标题： 纠缠Yang-Baxter映射在Grassmann代数上 显示英文标题

标题： Entwining Yang-Baxter maps over Grassmann algebras

摘要： 我们构造了集合论缠绕Yang-Baxter方程的新解。 这些解是涉及非交换动力学变量的有理映射，这些变量是阶数为$n$的Grassmann代数的元素。 这些映射来源于与非线性Schrödinger方程相关的Lax超矩阵的重构问题。 在非交换设置中，我们使用其单色超矩阵的特征函数为每个得到的映射构造了一个谱曲线。 我们从谱曲线的模空间中找到了缠绕Yang-Baxter映射的不变量（首次积分）生成函数。 此外，我们表明通过固定Grassmann代数的阶数$n$，可以得到一个具有交换变量的有理缠绕Yang-Baxter映射层次结构。 我们在情况$n=1$（对偶数）和$n=2$中展示了该层次结构的成员，并讨论了它们的动力学和可积性性质，如Lax矩阵、不变量和测度保持性。 显示英文摘要

摘要： We construct novel solutions to the set-theoretical entwining Yang-Baxter equation. These solutions are birational maps involving non-commutative dynamical variables which are elements of the Grassmann algebra of order $n$. The maps arise from refactorisation problems of Lax supermatrices associated to a nonlinear Schr\"odinger equation. In this non-commutative setting, we construct a spectral curve associated to each of the obtained maps using the characteristic function of its monodromy supermatrix. We find generating functions of invariants (first integrals) for the entwining Yang-Baxter maps from the moduli of the spectral curves. Moreover, we show that a hierarchy of birational entwining Yang-Baxter maps with commutative variables can be obtained by fixing the order $n$ of the Grassmann algebra. We present the members of the hierarchy in the case $n=1$ (dual numbers) and $n=2$, and discuss their dynamical and integrability properties, such as Lax matrices, invariants, and measure preservation.