Title: Lie algebras associated to fiber-type arrangements
Authors: Daniel C. Cohen, Frederick R. Cohen, Miguel Xicotencatl
AMS Classification numbers:
Primary 52C35, 55P35; Secondary 20F14, 20F36, 20F40
math.AT/0005091
Addresses of Authors
D. Cohen, Department of Mathematics, Louisiana State University,
Baton Rouge, LA 70803
F. Cohen, Department of Mathematics, University of Rochester,
Rochester, NY 14627
M. Xicotencatl, Depto. de Mathematicas, Cinvestav del IPN, Mexico City
Max-Plank-Institut fur Mathematik, P.O. Box 7280, D-53072 Bonn, Germany
Email address of Authors
cohen@math.lsu.edu
cohf@math.rochester.edu
xico@@mpim-bonn.mpg.de
Abstract:
Given a hyperplane arrangement in a complex vector space of dimension n,
there is a natural associated arrangement of codimension k subspaces in a
complex vector space of dimension k*n. Topological invariants of the
complement of this subspace arrangement are related to those of the
complement of the original hyperplane arrangement. In particular, if the
hyperplane arrangement is fiber-type, then, apart from grading, the Lie
algebra obtained from the descending central series for the fundamental
group of the complement of the hyperplane arrangement is isomorphic to the
Lie algebra of primitive elements in the homology of the loop space for the
complement of the associated subspace arrangement. Furthermore, this last
Lie algebra is given by the homotopy groups modulo torsion of the loop
space of the complement of the subspace arrangement. Looping further
yields an associated Poisson algebra, and generalizations of the
"universal infinitesimal Poisson braid relations."