nLab crystal basis

Crystal bases are a construction in the representation theory of quantum groups (which in a specialization exist hence for usual Lie groups as well): roughly speaking they provide a uniform description not only of irreducible finite-dimensional modules but also a uniform choice of bases in all of them as well as in tensor products.

There are two mutually dual versions due Masaki Kashiwara and George Lusztig.

History and references

The terminology is due to the involved $q=0$ limit of quantum groups used in the construction (the classical case is $q=1$). In a thermodynamic parlance zero temperature would involve passing to crystalization.

• Masaki Kashiwara, Crystalizing the q-analogue of universal enveloping algebras, Commun. Math. Phys. 133 (1990) 249-260, proj euclic, pdf

• M. Kashiwara, On crystal bases, Representations of Groups, Proc. of the 1994 Annual Seminar of the Canadian Math. Soc. Ban 16 (1995) 155–197, Amer. Math. Soc., Providence, RI. (pdf, ps.gz)

• Masaki Kashiwara, Global crystal bases of quantum groups, Duke Math. J. 69 (1993), no. 2, 455–485, link.

• M. Kashiwara, The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J. 1993.

• Masaki Kashiwara, Yoshihisa Saito, Geometric construction of crystal bases, Duke Math. J. 1996, pdf cached

• S-J.Kang, M. Kashiwara, K.Misra, Crystal bases of Verma modules for quantum affine Lie algebras, Compositio Math. 92 (1994) 299–325, numdam

• Hong and Kang, Introduction to quantum groups and crystal bases, Grad. Studies in Math. 42, AMS 2002, 307 pp.

• John R. Stembridge, A local characterization of simply-laced crystals, Trans. Amer. Math. Soc. 355 (2003), 4807–4823.