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Reductions and degenerate limits of Yang-Baxter maps with $3\times 3$ Lax matrices

P. Adamopoulou, T. E. Kouloukas, G. Papamikos

TL;DR

The paper addresses the problem of classifying and understanding reductions and degenerations of Yang-Baxter maps that arise from strong $3\times3$ Lax matrices, preserving the Sklyanin Poisson structure. It constructs the principal parametric YB map from a 3×3 Lax refactorisation, then performs hierarchical reductions to 12-, 8-, and 4-dimensional maps, all carrying invariant quantities that commute under the Sklyanin bracket and render the reduced maps Liouville integrable. Degenerate limits ($a_3\to0$ and a double limit $a_2,a_3\to0$) yield birational non-quadrirational maps and a vectorial Adler–Yamilov type map, respectively, while preserving integrability properties. The work provides a unified framework for a broad class of $3\times3$ binomial Lax-matrix Yang-Baxter maps, with potential extensions to other Jordan forms and associated lattice equations, and highlights the role of monodromy invariants as a source of commuting integrals for transfer maps.

Abstract

We generalise a family of quadrirational parametric Yang-Baxter maps with $3\times 3$ Lax matrices by introducing additional essential parameters. These maps preserve a prescribed Poisson structure which originates from the Sklyanin bracket. We investigate various low-dimensional reductions of this family, as well as degenerate limits with respect to the parameters that were introduced. As a result, we derive several birational Yang-Baxter maps, and we discuss some of their integrability properties. This work is part of a more general classification of Yang-Baxter maps admitting a strong $3\times 3$ Lax matrix with a linear dependence on the spectral parameter.

Reductions and degenerate limits of Yang-Baxter maps with $3\times 3$ Lax matrices

TL;DR

The paper addresses the problem of classifying and understanding reductions and degenerations of Yang-Baxter maps that arise from strong Lax matrices, preserving the Sklyanin Poisson structure. It constructs the principal parametric YB map from a 3×3 Lax refactorisation, then performs hierarchical reductions to 12-, 8-, and 4-dimensional maps, all carrying invariant quantities that commute under the Sklyanin bracket and render the reduced maps Liouville integrable. Degenerate limits ( and a double limit ) yield birational non-quadrirational maps and a vectorial Adler–Yamilov type map, respectively, while preserving integrability properties. The work provides a unified framework for a broad class of binomial Lax-matrix Yang-Baxter maps, with potential extensions to other Jordan forms and associated lattice equations, and highlights the role of monodromy invariants as a source of commuting integrals for transfer maps.

Abstract

We generalise a family of quadrirational parametric Yang-Baxter maps with Lax matrices by introducing additional essential parameters. These maps preserve a prescribed Poisson structure which originates from the Sklyanin bracket. We investigate various low-dimensional reductions of this family, as well as degenerate limits with respect to the parameters that were introduced. As a result, we derive several birational Yang-Baxter maps, and we discuss some of their integrability properties. This work is part of a more general classification of Yang-Baxter maps admitting a strong Lax matrix with a linear dependence on the spectral parameter.
Paper Structure (11 sections, 3 theorems, 67 equations, 1 figure)

This paper contains 11 sections, 3 theorems, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Yang-Baxter maps and Lax matrices
  3. Poisson Yang-Baxter maps with binomial Lax matrices
  4. Yang-Baxter maps with $3\times 3$ Lax matrices
  5. The principal parametric Yang-Baxter map
  6. Reduction to $8$-dimensional Yang-Baxter map
  7. Reduction to a $4$-dimensional Yang-Baxter map
  8. Degenerate Limits
  9. A single degenerate limit
  10. A double limit to a vectorial Adler-Yamilov map
  11. Conclusions

Key Result

Proposition 2.1

$\mathcal{M}$ is a Poisson submanifold of $\mathcal{L}$ of rank four. Furthermore, the discriminant of the cubic polynomial in $\lambda$ of $\iota^*p^a_{\lambda}$ vanishes, i.e.

Figures (1)

  • Figure :

Theorems & Definitions (9)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Remark 2.3
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4