Nichols Algebras over Simple Groups

Definitions and mathematical statement

Let $G$ be a finite simple group, where simplicity means that $G\neq\{e\}$ and its only normal subgroups are $\{e\}$ and $G$. This convention includes cyclic groups of prime order; the supplied statement does not specify that $G$ is nonabelian.

Using the left--left convention, a Yetter--Drinfeld module over ${\mathbb{C}} G$ is a complex vector space $V$ with a linear $G$-action and a grading \[ V=\bigoplus_{h\in G}V_h, \qquad g\cdot V_h=V_{ghg^{-1}}\quad(g,h\in G). \] Its braiding is determined by \[ c(v\otimes w)=h\cdot w\otimes v \qquad(v\in V_h,\ w\in V). \] Let $T(V)=\bigoplus_{n\geq0}V^{\otimes n}$ be the tensor algebra with its braided Hopf algebra structure in which every element of $V$ is primitive. The Nichols algebra of $V$ is \[ \mathcal B(V)=T(V)/I(V), \] where $I(V)$ is the largest homogeneous braided Hopf ideal contained in $\bigoplus_{n\geq2}V^{\otimes n}$.

Question as supplied. Is it true that, for every finite simple group $G$ and every nonzero Yetter--Drinfeld module $V$ over ${\mathbb{C}} G$, \[ \dim_{{\mathbb{C}}}\mathcal B(V)=\infty? \] No finite-dimensionality assumption on $V$ is supplied.

Unresolved formulation qualification. The record does not establish whether "finite simple group" was intended to mean "finite nonabelian simple group." These give different quantified questions. The displayed question retains the literal supplied wording; its identification with the intended source conjecture requires review. The supplied original excerpt names Conjecture 1 but omits its mathematical content and the identity of its reference [1].

Short English statement

The supplied question asks whether every nonzero Yetter--Drinfeld module over the complex group algebra of a finite simple group has an infinite-dimensional Nichols algebra. Whether the group must be nonabelian remains unresolved in this draft. The record labels the problem open; this is supplied curation, not an independently verified status.

Sources

• Local UnsolvedMath registry, permanent ID $30006499$, title "Nichols Algebras over Simple Groups," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The registry supplies proposed year 2025 and no named proposer.
• Supplied report citation: Mini-Workshop: The Yang--Baxter Equation and Representations of Braid Groups (2026), Oberwolfach Reports, pp. 2633--2672, DOI 10.4171/owr/2025/49. https://doi.org/10.4171/owr/2025/49. The literature-source label is "Oberwolfach Report 49/2025." The differing year indications are retained as supplied.
• Supplied original-excerpt attribution: Leandro Vendramin, "One is never too old to learn about Nichols algebras," joint work with A. Andruskiewitsch, I. Heckenberger, and E. Meir. The excerpt attributes the conjecture to reference [1], whose bibliographic identity is not supplied; this attribution does not identify a conjecture proposer.
• Supplied literature source: Finite-dimensional Nichols algebras of prime dimension. https://arxiv.org/abs/2306.02989. No authors or specific supporting passage are supplied in the record.
• Literature assessment dated 2026-08-22: the supplied curation labels the conjecture open and reports that alternating, most sporadic, and many Lie-type cases are known, without covering all simple groups. This is supplied dated curation derived from the OWR record, not independently verified literature assessment; its status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked, the question isolated, and its notation repaired. These are supplied source-verification claims, not verification performed for this draft.