# 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 notonly 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.

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

Revised on October 11, 2011 22:11:17 by Zoran Škoda (161.53.130.104)