On the Fomin-Kirillov algebras

The Fomin-Kirillov algebras ${\mathcal{F}\mathcal{K}}_n$ are shown to be infinite-dimensional for $n\ge 6$. By computer calculations, we find an explicit nonzero primitive element of degree 14 in the identity component of ${\mathcal{F}\mathcal{K}}_6$. For every $n\ge 6$, the existence of such an element implies that ${\mathcal{F}\mathcal{K}}_n$ contains a polynomial subalgebra and is therefore infinite-dimensional, and that the canonical homomorphism from ${\mathcal{F}\mathcal{K}}_n$ onto its associated Nichols algebra is not an isomorphism.