Genuine Finite-Growth Nichols Algebras over Nonabelian Groups

Definitions and mathematical statement

Let $k$ be a field and $G$ a group. The supplied record does not specify the characteristic or other hypotheses on $k$; the formulation below is parameterized by $k$, without asserting that the question has the same answer over every field.

A left Yetter--Drinfeld module over $kG$ is a $G$-graded vector space $V=\bigoplus_{g\in G}V_g$ with a left $G$-action satisfying \[ h\cdot V_g\subseteq V_{hgh^{-1}}\qquad(g,h\in G). \] Its braiding is \[ c(v\otimes w)=(g\cdot w)\otimes v\qquad(v\in V_g,\ w\in V). \] Define its support by $\operatorname{supp}(V)=\{g\in G:V_g\ne0\}$. Interpret nonabelian support as the condition \[ \exists g,h\in\operatorname{supp}(V)\quad gh\ne hg, \] equivalently, that the subgroup generated by the support is nonabelian. This interpretation requires confirmation from the source context.

The Nichols algebra of $V$ is \[ \mathcal B(V)=T(V)/I(V), \] where $T(V)$ has its braided Hopf algebra structure with $V$ primitive, and $I(V)$ is the largest homogeneous braided Hopf ideal contained in $\bigoplus_{n\ge2}V^{\otimes n}$. For a unital $k$-algebra $A$, define \[ \operatorname{GKdim}_k A =\sup_W\limsup_{n\to\infty} \frac{\log\left(\dim_k W^n\right)}{\log n}, \] where $W$ ranges over finite-dimensional $k$-subspaces of $A$ containing $1$, and $W^n$ is the span of products of $n$ elements of $W$.

Existence question. Find a nonabelian group $G$ and a Yetter--Drinfeld module $V$ over $kG$ such that \[ \exists g,h\in\operatorname{supp}(V)\ (gh\ne hg),\qquad \dim_k\mathcal B(V)=\infty,\qquad \operatorname{GKdim}_k\mathcal B(V)<\infty, \] and such that $\mathcal B(V)$ is genuine. The supplied passage explains genuine only as excluding any trivial example; it does not specify which examples are excluded. Thus the displayed conditions formalize the explicit algebraic requirements, but do not fully formalize this additional qualification. The record also does not specify whether $G$ must be finite or $V$ must be finite-dimensional.

Specific question. Let \[ X=\{(12),(13),(23)\}\subset S_3, \qquad s\triangleright t=sts^{-1}. \] The constant rack cocycle $q(s,t)=1$ gives the braided vector space $V_X$ with basis $\{e_s:s\in X\}$ and braiding \[ c(e_s\otimes e_t)=e_{sts^{-1}}\otimes e_s. \] Equivalently, give $e_s$ degree $s$ and let $h\cdot e_s=e_{hsh^{-1}}$ for $h\in S_3$. Determine the Nichols algebra $\mathcal B(V_X)$. Its intended ground field remains unspecified. The supplied wording does not assert that this example satisfies the infinite-dimension and finite-Gelfand--Kirillov-dimension requirements.

Short English statement

The question asks for a genuine Nichols algebra with noncommuting support, infinite vector-space dimension, and finite Gelfand--Kirillov dimension. It also asks to determine the Nichols algebra of the transposition rack in $S_3$ with constant cocycle $1$. The ground field and the exclusion of trivial examples need clarification. The supplied record labels the problem open; this draft does not independently verify that label.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30004842$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. Project registry paths are data/unsolvedmath/problems.json and the supplementary catalog/problems.json; their contents were not independently inspected for this draft.
• Supplied primary 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 original supplied passage attributes the question to Nicolás Andruskiewitsch, with the initials N.A.; the structured proposer field is empty and the proposed year is 2021.
• Supplied literature source: Pointed Hopf algebras over nonabelian groups. https://doi.org/10.1112/plms.12559.
• Supplied literature source: Finite-GK pre-Nichols algebras. https://arxiv.org/abs/2504.12206.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that finite-dimensional and selected nonabelian Nichols algebras have been determined, but that no genuine example meeting the requested nonabelian-support, infinite-dimension, and finite-GK conditions has been identified or classified in general. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified. Its verification note reports inspection of the linked report, removal of the author label, and restoration of the omitted example. These are supplied provenance claims; no source was visited for this draft.