Koszul algebras from graphs and hyperplane arrangements

This work was started as an attempt to apply theory from noncommutative graded algebra to questions about the holonomy algebra of a hyperplane arrangement. We soon realized that these algebras and their deformations, which form a class of quadratic graded algebras, have not been studied much and yet are interesting to algebra, arrangement theory and combinatorics. Let X be a topological space having homotopy type of a finite cell complex. Let Hk(X ) be the homology coalgebra with coefficients in a field and comultiplication dual to the cup product. Then the holonomy Lie algebra G X of X is the quotient of the free Lie algebra on H " (X ) over the ideal generated by the image of the comultiplication H # (X )!Λ#(H " (X )). The universal enveloping algebra U(X ) of G X is called the holonomy algebra of X. Holonomy algebras were introduced to arrangement theory by T. Kohno in [14, 15]. Let ! be an arrangement over #, that is, a set 2H " ,...,H n ́ of linear hyperplanes in a linear space #l. Let X be the complement of Vn i=" H i in #l and let U(!) ̄U(X ). In [14], U(!) is defined explicitly by generators and relations that can be obtained from the combinatorics of !, see Section 4. Recall that there is another graded algebra defined by the combinatorics of !, the Orlik–Solomon algebra A(!) [19]. A well-known theorem of Brieskorn–Orlik–Solomon says A(!) is isomorphic to H*(X,#). In his papers, Kohno studied a complex, Kh (the Aomoto–Kohno complex), of free modules over U(!), defined by K p ̄Hom#(A(!)p,U(!)) for p ̄ 0, 1,..., and K −" ̄#. He proved the acyclicity of this complex for certain classes of reflection arrangements. He also proved that if this complex is acyclic then the Lower Central Series (LCS) formula holds: