Isotopes of biracks and Zhang twists of algebras

Abstract

In this paper, we introduce the notion of an ${\mathbb{N}}^p$-graded birack and construct its isotope. Every involutive ${\mathbb{N}}^p$-graded birack gives rise to an ${\mathbb{N}}^p$-graded Yang-Baxter algebra. We study the relation between isotopes of involutive ${\mathbb{N}}^p$-graded biracks and Zhang twists of ${\mathbb{N}}^p$-graded Yang-Baxter algebras. As an example, Yang-Baxter algebras determined by distributive solutions are proved to be Zhang twists of polynomial algebras.