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Non-abelian GKM theory

Abstract

We describe a generalization of GKM theory for actions of arbitrary compact connected Lie groups. To an action satisfying the non-abelian GKM conditions we attach a graph encoding the structure of the non-abelian 1-skeleton, i.e., the subspace of points with isotopy rank at most one less than the rank of the acting group. We show that the algebra structure of the equivariant cohomology can be read off from this graph. In comparison with ordinary abelian GKM theory, there are some special features due to the more complicated structure of the non-abelian 1-skeleton.

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Notes

  1. Note that there is a slight error in [7, Proposition 4.2]: on the right hand side of the equation one has to consider the subgroup of \(N_G(\mathfrak {t}_p)\) consisting of those elements which leave invariant \(M^{\mathfrak {t}_p,p}\) instead of the whole normalizer \(N_G(\mathfrak {t}_p)\).

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Acknowledgments

We wish to thank the anonymous referee for suggesting several improvements.

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Correspondence to Augustin-Liviu Mare.

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Goertsches, O., Mare, AL. Non-abelian GKM theory. Math. Z. 277, 1–27 (2014). https://doi.org/10.1007/s00209-013-1242-x

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  • DOI: https://doi.org/10.1007/s00209-013-1242-x

Mathematics Subject Classification (2010)

  • 57S15
  • 55N91
  • 57R91