an:00839325
Zbl 0840.17010
Ringel, Claus Michael
PBW-bases of quantum groups
EN
J. Reine Angew. Math. 470, 51-88 (1996).
00030311
1996
j
17B37 16S36 20F36
Poincar??-Birkhoff-Witt bases; quantum group; Hall algebra; iterated skew polynomial ring; action of the braid group; Coxeter reflection functor
An explicit construction of Poincar??-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.
S.Khoroshkin (Moskva)