Multinomial expansion and Nichols algebras associated to non-degenerate involutive solutions of the Yang-Baxter equation

TL;DR

This paper explores the structure of Nichols algebras linked to involutive solutions of the Yang-Baxter equation, revealing new finite-dimensional examples and connections to multinomial expansion, generalizing previous work related to Pascal's triangle.

Contribution

It introduces a novel connection between Nichols algebras and multinomial expansion, extending prior results to broader classes of solutions.

Findings

Constructed infinite families of finite-dimensional Nichols algebras.

Established a relationship between Nichols algebras and multinomial coefficients.

Generalized previous work connecting Nichols algebras to Pascal's triangle.

Abstract

In this paper, we investigate the Nichols algebra $\mathfrak{B}(W_{X,r})$ associated to any non-degenerate involutive solution $(X, r)$ of the Yang-Baxter equation. Infinite examples of finite dimensional Nichols algebras are obtained, including those of dimension $n^m$ with $m$, $n\in\Bbb Z^{\geq 2}$. It turns out that the Nichols algebra $\mathfrak{B}(W_{X, r})$ has interesting relations with multinomial expansion. This is a generalization of the work in arXiv:2103.06489, which built a connection between the Nichols algebras of squared dimension and Pascal's triangle.