Integral homology of loop groups via Langlands dual groups

2009/09/30 by Zhiwei Yun, Yun, Zhiwei, Xinwen Zhu +1 · 3 citations
Mathematics · #20G07 #57T10 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #math.AT #math.RT #msc:20G07 #msc:57T10

paper · pdf · doi:10.48550/arxiv.0909.5487

23 pages

arxiv created 2009/09/30 · openalex publication_date 2009/09/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a connected compact Lie group, and G be its complexification. The homology of the based loop group ΩK with integer coefficients is naturally a \ZZ-Hopf algebra. After possibly inverting 2 or 3, we identify H_*(ΩK,\ZZ) with the Hopf algebra of algebraic functions on B^\vee_e, where B^\vee is a Borel subgroup of the Langlands dual group scheme G^\vee of G and B^\vee_e is the centralizer in B^\vee of a regular nilpotent element e∈\Lie B^\vee. We also give a similar interpretation for the equivariant homology of ΩK under the maximal torus action.

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