Isotopes of biracks and Zhang twists of algebras

TL;DR

The paper develops aN^p-graded birack theory and introduces a phi-isotope construction that preserves key properties, enabling a precise link between isotopes of involutive biracks and Zhang twists of their Yang–Baxter algebras. It proves that, under mild hypotheses, the YB algebra associated to an isotope is isomorphic to a Zhang twist of the original algebra, providing a powerful method to relate complex YB algebras to simpler ones. As an application, distributive YBE solutions yield Yang–Baxter algebras that are Zhang twists of polynomial algebras, suggesting structural parallels with commutative cases and guiding further study of their homological and growth properties.

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.

Theorems & Definitions (12)

• Lemma 1.2
• Remark 1.3
• Lemma 1.5
• Lemma 2.1
• Lemma 2.2
• Remark 2.3
• Lemma 2.4
• Proposition 2.5
• Remark 3.1
• Theorem 3.2