## Duke Mathematical Journal

### Crystal bases and two-sided cells of quantum affine algebras

#### Abstract

Let $\mathfrak{g}$ be an affine Kac-Moody Lie algebra. Let $\mathbf{U}^+$ be the positive part of the Drinfeld-Jimbo quantum enveloping algebra associated to $\mathfrak{g}$. We construct a basis of $\mathbf{U}^+$ which is related to the Kashiwara-Lusztig global crystal basis (or canonical basis) by an upper-triangular matrix (with respect to an explicitly defined ordering) with 1's on the diagonal and with above-diagonal entries in $q_s^{-1} \mathbf{Z}[q_s^{-1}]$. Using this construction, we study the global crystal basis $\mathscr{B}(\widetilde{\mathbf{U}})$ of the modified quantum enveloping algebra defined by Lusztig. We obtain a Peter-Weyl-like decomposition of the crystal $\mathscr{B}(\widetilde{\mathbf{U}})$ (Th. 4.18), as well as an explicit description of two-sided cells of $\mathscr{B}(\widetilde{\mathbf{U}})$ and the limit algebra of $\widetilde{\mathbf{U}}$ at $q=0$ (Th. 6.44).

#### Article information

Source
Duke Math. J., Volume 123, Number 2 (2004), 335-402.

Dates
First available in Project Euclid: 11 June 2004

https://projecteuclid.org/euclid.dmj/1086957676

Digital Object Identifier
doi:10.1215/S0012-7094-04-12325-2X

Mathematical Reviews number (MathSciNet)
MR2066942

Zentralblatt MATH identifier
1062.17006

#### Citation

Beck, Jonathan; Nakajima, Hiraku. Crystal bases and two-sided cells of quantum affine algebras. Duke Math. J. 123 (2004), no. 2, 335--402. doi:10.1215/S0012-7094-04-12325-2X. https://projecteuclid.org/euclid.dmj/1086957676

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