Integrable extensions of Adler's map via Grassmann algebras

TL;DR

This paper extends Adler's map using Grassmann algebras, demonstrating its integrability, invariants, and explicit solutions, thereby enriching the understanding of integrable systems in algebraic structures.

Contribution

It introduces a new Grassmann algebra extension of Adler's map, proves its Yang-Baxter property, invariants, and Liouville integrability, and provides explicit solutions.

Findings

Map satisfies Yang-Baxter equation

Map admits three invariants

Map is Liouville integrable

Abstract

We study certain extensions of the Adler map on Grassmann algebras $\Gamma(n)$ of order $n$. We consider a known Grassmann-extended Adler map, and assuming that $n=1$ we obtain a commutative extension of Adler's map in six dimensions. We show that the map satisfies the Yang--Baxter equation, admits three invariants and is Liouville integrable. We solve the map explicitly, viewed as a discrete dynamical system.