Matrix Dressing Beyond Pfaff-Toda

@misc{carpentier2026matrix,
  title = {{Matrix Dressing Beyond Pfaff-Toda}},
  author = {Sylvain Carpentier and Marta Dell'Atti},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12970},
  eprint = {2609.12970},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12970},
}

TY  - UNPB
TI  - Matrix Dressing Beyond Pfaff-Toda
AU  - Sylvain Carpentier
AU  - Marta Dell'Atti
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12970
UR  - https://arxiv.org/abs/2609.12970
AB  - 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 2×2 matrix pseudodifference Lax hierarchy. We introduce the Matrix Pfaff-Toda hierarchy by dressing the matrix Laurent algebra M_2N(ℂ[𝒮,𝒮^−1]) 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.
ER  -