Crystal graphs for representations of the \(q\)-analogue of classical Lie algebras. (English) Zbl 0808.17005

Kashiwara, Masaki; Nakashima, Toshiki

J. Algebra 165, No. 2, 295-345 (1994).

The notion of a crystal base was introduced by Kashiwara and proved to exist uniquely for any integrable highest weight representation of the \(q\)-analogue of symmetrizable Kac-Moody Lie algebras. In the present paper the authors give their explicit description for finite dimensional irreducible representations of \(A_ n\), \(B_ n\), \(C_ n\) and \(D_ n\). In particular, in the \(A_ n\)-case the crystal bases are labelled by the semistandard tableaux. The description is concrete and self-contained.

Reviewer: H.Yamada (Tokyo)

MSC:

17B37

Quantum groups (quantized enveloping algebras) and related deformations

Keywords:

crystal graph; \(q\)-analogue of classical Lie algebras; crystal base; irreducible representations; semistandard tableaux