Ringel, Claus Michael PBW-bases of quantum groups. (English) Zbl 0840.17010 J. Reine Angew. Math. 470, 51-88 (1996). An explicit construction of Poincaré-Birkhoff-Witt bases for the positive part \(U^+_q ({\mathfrak g})\) of a quantum group via Hall algebra approach is presented. The Hall algebra description of \(U^+_q ({\mathfrak g})\), introduced by C. M. Ringel, starts from an hereditary algebra naturally attached to an oriented Dynkin graph. The properties of the category of representations of this hereditary algebra allow one to describe various algebraic properties of PBW bases. In particular, \(U^+_q ({\mathfrak g})\) is described as iterated skew polynomial ring; the action of the braid group is interpreted in terms of the Coxeter reflection functor; different PBW data are connected with exceptional sequences in the category of representations of hereditary algebra and so on. The precise correspondence with other constructions of PBW bases is given in details. Reviewer: S.Khoroshkin (Moskva) Cited in 3 ReviewsCited in 76 Documents MSC: 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16S36 Ordinary and skew polynomial rings and semigroup rings 20F36 Braid groups; Artin groups Keywords:Poincaré-Birkhoff-Witt bases; quantum group; Hall algebra; iterated skew polynomial ring; action of the braid group; Coxeter reflection functor PDFBibTeX XMLCite \textit{C. M. Ringel}, J. Reine Angew. Math. 470, 51--88 (1996; Zbl 0840.17010) Full Text: DOI Crelle EuDML