We describe the extension, beyond fundamental representations of the Yang-Baxter algebra, of our new construction of separation of variables bases for quantum integrable lattice models. The key idea underlying our approach is to use the commuting conserved charges of the quantum integrable models to generate bases in which their spectral problem is separated, i.e. in which the wave functions are factorized in terms of specific solutions of a functional equation. For the so-called "non-fundamental" models we construct two different types of SoV bases. The first is given from the fundamental quantum Lax operator having isomorphic auxiliary and quantum spaces and that can be obtained by fusion of the original quantum Lax operator. The construction essentially follows the one we used previously for fundamental models and allows us to derive the simplicity and diagonalizability of the transfer matrix spectrum. Then, starting from the original quantum Lax operator and using the full tower of the fused transfer matrices, we introduce a second type of SoV bases for which the proof of the separation of the transfer matrix spectrum is naturally derived. We show that, under some special choice, this second type of SoV bases coincides with the one associated to Sklyanin's approach. Moreover, we derive the finite difference type (quantum spectral curve) functional equation and the set of its solutions defining the complete transfer matrix spectrum. This is explicitly implemented for the integrable quantum models associated to the higher spin representations of the general quasi-periodic Y(gl2) Yang-Baxter algebra. Our SoV approach also leads to the construction of a Q-operator in terms of the fused transfer matrices. Finally, we show that the Q-operator family can be equivalently used as the family of commuting conserved charges enabling to construct our SoV bases.

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关于基本表示之外的变量的量子分离

我们描述了超出杨-巴克斯特代数基本表示形式的扩展，即量子可积格模型的变量基分离的新构造。我们方法所基于的关键思想是使用量子可积模型的交换守恒电荷来生成碱基，在该碱基中其光谱问题得以分离，即根据函数方程的特定解将波函数分解。对于所谓的“非基本”模型，我们构建两种不同类型的SoV碱基。第一个是从具有同构辅助空间和量子空间的基本量子Lax算子给出的，可以通过合并原始量子Lax算子来获得。构造基本上遵循我们先前用于基本模型的构造，并允许我们推导传递矩阵谱的简单性和对角线化性。然后，从原始的量子Lax算子开始，并使用熔融转移矩阵的全塔，我们引入了第二种SoV碱基，自然可以得出转移矩阵光谱分离的证明。我们证明，在某些特殊选择下，第二种SoV碱基与Sklyanin的方法有关。此外，我们导出了有限差分类型（量子谱曲线）函数方程及其定义完整传递矩阵谱的解集。这是为与一般准周期Y（gl2）Yang-Baxter代数的较高自旋表示相关的可积分量子模型明确实现的。我们的SoV方法还可以根据融合的传递矩阵构造Q算子。最后，我们证明了Q算子家族可以等效地用作通勤的保守电荷家族，从而可以构建我们的SoV基地。