Virtual Abelianity of Unitary Yang–Baxter Braid Images

Definitions and mathematical statement

We use the finite-dimensional complex operator interpretation of the supplied statement. The record does not explicitly specify the scalar field or require finite-dimensionality, so these conventions require confirmation against the source before acceptance.

Let $V$ be a nonzero finite-dimensional complex Hilbert space. A unitary Yang--Baxter operator on $V$ is a linear map $R:V\otimes V\to V\otimes V$ satisfying \[ R^*R=RR^*=I_{V\otimes V} \] and the braid-form Yang--Baxter equation \[ (R\otimes I_V)(I_V\otimes R)(R\otimes I_V) =(I_V\otimes R)(R\otimes I_V)(I_V\otimes R) \quad\text{on }V^{\otimes 3}. \] For each integer $n\geq 2$, let $B_n$ be the braid group with generators $\sigma_1,\ldots,\sigma_{n-1}$ and relations \[ \sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1} \quad(1\leq i\leq n-2), \qquad \sigma_i\sigma_j=\sigma_j\sigma_i \quad(|i-j|\geq 2). \] The associated representation is \[ \rho_{R,n}:B_n\longrightarrow U(V^{\otimes n}), \qquad \rho_{R,n}(\sigma_i) =I_V^{\otimes(i-1)}\otimes R\otimes I_V^{\otimes(n-i-1)}, \] where a zeroth tensor power of an identity is omitted. Write $G_{R,n}=\rho_{R,n}(B_n)$. Following the source's explicit formulation, virtual abelianity means that there exists an abelian normal subgroup $A\trianglelefteq G_{R,n}$ with finite group index $[G_{R,n}:A]$.

Supplied decategorified conjecture, under the conventions above. For every such $V$, every unitary Yang--Baxter operator $R$ on $V$, and every integer $n\geq 2$, does there exist an abelian normal subgroup $A_{R,n}\trianglelefteq G_{R,n}$ such that \[ [G_{R,n}:A_{R,n}]<\infty? \] The subgroup and its index may depend on $V$, $R$, and $n$. The image here is the abstract group image, without taking a topological closure. If $n=1$ is included, $B_1$ and its image are trivial. Whether infinite-dimensional Hilbert spaces are intended is unresolved in the supplied record; the displayed formulation does not address that extension.

Short English statement

The supplied conjecture asks whether every braid-group representation obtained by placing a fixed unitary Yang--Baxter operator on adjacent tensor factors has an abelian normal subgroup of finite index in its image, for every number of strands. The finite-dimensional complex Hilbert-space convention needs source confirmation. The record labels the general conjecture open; that label is supplied dated curation, not an independently verified assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006496$, source problem code catalogue entry, title: Virtual Abelianity of Unitary Yang--Baxter Braid Images. The project identifies data/unsolvedmath/problems.json and catalog/problems.json as local registries; this draft uses the supplied complete record without independently inspecting those files.
• 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 supplied excerpt bears page number 2665 and calls the statement a "Decategorified Conjecture," referring to its internal reference [3]. No proposer or bibliographic details for [3] are supplied. The citation's 2026 year differs from the report identifier and proposed year 2025.
• Supplied literature source: Oberwolfach Report 49/2025. https://doi.org/10.4171/owr/2025/49.
• Supplied literature source: Localization of unitary braid group representations. https://arxiv.org/abs/1009.0241.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the general conjecture open, describes substantial evidence and special-case proofs, and says the general statement remains unresolved. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry marks the statement corrected_verified and reports that the official report was checked and notation repaired. These are supplied provenance claims; no source visit or independent verification is asserted here.