Communications in Mathematical Physics

, Volume 133, Issue 2, pp 249–260

Crystalizing theq-analogue of universal enveloping algebras

  • Masaki Kashiwara

DOI: 10.1007/BF02097367

Cite this article as:
Kashiwara, M. Commun.Math. Phys. (1990) 133: 249. doi:10.1007/BF02097367


For an irreducible representation of theq-analogue of a universal enveloping algebra, one can find a canonical base atq=0, named crystal base (conjectured in a general case and proven forAn, Bn, Cn andDn). The crystal base has a structure of a colored oriented graph, named crystal graph. The crystal base of the tensor product (respectively the direct sum) is the tensor product (respectively the union) of the crystal base. The crystal graph of the tensor product is also explicitly described. This gives a combinatorial description of the decomposition of the tensor product into irreducible components.

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Masaki Kashiwara
    • 1
  1. 1.RIMSKyoto UniversityKyotoJapan

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