Anticommutative extension of the Adler map

TL;DR

This paper introduces a noncommutative Grassmann extension of the Adler Yang-Baxter map, maintaining key properties like the Yang-Baxter equation, reversibility, and birationality, but not involutivity.

Contribution

It presents the first noncommutative extension of the Adler map that preserves core Yang-Baxter properties while highlighting the loss of involutivity.

Findings

The extension satisfies the Yang-Baxter equation.

The map remains reversible and birational.

Involutivity is not preserved in the extension.

Abstract

We construct a noncommutative (Grassmann) extension of the well known Adler Yang-Baxter map. It satisfies the Yang-Baxter equation, it is reversible and birational. Our extension preserves all the properties of the original map except the involutivity.