Entwining Yang–Baxter maps over Grassmann algebras

Abstract

In this work 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 Schrödinger 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 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, and we present the cases $n=1$ (dual numbers) and $n=2$. Then we discuss the integrability properties, such as Lax matrices, invariants, and measure preservation, for the obtained discrete dynamical systems.