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


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