AI 中文总结
本文提出矩阵伪差分算子框架统一Adler-Pfaff与Pfaff-Toda层级，引入矩阵Pfaff-Toda层级，恢复连续与多分量情形，并导出相关tau函数与约化结果。
AI 中文摘要
我们发展了一个矩阵伪差分算子框架，将Adler-Pfaff和Pfaff-Toda层级置于一个共同的代数背景中。双无限Adler-Pfaff层级因此成为一个$2\times2$矩阵伪差分Lax层级。我们通过使用矩阵伪差分代数的Pfaff-Toda分裂对矩阵Laurent代数$M_{2N}\bigl(\mathbb{C}[\mathcal{S},\mathcal{S}^{-1}]\bigr)$进行dressing，引入了矩阵Pfaff-Toda层级。在单分量情形下，其对角扇区恢复了连续Pfaff-Toda层级，而对角线外的方向提供了缺失的奇流，单个裸算子的幂重构了完整的Adler-Pfaff层级。对于一般的$N$，在正则分解轨迹上，Savchenko和Zabrodin的多分量Pfaff-Toda tau函数实现了交换对角扇区，其dressing方程直接从费米双线性恒等式导出。约化到块对角和标量情形恢复了多分量和标量$2D$ Toda，将偶Pfaff层级等同于反对角Toda子层级，并得到Krichever-Zabrodin C-Toda层级。
英文摘要
We develop a matrix pseudodifference operator framework that places theAdler-Pfaff and Pfaff-Toda hierarchies in a common algebraic setting. The bi-infinite Adler-Pfaff hierarchy then becomes a $2\times2$ matrix pseudodifference Lax hierarchy. We introduce the Matrix Pfaff-Toda hierarchy by dressing the matrix Laurent algebra $M_{2N}\bigl(\mathbb{C}[\mathcal{S},\mathcal{S}^{-1}]\bigr)$ using a Pfaff-Toda splitting of the matrix pseudodifference algebra. In the one-component case, its diagonal sector recovers the continuous Pfaff-Toda hierarchy, while the off-diagonal directions supply the missing odd flows, and the powers of a single bare operator reconstruct the full Adler-Pfaff hierarchy. For general $N$, on the regular factorization locus, the multicomponent Pfaff-Toda tau-functions of Savchenko and Zabrodin realize the commuting diagonal sector, with its dressing equations derived directly from the fermionic bilinear identity. Reductions to the block-diagonal and scalar cases recover multicomponent and scalar $2D$ Toda, identifying the even Pfaff hierarchy with an anti-diagonal Toda subhierarchy, and yielding the Krichever-Zabrodin C-Toda hierarchy.
Comments54 pages
