Polyhedral Realizations of Crystal Bases for Quantized Kac-Moody Algebras

@article{Nakashima1997PolyhedralRO,
  title={Polyhedral Realizations of Crystal Bases for Quantized Kac-Moody Algebras},
  author={Toshiki Nakashima and Andrei Zelevinsky},
  journal={Advances in Mathematics},
  year={1997},
  volume={131},
  pages={253-278}
}
Let B(\infty) be the crystal corresponding to the nilpotent part of a quantized Kac-Moody algebra. We suggest a general way to represent B(\infty) as the set of integer solutions of a system of linear inequalities. As an application, we treat in a unified manner all Kac-Moody algebras of rank 2 (sharpening the result by Kashiwara), as well as the algebras of types A_n and A_{n-1}^{(1)}. 
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