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The Face Semigroup Algebra of a Hyperplane Arrangement

Published online by Cambridge University Press:  20 November 2018

Franco V. Saliola
Franco V. Saliola*
Affiliation:
Laboratoire de Combinatoire et d’Informatique, Mathématique (LaCIM), Université du Québec à Montréal, Montréal, QC, e-mail:saliola@gmail.com
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Abstract

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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.

Keywords

52C3505E2516S37
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 4 , 01 August 2009 , pp. 904 - 929
Copyright
Copyright © Canadian Mathematical Society 2012

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