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Canonical bases and standard monomials. (English) Zbl 0904.17016

Let \(U\) be the quantized enveloping algebra associated to a simple Lie algebra \({\mathfrak g}\) by Drinfel’d and Jimbo. Let \(\lambda\) be a classical fundamental weight for \({\mathfrak g}\) and \(V(\lambda)\) the irreducible, finite-dimensional type 1 highest weight \(U\)-module with highest weight \(\lambda\). Then \(V(\lambda)\) possesses a canonical basis (see §0 of M. Kashiwara [Duke Math J. 63, 465-516 (1991; Zbl 0739.17005)] and §14.4.12 in G. Lusztig [Introduction to quantum groups (Prog. Math. 110, Birkhäuser, Boston) (1993; Zbl 0788.17010)]) and a standard monomial basis (see §§2.4 and 2.5 in V. Lakshmibai and N. Reshetikhin [Contemp. Math. 134, 145-181 (1992; Zbl 0792.17012)]).
We show that the canonical basis and the standard monomial basis for \(V(\lambda)\) coincide. We also show that this can fail for non-classical fundamental modules.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
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References:

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