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

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 $m$, $n\in {\mathbb{Z}}^{\ge 2}$. 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.

23 Pages. The original title is "Nichols algebras of dimension $n^m$ and Multinomial expansion"

References in corpus (5)

• Nichols algebras of group type with many quadratic relations
• Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups
• Enumeration of set-theoretic solutions to the Yang-Baxter equation
• Decomposition theorems for involutive solutions to the Yang-Baxter equation
• Finite dimensional Nichols algebras over Suzuki algebra I: simple Yetter-Drinfeld modules of $A_{N2n}^{\mu \lambda }$