A PBW basis for Lusztig's form of untwisted affine quantum groups

Abstract

Let $\mathfrak{g}$ be an untwisted affine Kac-Moody algebra over the field $K$, and let $U_q(\mathfrak{g})$ be the associated quantum enveloping algebra; let ${\mathfrak{U}}_q(g)$ be the Lusztig's integer form of $U_q(\mathfrak{g})$, generated by $q$-divided powers of Chevalley generators over a suitable subring $R$ of $K(q)$. We prove a Poincaré-Birkhoff-Witt like theorem for ${\mathfrak{U}}_q(\mathfrak{g})$, yielding a basis over $R$ made of ordered products of $q$-divided powers of suitable quantum root vectors.

22 pages, AMS-TeX C, Version 2.1c. This is the author's final version, corresponding to the printed journal version

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