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Crystalizing the q-analogue of universal enveloping algebras. (English) Zbl 0724.17009

Let \(U_ q\) denote the quantized enveloping algebra over \({\mathbb{Q}}(q)\) associated to a symmetrizable Kac-Moody algebra \({\mathfrak g}\). For any integrable \(U_ q\)-module M the author defines a crystal base for M to be a pair (L,B) consisting of a lattice L of M and a \({\mathbb{Q}}\)-basis B of L/qL with certain nice properties. In the paper under review the author proves the existence and uniqueness of crystal bases for the case where \({\mathfrak g}\) is a finite dimensional classical Lie algebra, and in the paper reviewed above he announces the extension to the general case [see also Preprint 728, Res. Inst. Math. Sci., Kyoto Univ. for details and proofs)].
G. Lusztig has constructed a so-called canonical basis for the \(+\) part of \(U_ q\) (for types A, D and E), see [J. Am. Math. Soc. 3, 447- 498 (1990; Zbl 0703.17008) and Prog. Theor. Phys. Suppl. 102, 175-201 (1990)]. On the irreducible integrable highest weight modules the two constructions lead to the same bases.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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[1] Date, E., Jimbo, M., Miwa, T.: Representations ofU q(gl(n, C)) atq=0 and the Robinson-Schensted correspondence, to appear in Physics and Mathematics of Strings, Memorial Volume of Vadim Knizhnik. Brink, L., Friedan, D., Polyakov A. M. (eds.). Singapore: World Scientific · Zbl 0743.17018
[2] Drinfeld, V. G.: Hopf algebra and the Yang-Baxter equation. Soviet Math. Dokl.32, 254–258 (1985)
[3] Jimbo, M.: Aq-difference analogue of UG and the Yang-Baxter equation. Lett. Math. Phys.10, 63–69 (1985) · Zbl 0587.17004
[4] Lusztig, G.: Quantum groups at roots of 1, preprint · Zbl 0714.17013
[5] Rosso, M.: Analogues de la forme de Killing et, du théorème d’Harish-Chandra pour les groupes quantiques, preprint · Zbl 0721.17012
[6] Reshetikhin, N. Yu.: Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I, preprint
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