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

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.