Kashiwara, Masaki; Nakashima, Toshiki Crystal graphs for representations of the \(q\)-analogue of classical Lie algebras. (English) Zbl 0808.17005 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) Cited in 11 ReviewsCited in 202 Documents 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 PDF BibTeX XML Cite \textit{M. Kashiwara} and \textit{T. Nakashima}, J. Algebra 165, No. 2, 295--345 (1994; Zbl 0808.17005) Full Text: DOI