Abstract
A family of non-parametric Yang–Baxter (YB) maps is constructed by re-factorization of 2 × 2 matrix polynomials of first degree. These maps are Poisson with respect to the Sklyanin bracket. For each Casimir function a parametric Poisson YB map is generated by reduction on the corresponding level set. By considering a complete set of Casimir functions symplectic multi-parametric YB maps are derived. These maps are quadrirational with explicit formulae in terms of matrix operations. Their Lax matrices are, by construction, 2 × 2 first-degree-polynomial in the spectral parameter and are classified by THE Jordan normal form of the leading term. Degenerate parametric YB maps constructed as limits of the non-degenerate ones are connected to known integrable systems on quadgraphs.
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