Nichols Algebras over Nonabelian Simple Groups

Definitions and mathematical statement

Let $G$ be a finite nonabelian simple group. A left Yetter--Drinfeld module over ${\mathbb{C}} G$ is a complex vector space $V$ with a left $G$-action and a grading \[ V=\bigoplus_{g\in G}V_g, \qquad h\cdot V_g\subseteq V_{hgh^{-1}}\quad(g,h\in G). \] Its support is $\operatorname{supp}(V)=\{g\in G:V_g\ne0\}$. The associated braiding is \[ c(v\otimes w)=g\cdot w\otimes v \qquad(v\in V_g,\ w\in V). \] Write ${}_{{\mathbb{C}} G}^{{\mathbb{C}} G}\mathcal{YD}$ for this category. The Nichols algebra of $V$ is \[ \mathcal B(V)=T(V)/I(V),\qquad T(V)=\bigoplus_{n\ge0}V^{\otimes n}, \] where $T(V)$ has its braided Hopf algebra structure with every element of $V$ primitive, and $I(V)$ is the largest homogeneous braided Hopf ideal contained in $\bigoplus_{n\ge2}V^{\otimes n}$.

Question. Is the following assertion true? \[ \text{For every finite nonabelian simple group }G \text{ and every }0\ne V\in{}_{{\mathbb{C}} G}^{{\mathbb{C}} G}\mathcal{YD}, \qquad \dim_{{\mathbb{C}}}\mathcal B(V)=\infty. \] The supplied clean statement imposes neither irreducibility nor finite dimensionality on $V$, and it imposes no condition that its support generate $G$.

Scope qualification. The original excerpt also asks a broader question about Nichols algebras of irreducible Yetter--Drinfeld modules. The displayed question follows the supplied clean statement concerning finite nonabelian simple groups; it does not formulate a classification for arbitrary groups. The excerpt includes supporting results, but its transcription of some group families is corrupted, and its prime-dimensional equivalence requires source checking for potentially missing hypotheses. These contextual assertions are not adopted as additional hypotheses or as verified theorems here.

Short English statement

For every finite nonabelian simple group $G$, must the Nichols algebra of every nonzero complex Yetter--Drinfeld module over ${\mathbb{C}} G$ be infinite-dimensional? The supplied formulation includes modules that are not irreducible. Its contextual theorem statements and supplied partial-solution status require human review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005781$, title: Nichols Algebras over Nonabelian Simple Groups; exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year $2024$; no individual proposer is supplied.
• Mini-Workshop: Bridging Number Theory and Nichols Algebras via Deformations (2024), Oberwolfach Reports, pp. 235--272, DOI 10.4171/owr/2024/5: https://doi.org/10.4171/owr/2024/5. The supplied excerpt contains an embedded page marker $263$. Oberwolfach Reports are described in the record as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Angiono, pointed Hopf algebras over non-abelian groups, https://doi.org/10.1112/plms.12559. This is the supplied descriptive label; complete bibliographic metadata is not provided.
• Supplied literature source: Heckenberger--Vendramin, classification over non-abelian groups, https://arxiv.org/abs/1412.0857. This is the supplied descriptive label; complete bibliographic metadata is not provided.
• Contextual attributions in the excerpt: Theorem 3 is attributed to Andruskiewitsch--Carnovale--García and to report references [1, 2]; Theorem 4 is attributed to Carnovale--Costantini and report reference [6]. Theorem 5 is described as joint work "with Heckenberger and Meir", and report reference [5] is cited for finite-dimensional examples. The supplied record does not identify the speaker associated with "with" or resolve these numbered references bibliographically.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not an independently verified assessment. The record labels the problem partially_solved and states that finite-dimensionality is ruled out for many nonabelian simple groups, that classified prime-dimensional cases establish finite-dimensionality, and that no complete description for irreducible Yetter--Drinfeld modules is known. The record explicitly asks that time-sensitive status and cited claims be verified independently.
• The supplied statement status is corrected_verified, with a registry assertion that the statement was reconstructed and checked against the official report. This assertion is source metadata, not independent verification by this draft.