, and the Polish National Agency for Academic Exchange, POLONIUM program PPN/BIL/2018/1/00197/U/00021. These acknowledgements do not identify the conjecture's proposer.