AI 中文总结
该研究针对酉杨-巴克斯特算子分类缺失的问题，提出三类生成源的猜想性分类，通过计算搜索得到特殊解，回答了Lechner的存在性问题并验证了Rowell-Wang局域化猜想的开放案例。
AI 中文摘要
存在一系列著名的猜想，将杨-巴克斯特方程的酉解、酉辫融合范畴、拓扑量子计算和纽结不变量关联起来。由于缺乏酉杨-巴克斯特算子的分类，相关进展受到限制。我们提出一个猜想性分类，包含三类生成源：单项式解、由Yetter-Drinfeld模产生的群型解，以及来自扭曲群代数塔的解。我们猜想，在酉标量和局域酉基变换下，所有解都由这些源生成。我们通过在克利福德群中对酉杨-巴克斯特算子的计算搜索，以及多项式-泡利近似，为该分类猜想提供证据。该搜索仅得到少数似乎不等价于单项式或群型解的解。直接计算表明，它们的纽结不变量与HOMFLYPT不变量和BMW/考夫曼不变量的特化对应；后者是本原六次单位根处的平方琼斯特化，对应的三态算子严格局域化了与辫融合范畴SO(4)₂相关的BMW代数特化𝒞ₙ(q³,e^(πⅈ/6))的半单商。四元数III族特化实现了二类双本征值，并给出与SU(3)₃相关的半单赫克代数塔的张量积实现。重要的是，这回答了Lechner近期提出的存在性问题，并验证了2010年Rowell-Wang局域化猜想的一个开放案例。
英文摘要
There is a well-known circle of conjectures relating unitary solutions of the Yang--Baxter equation, unitary braided fusion categories, topological quantum computation, and link invariants. Progress is limited by the lack of a classification of unitary Yang--Baxter operators. We propose a conjectural classification with three generating sources: monomial solutions, group-type solutions arising from Yetter--Drinfeld modules, and solutions from twisted group-algebra towers. We conjecture that, up to unit scalars and local unitary basis changes, all solutions are generated by these sources. We provide evidence for our classification conjecture by means of computational searches for unitary Yang--Baxter operators among Clifford groups and a polynomial-in-Pauli ansatz. This search yields only a few solutions that do not appear to be equivalent to monomial or group type solutions. Direct calculations identify their link invariants with specializations of the HOMFLYPT and BMW/Kauffman invariants; the latter is a squared Jones specialization at a primitive sixth root, and the corresponding qutrit operator strictly localizes the semisimple quotient of the BMW algebra specializations $\mathcal{C}_n(q^3,e^{π\ii/6})$ associated with the braided fusion category $SO(4)_2$. The quaternionic Family III specialization realizes a two-eigenvalue class and gives tensor-power realization of the semisimple Hecke algebra tower associated with \(SU(3)_3\). Significantly, this answers a recent existence question of Lechner, and verifies an open case of the Rowell-Wang localization conjecture from 2010.
CommentsPreliminary version for expedited posting
