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Global crystal bases of quantum groups. (English) Zbl 0774.17018
The author constructs the global crystal bases of the $$q$$-analogue $$A_ q({\mathfrak g})$$ of the coordinate ring of the reductive algebraic group associated with the Lie algebra $${\mathfrak g}$$ by using the same method as in his paper [Duke Math. J. 63, 465-516 (1991; Zbl 0739.17005)]. The main theorem of this paper is in §7.4:
Theorem 1. (i) $$A_ q^{\mathbb{Q}}({\mathfrak g})\cap L(A_ q({\mathfrak g}))\cap \overline{L}(A_ q({\mathfrak g})) \to L(A_ q({\mathfrak g}))/qL(A_ q({\mathfrak g}))$$ is an isomorphism. (ii) Letting $$G$$ be the inverse of the isomorphism above, we have $$A_ q^{\mathbb{Q}}({\mathfrak g})= \oplus_{b\in B(A_ q({\mathfrak g}))} \mathbb{Q}[q,q^{-1}] G(b)$$. The author also gives an explicit form of the global crystal base of $$A_ q(sl_ 2)$$ and examines Berenstein and Zelevinsky’s conjecture as an example.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations
Zbl 0739.17005
Full Text:
##### References:
 [1] A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of $$\mathrm GL_ n$$ , Duke Math. J. 61 (1990), no. 2, 655-677. · Zbl 0713.17012 [2] M. Kashiwara, Crystalizing the $$q$$-analogue of universal enveloping algebras , Comm. Math. Phys. 133 (1990), no. 2, 249-260. · Zbl 0724.17009 [3] M. Kashiwara, On crystal bases of the $$q$$-analogue of universal enveloping algebras , Duke Math. J. 63 (1991), no. 2, 465-516. · Zbl 0739.17005 [4] M. Kashiwara, Crystallizing the $$q$$-analogue of universal enveloping algebras , Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, pp. 791-797. · Zbl 0749.17017 [5] G. Lusztig, Canonical bases in tensor products , Proc. Nat. Acad. Sci. U.S.A. 89 (1992), no. 17, 8177-8179. JSTOR: · Zbl 0760.17011
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