bibkey: adamopoulou2025entwininggrassmann
authors: P. Adamopoulou, G. Papamikos
year: 2025
title: "Entwining Yang–Baxter maps over Grassmann algebras"
doi: 10.1016/j.physd.2024.134469
url: https://arxiv.org/abs/2311.18673v1
claim: "Over the Grassmann algebra Γ(n) the authors construct birational parametric entwining Yang–Baxter maps R_{a,b}, S_{a,b}, T_{a,b} from refactorisations of Lax supermatrices of a Grassmann NLS equation (Eqs. (17)–(19)), and expand them into 2^{n+2}-dimensional maps with commutative variables. They show for n = 1 and n = 2 that S_{a,b} preserves the measure m = y₁^{−1} and m = y₁₁^{−2} (the Jacobian determinant equals m(z)/m(S(z))), and conjecture that in Γ(n) R_{a,b} is volume preserving while S_{a,b} and T_{a,b} preserve measures of the form y₁₁^{−n} and x₁₁^{−n}."
strata_touched:
  - D5/S3/Quantum/Algebra/GrassmannEntwiningMeasureRefutation
license: citation-only
triage: anchor

Adamopoulou–Papamikos, entwining Yang–Baxter maps over Grassmann algebras

P. Adamopoulou and G. Papamikos, Entwining Yang–Baxter maps over Grassmann algebras, Physica D 472, 134469 (2025), DOI 10.1016/j.physd.2024.134469; arXiv:2311.18673 [nlin.SI] (v1, the only arXiv version).

Verified locator

DOI: 10.1016/j.physd.2024.134469 (as listed by Crossref, published February 2025). Primary version read: https://arxiv.org/abs/2311.18673v1 (the only arXiv version). The journal text was not read. Source: https://arxiv.org/pdf/2311.18673v1 (text read from the PDF):

• §2.1, the Grassmann algebra Γ(n), Eq. (4);
• §3, Eqs. (17)–(19), the maps R_{a,b}, S_{a,b}, T_{a,b}, and Theorem 3.1;
• §4 (Γ(1)), the definition of a measure-preserving map and the measures 1, 1/y₁, 1/x₁;
• §5 (Γ(2)), the measures 1, 1/y₁₁², 1/x₁₁² and the conjecture.

Source statements

§3, Eq. (18): “map S_{a,b} is given by u₁ = y₁²(1 + χ₁ψ₂)/(b x₁) + y₁(y₂ − a − ψ₁ψ₂)/b, v₁ = x₁, u₂ = b/y₁, v₂ = a + x₁x₂ − y₁/x₁ + χ₁χ₂, ξ₁ = ψ₁ − (y₁/x₁)χ₁, η₁ = χ₁, ξ₂ = ψ₂, η₂ = χ₂ + (y₁/x₁)ψ₂”, with even variables x₁, x₂, y₁, y₂ and odd variables χ₁, χ₂, ψ₁, ψ₂ in Γ(n).

§4: “we show that these maps are measure preserving. This means that for each of them there exists a function m of the dynamical variables such that the Jacobian determinant J of the map can be written as J = m(x₁, x₂, χ₁, χ₂, y₁, y₂, ψ₁, ψ₂)/m(u₁, u₂, ξ₁, ξ₂, v₁, v₂, η₁, η₂). It can be verified that maps R_{a,b}, S_{a,b}, T_{a,b} in (29)–(31) are measure preserving with m … equal to 1, 1/y₁ and 1/x₁, respectively.”

§5: “each of the maps R_{a,b}, S_{a,b}, T_{a,b} in (43)–(45) admits an invariant measure m. These measures are m = 1 for R_{a,b}, m = 1/y₁₁² for S_{a,b}, and m = 1/x₁₁² for T_{a,b}. … We conjecture that in Γ(n) the map R_{a,b} is volume preserving for every n, while the maps S_{a,b} and T_{a,b} preserve measures of the form y₁₁^{−n} and x₁₁^{−n}, respectively, with x₁₁ and y₁₁ defined similar to the cases n = 1 and n = 2.”

Reading used by the repository

Γ(3) is the exterior algebra on three generators with its eight monomials. Expanding the eight variables of Eq. (18) in their monomial coefficients gives a rational map on ℝ³², and J is the determinant of its derivative. Quotients by even elements are products with the two-sided inverse, which exists when the scalar part is nonzero. y₁₁ is the scalar coefficient of y₁.