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Quivers with relations for symmetrizable Cartan matrices IV: crystal graphs and semicanonical functions

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Abstract

We generalize Lusztig’s nilpotent varieties, and Kashiwara and Saito’s geometric construction of crystal graphs from the symmetric to the symmetrizable case. We also construct semicanonical functions in the convolution algebras of generalized preprojective algebras. Conjecturally these functions yield semicanonical bases of the enveloping algebras of the positive part of symmetrizable Kac–Moody algebras.

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Acknowledgements

We thank Peter Tingley and Vinoth Nandakumar for providing us with a preliminary version of their preprint [15]. The second and third author thank the Mittag-Leffler Institute for kind hospitality in February/March 2015. The third author thanks the SFB/Transregio TR 45 for financial support. The first author thanks the Mathematical Institute of the University of Bonn for one month of hospitality in June/July 2016.

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Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, 04510, Mexico, D.F., Mexico

    Christof Geiss

  2. CNRS, UMR 6139, LMNO, Univ. de Caen, 14032, Caen Cedex, France

    Bernard Leclerc

  3. Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Jan Schröer

Authors
  1. Christof Geiss
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  2. Bernard Leclerc
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  3. Jan Schröer
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Correspondence to Jan Schröer.

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Geiss, C., Leclerc, B. & Schröer, J. Quivers with relations for symmetrizable Cartan matrices IV: crystal graphs and semicanonical functions. Sel. Math. New Ser. 24, 3283–3348 (2018). https://doi.org/10.1007/s00029-018-0412-4

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  • DOI: https://doi.org/10.1007/s00029-018-0412-4

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