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Cited by 26Homotopy and Cohomology in Algebraic TopologyGeometric and Algebraic TopologyAdvanced Combinatorial Mathematics

Lie algebras associated to fiber-type arrangements

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Daniel C. Cohen, Fred Cohen, Miguel A. Xicoténcatl·2003·International Mathematics Research Notices
DOI 10.1155/s1073792803208102
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Abstract

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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Authors
Cohen, D. C., Cohen, F., & Xicoténcatl, M. A.
Affiliations
Max Planck Society (DE)
ORCID
Venue
International Mathematics Research Notices
Publisher
Oxford University Press (OUP)
Published
2003
Volume
2003
Issue
29
Pages
1591
ISSN
1073-7928 (ISSN-L)
DOI
10.1155/s1073792803208102
OpenAlex
W2098754215
Cited by
26
References
1
Citations by year
2025: 2 · 2019: 2 · 2017: 1 · 2016: 1
FWCI
3.91
Type
journal article
Language
en
Open access
Locations
  • International Mathematics Research Notices (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • Civil War Book Review (submittedVersion)
  • Max Planck Institute for Plasma Physics (submittedVersion)
  • MPG.PuRe (Max Planck Society) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • National Security Agency
  • Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional
Topics
  • Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
  • Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
  • Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Advanced Combinatorial Mathematics
Keywords
Lie algebras, subspace arrangements, descending central series, fundamental group, loop space homology, primitive elements, homotopy groups, Poisson algebras

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