PBW basis of quantized universal enveloping algebras
Let \(U_ q= U_ q({\mathfrak g})\) be the quantized universal enveloping algebra of the Lie algebra \({\mathfrak g}\) and \(U_ q^ +\) (resp. \(U_ q^ -\)) be the subalgebra generated by \(e_ i\)'s (resp. \(f_ i\)'s). \textit{G. Lusztig} constructed, by using his PBW base, the canonical base of \(U_ q^ -\) in the case \({\mathfrak g}\) is a finite dimensional Lie algebra of ADE type [J. Am. Math. Soc. 3, 447-498 (1990; Zbl 0703.17008)]. On the other hand \textit{M. Kashiwara} constructed the global crystal base of \(U_ q^ -\) for an arbitrary symmetrizable Kac-Moody Lie algebra \({\mathfrak g}\) [Duke Math. J. 63, 465-516 (1991; Zbl 0739.17005)]. It was proved by \textit{G. Lusztig} [Prog. Theor. Phys. 102, Suppl., 175-201 (1990; Zbl 0776.17012)] that these two bases are the same in the ADE-case. In this paper the author shows that the monomials of the root vectors form a PBW base of \(U_ q^ -\) and give a crystal base at \(q=0\) for the case of finite dimensional \({\mathfrak g}\).
scientific article; zbMATH DE number 621957