LIE ALGEBRAS ASSOCIATED TO FIBER-TYPE ARRANGEMENTS

Abstract

Given a hyperplane arrangement in a complex vector space of dimen- sion ', there is a natural associated arrangement of codimension k subspaces in a complex vector space of dimension k'. Topological invariants of the comple- ment of this subspace arrangement are related to those of the complement of the original hyperplane arrangement. In particular, if the hyperplane arrangement is ber-type, then, apart from grading, the Lie algebra obtained from the descend- ing central series for the fundamental group of the complement of the hyperplane arrangement is isomorphic to the Lie algebra of primitive elements in the homol- ogy of the loop space for the complement of the associated subspace arrangement. Furthermore, this last Lie algebra is given by the homotopy groups modulo tor- sion of the loop space of the complement of the subspace arrangement. Looping further yields an associated Poisson algebra, and generalizations of the \universal innitesimal Poisson braid relations.