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