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

Abstract

We improve a theorem of Flores de Chela and Green on the determinants of braided symmetrizers appearing with quantum symmetric algebras (Nichols algebras). As an application we characterize the freeness of Nichols algebras of diagonal type. We also determine the dimensions of the kernels of certain shuffle maps considered as an operator acting on the free algebra. Our approach 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.