  Let $W$ be a finite Coxeter group. We give an algebraic presentation of what
we refer to as ``the non-crossing algebra'', which is associated to the
hyperplane complement of $W$ and to the cohomology of its Milnor fibre. This is
used to produce simpler and more general chain (and cochain) complexes which
compute the integral homology and cohomology groups of the Milnor fibre $F$ of
$W$. In the process we define a new, larger algebra $\widetilde{A}$, which
seems to be ``dual'' to the Fomin-Kirillov algebra, and in low ranks is
linearly isomorphic to it. There is also a mysterious connection between
$\widetilde{A}$ and the Orlik-Solomon algebra, in analogy with the fact that
the Fomin-Kirillov algebra contains the coinvariant algebra of $W$. This
analysis is applied to compute the multiplicities $\langle \rho,
H^k(F,\mathbb{C})\rangle_W$ and $\langle \rho, H^k(M,\mathbb{C})\rangle_W$,
where $M$ and $F$ are respectively the hyperplane complement and Milnor fibre
associated to $W$ and $\rho$ is a representation of $W$.
