Abstract
Given any Coxeter group, we define rigid reflections and rigid roots using non-self-intersecting curves on a Riemann surface with labeled curves. When the Coxeter group arises from an acyclic quiver, they are related to the rigid representations of the quiver. For a family of rank 3 Coxeter groups, we show that there is a surjective map from the set of reduced positive roots of a rank 2 Kac-Moody algebra onto the set of rigid reflections. We conjecture that this map is bijective.
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Acknowledgements
The first author was supported by a grant from the Simons Foundation (Grant No. 318706). The second author was supported by National Science Foundation of USA (Grant No. DMS 1800207), the University of Nebraska-Lincoln, and Korea Institute for Advanced Study. The authors thank the anonymous referees for their helpful comments which improved the exposition of this paper.
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Lee, KH., Lee, K. Rigid reflections and Kac-Moody algebras. Sci. China Math. 62, 1317–1330 (2019). https://doi.org/10.1007/s11425-018-9530-4
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DOI: https://doi.org/10.1007/s11425-018-9530-4