The Face Semigroup Algebra of a Hyperplane Arrangement
By Franco V. Saliola. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2005
53 citations recorded by the source corpus.
Abstract 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.
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/1f3dfd0f018251a792c4b4990856e017f7350521
How researchers see this paper
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In Algebra and Number Theory
- Ben Steinberg
- Stuart W. Margolis
- Swapneel Mahajan
- Marcelo Aguiar
- Franco V. Saliola
- Li Guo
- Kurusch Ebrahimi-Fard
- Apurba Kumar Das
In Geometry and Topology
- Stuart W. Margolis
- Ben Steinberg
- Swapneel Mahajan
- Marcelo Aguiar
- Franco V. Saliola
- Li Guo
- Kurusch Ebrahimi-Fard
- Mark V. Lawson
Citations in the corpus
- Combinatorial Topology and the Global Dimension of Algebras Arising in Combinatorics (2012, methodology): “The second author [80] computed the Ext-spaces between simple modules in the case of the algebras of face monoids of hyperplane arrangements using the resolutions of Brown and Diaconis coming from the topology of hyperplane arrangements.”
- Quivers of monoids with basic algebras (2011, methodology): “Saliola computed the quiver, first of hyperplane face semigroups [83], and then of a left regular band algebra, and the projective indecomposable modules [81]; for the former he used homological methods, whereas in the latter case he computed primitive idempotents.”
- HOPF MONOIDS IN THE CATEGORY OF SPECIES (2012, background): “ic comonoid may then be reformulated as follows. Coassociativity states that (62) (x|R⊔S)|R= x|R, (x|R⊔S)/R= (x/R)|S, x/R⊔S= (x/R)/S, for any decomposition I= R⊔S⊔Tand x∈Q[I]. Counitality states that (63) x|I= x= x/∅ for any x∈Q[I]. In particular, it follows from (63) that for x∈Q[∅] we have ∆∅,∅(x)”