多参数量子仿射空间与不等边杨-巴克斯特方程
Multiparameter Quantum Affine Spaces and the Scalene Yang--Baxter Equation
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中文总结 AI 辅助
该研究从二次非交换代数构造非编织不等边杨-巴克斯特方程的与表示无关的解族，选出多参数量子仿射空间的特殊子类，构建有限与无限维实现，为相关研究提供代数框架。
中文摘要 AI 辅助
我们从二次非交换代数出发，构造了非编织不等边杨-巴克斯特方程的与表示无关的解族。从两个反交换生成元开始，我们借助量子平面代数得到了一种连续形变，并将该构造扩展到任意数量的生成元。在后一种情况下，不等边杨-巴克斯特关系将多参数交换矩阵确定为精确的乘法形式，从而选出了多参数量子仿射空间的一个特殊子类。我们利用奇异矩阵和有限海森堡-外尔算子构造了有限维实现，还借助双边加权移位和乘法-移位算子构造了无限维实现。这些显式实现一般是**纯不等边的**：尽管有序三元组满足不等边杨-巴克斯特方程，但其单个组成算子并不满足普通的非编织杨-巴克斯特方程。这些结果为构造杨-巴克斯特不可约的不等边三元组提供了与表示无关的代数框架，也为研究其在量子可积性中的潜在应用奠定了基础。
英文摘要
We construct representation-independent families of solutions of the non-braided scalene Yang--Baxter equation from quadratic noncommutative algebras. Beginning with two anticommuting generators, we obtain a continuous deformation in terms of quantum-plane algebras and extend the construction to an arbitrary number of generators. In the latter case the scalene Yang--Baxter relation fixes the multiparameter exchange matrix to an exact multiplicative form, thereby selecting a distinguished subclass of multiparameter quantum affine spaces. We construct finite-dimensional realizations using singular matrices and finite Heisenberg--Weyl operators, as well as infinite-dimensional realizations in terms of bilateral weighted shifts and multiplicative-shift operators. The explicit realizations are generically \emph{purely scalene}: although the ordered triple satisfies the scalene Yang--Baxter equation, its individual constituent operators do not satisfy the ordinary non-braided Yang--Baxter equation. These results provide a representation-independent algebraic framework for constructing Yang--Baxter-irreducible scalene triples and a starting point for investigating their possible applications to quantum integrability.
