Journal Article

Lie algebras associated to fiber-type arrangements

Daniel C. Cohen, Frederick R. Cohen and Miguel Xicoténcatl

in International Mathematics Research Notices

Volume 2003, issue 29, pages 1591-1621
Published in print January 2003 | ISSN: 1073-7928
Published online January 2003 | e-ISSN: 1687-0247 | DOI: https://dx.doi.org/10.1155/S1073792803208102

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Given a hyperplane arrangement in a complex vector space of dimension ℓ, there is a natural associated “redundant” arrangement of codimension k subspaces in a complex vector space of dimension kℓ. 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 which satisfies generalizations of the “universal infinitesimal Poisson braid relations.”

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Subjects: Mathematics