This article presents a study of an algebra spanned by the faces of a
hyperplane arrangement. The quiver with relations of the algebra is computed
and the algebra is shown to be a Koszul algebra. It is shown that the algebra
depends only on the intersection lattice of the hyperplane arrangement. A
complete system of primitive orthogonal idempotents for the algebra is
constructed and other algebraic structure is determined including: a
description of the projective indecomposable modules; the Cartan invariants;
projective resolutions of the simple modules; the Hochschild homology and
cohomology; and the Koszul dual algebra. A new cohomology construction on
posets is introduced and it is shown that the face semigroup algebra is
isomorphic to the cohomology algebra when this construction is applied to the
intersection lattice of the hyperplane arrangement.Comment: 37 pages, LaTeX; Added section 8.3B; Changed the wording of a few
paragraphs in the introduction and abstract. No major change
