A characterization of Nichols algebras of diagonal type which are free algebras

TL;DR

This paper investigates when Nichols algebras of diagonal type are free, analyzing the kernel of the shuffle map and providing criteria based on Lyndon words and Diophantine equations.

Contribution

It offers a new characterization of freeness for Nichols algebras of diagonal type using combinatorial and algebraic methods.

Findings

Identifies conditions for Nichols algebra freeness

Determines the kernel dimension of the shuffle map

Connects freeness to solutions of quadratic Diophantine equations

Abstract

This paper is devoted to explore the freeness of Nichols algebras of diagonal type and to determine the dimension of the kernel of the shuffle map considered as an operator acting on the free algebra. Our proof is based on an inequality for the number of Lyndon words and on an identity for the shuffle map. For a particular family of examples, the freeness of the Nichols algebra is characterized in terms of solutions of a quadratic diophantine equation.