Universal PBW-bases of Hall-Ringel algebras and Hall polynomials
By results of C. M. Ringel and of J. A. Green, the positive part \(U^+\) of a quantized enveloping algebra of a symmetrizable Kac-Moody algebra is isomorphic to the composition algebra \(\mathcal C\) of an associated quiver \(\Gamma\). This composition algebra is contained in and in the case of a Dynkin diagram isomorphic to the Hall algebra \(\mathcal H\) (defined by Ringel and hence also named `Hall-Ringel algebra') of the same quiver. There is a more general definition of such a Hall algebra which works for finitary rings. The authors generalize several results which were known before in more special situations. The Hall algebra by definition has a basis given by isomorphism classes of representations of \(\Gamma\). The authors show that over the field of rational numbers there is a PBW-like basis consisting of ordered products of indecomposable representations. Moreover, it is shown that \(\mathcal C\) coincides with \(\mathcal H\) if the category of representations of \(\Gamma\) has no short cycle (in which case indecomposable modules are determined by their composition factors). Furthermore, the existence of Hall polynomials is proved in some cases, in particular in the case of module categories without short cycles.
scientific article; zbMATH DE number 1106613