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 under the assumption that \(n=1\), obtain a commutative extension of the Adler 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 by regarding it as a discrete dynamical system.

中文翻译：

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通过 Grassmann 代数对 Adler 映射进行可积扩展

摘要

我们研究了 Adler 映射在阶为\(n\) 的Grassmann 代数\(\Gamma(n)\)上的某些扩展。我们考虑已知的 Grassmann-extended Adler 映射，并在\(n=1\)的假设下，获得 Adler 映射在六维的交换扩展。我们证明该映射满足 Yang-Baxter 方程，承认三个不变量，并且是 Liouville 可积的。我们通过将其视为离散动态系统来明确地解决该地图。