Connections Among Weyl Groupoids, Nichols Algebras, and Cluster Algebras

Definitions and mathematical statement

The objects named in the supplied question are Weyl groupoids, Nichols algebras, and cluster algebras. A groupoid is a category in which every morphism is invertible; Weyl groupoids are groupoids generated by reflections associated with Cartan data. A Nichols algebra of a braided vector space $V$ is the quotient of its braided tensor algebra $T(V)$ by the largest homogeneous braided Hopf ideal contained in $\bigoplus_{n\geq 2}V^{\otimes n}$. A cluster algebra is a commutative algebra generated by cluster variables obtained by iterated seed mutation. These descriptions identify the terminology without selecting the classes of objects intended by the source.

The supplied question is:

Is there a deeper connection between Weyl groupoids or Nichols algebras and cluster algebras?

The record does not define "deeper connection," specify a correspondence or construction to be sought, or give a property that such a connection must satisfy. It also does not specify the Cartan data, finiteness assumptions, ground field, braiding, coefficient system, or permitted seed data. Consequently, the question is a research direction rather than a fully specified mathematical proposition: no definite domains, quantifiers, or criterion for an affirmative answer can be recovered without additional information. In particular, the word "or" does not specify whether separate connections for the two families or a common framework involving both families is intended.

As supplied context, the dated curation reports a precise connection between rank-two reflection groupoids and type-$A$ cluster algebras, and reports that no general equivalence or deeper connection for Weyl groupoids or Nichols algebras has been established. These are supplied literature assessments, not independently verified mathematical or bibliographic conclusions. They do not specify the additional connection sought in this question.

Short English statement

The question asks for a deeper relationship between Weyl groupoids or Nichols algebras and cluster algebras. The intended relationship and the classes of objects are unspecified, so a precise conjecture cannot yet be extracted. The supplied partially solved label and rank-two background require independent review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002967$, source problem code catalogue entry, title: Connections Among Weyl Groupoids, Nichols Algebras, and Cluster Algebras. The record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied report citation: Mini-Workshop: Friezes (2016), Oberwolfach Reports, DOI 10.4171/owr/2015/48, pp. 2859--2894. https://doi.org/10.4171/owr/2015/48. The supplied background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. The citation's 2016 designation and the record's 2015 year have not been reconciled.
• Cuntz--Heckenberger, Reflection Groupoids of Rank Two. https://doi.org/10.1016/j.jcta.2010.12.003.
• Cuntz--Heckenberger, arXiv version. https://arxiv.org/abs/0911.3051.
• Oberwolfach Report on Weyl Groupoids. https://publications.mfo.de/bitstream/handle/mfo/3318/OWR_2012_47.pdf?isAllowed=y&sequence=1.
• Literature assessment date: 2026-08-21. The supplied dated curation labels the problem partially solved, reports a precise rank-two reflection-groupoid connection with type-$A$ cluster algebras, and reports no established general equivalence or deeper connection for Weyl groupoids or Nichols algebras. This is supplied dated curation derived from the supplied OWR material, not independently verified. Time-sensitive status and cited claims require independent verification.