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ROOT SYSTEMS AND SYMMETRIES OF TORUS MANIFOLDS

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Abstract

We associate a root system to a finite set in a free abelian group and prove that its irreducible subsystem is of type A, B, or D. Applying this general result to a torus manifold M, where a torus manifold is a 2n-dimensional connected closed smooth manifold with a smooth effective action of an n-dimensional compact torus having a fixed point, we introduce a root system R(M) for M and show that if the torus action on M extends to a smooth action of a connected compact Lie group G, then the root system of G is a subsystem of R(M) so that any irreducible factor of the Lie algebra of G is of type A, B, or D. Moreover, we show that only type A appears if H*(M) is generated by H 2(M) as a ring. We also discuss a similar problem for a torus manifold with an invariant stable complex structure. Only type A appears in this case, too.

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Authors and Affiliations

  1. Department of Applied Mathematics, Okayama University of Science, 1-1 Ridai-cho, Kita-ku, Okayama-shi, Okayama, 700-0005, Japan

    SHINTARÔ KUROKI

  2. Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka, 558-8585, Japan

    MIKIYA MASUDA

Authors
  1. SHINTARÔ KUROKI
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  2. MIKIYA MASUDA
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Correspondence to SHINTARÔ KUROKI.

Additional information

(SHINTARÔ KUROKI) Supported in part by JSPS KAKENHI Grant Number 15K17531, 24224002, and the JSPS Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation “Deepening and Evolution of Mathematics and Physics, Building of International Network Hub based on OCAMI”, and the bilateral program between Japan and Russia: “Topology and geometry of torus actions and combinatorics of orbit quotients”.

(MIKIYA MASUDA) Supported by Grant-in-Aid for Scientific Research 25400095.

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KUROKI, S., MASUDA, M. ROOT SYSTEMS AND SYMMETRIES OF TORUS MANIFOLDS. Transformation Groups 22, 453–474 (2017). https://doi.org/10.1007/s00031-016-9387-4

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