$q$-deformation of Aomoto complex

Abstract

A degree one element of the Orlik-Solomon algebra of a hyperplane arrangement defines a cochain complex known as the Aomoto complex. The Aomoto complex can be considerd as the ``linear approximation'' of the twisted cochain complex with coefficients in a complex rank one local system. In this paper, we discuss $q$-deformations of the Aomoto complex. The $q$-deformation is defined by replacing the entries of representation matrices of the coboundary maps with their $q$-analogues. While the resulting maps do not generally define cochain complexes, for certain special basis derived from real structures, the $q$-deformation becomes again a cochain complex. Moreover, it exhibits universality in the sense that any specialization of $q$ to a complex number yields the cochain complex computing the corresponding local system cohomology group.