Isotopes of biracks and Zhang twists of algebras

TL;DR

This paper introduces $ abla^p$-graded biracks, constructs their isotopes, and explores their connection to Zhang twists of $ abla^p$-graded Yang-Baxter algebras, including examples involving distributive solutions.

Contribution

It defines $ abla^p$-graded biracks, constructs isotopes, and links involutive biracks to Yang-Baxter algebra Zhang twists, advancing algebraic structures related to the Yang-Baxter equation.

Findings

Involutive $ abla^p$-graded biracks lead to $ abla^p$-graded Yang-Baxter algebras.

Isotopes of involutive $ abla^p$-graded biracks relate to Zhang twists of Yang-Baxter algebras.

Yang-Baxter algebras from distributive solutions are Zhang twists of polynomial 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.