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PBW-bases of quantum groups. (English) Zbl 0840.17010
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.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16S36 Ordinary and skew polynomial rings and semigroup rings
20F36 Braid groups; Artin groups
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