Integrable extensions of Adler's map via Grassmann algebras

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.