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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
Citations:
Zbl 0739.17005
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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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