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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Integral homology of loop groups via Langlands dual groups
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by Zhiwei Yun and Xinwen Zhu PDF
Represent. Theory 15 (2011), 347-369 Request permission

Abstract:

Let $K$ be a connected compact Lie group, and $G$ its complexification. The homology of the based loop group $\Omega K$ with integer coefficients is naturally a $\mathbb {Z}$-Hopf algebra. After possibly inverting $2$ or $3$, we identify $H_*(\Omega K,\mathbb {Z})$ 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 $e\in \operatorname {Lie} B^\vee$. We also give a similar interpretation for the equivariant homology of $\Omega K$ under the maximal torus action.
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Additional Information
  • Zhiwei Yun
  • Affiliation: Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
  • Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
  • MR Author ID: 862829
  • Email: zyun@math.mit.edu
  • Xinwen Zhu
  • Affiliation: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138
  • MR Author ID: 868127
  • Email: xinwenz@math.harvard.edu
  • Received by editor(s): September 29, 2009
  • Received by editor(s) in revised form: October 24, 2010
  • Published electronically: April 20, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 15 (2011), 347-369
  • MSC (2010): Primary 57T10, 20G07
  • DOI: https://doi.org/10.1090/S1088-4165-2011-00399-X
  • MathSciNet review: 2788897