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.


17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
Full Text: DOI


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.