Lie algebras associated to fiber-type arrangements

TL;DR

This paper explores the relationship between Lie algebras derived from fiber-type hyperplane arrangements and the topology of associated subspace arrangements, revealing new algebraic structures linked to homotopy and loop spaces.

Contribution

It establishes an isomorphism between the Lie algebra from the fundamental group of a fiber-type arrangement's complement and the primitive elements in the homology of a related loop space, extending understanding of their algebraic topology.

Findings

Lie algebra from fundamental group is isomorphic to primitive homology elements

Homotopy groups modulo torsion relate to the Lie algebra of the subspace arrangement

Looping yields a Poisson algebra with generalized braid relations

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."