On the double of the Hall algebra of a quiver
Let \(Q\) be a quiver and \(k\) be a finite field. Then Ringel defined the twisted Hall algebra \(H(Q)\) of \(Q\), and Green proved that \(H(Q)\) is a Hopf algebra in a certain twisted sense. Furthermore, one introduced certain Cartan elements to make \(H(Q)\) into a real Hopf algebra. In the paper under review the authors give a double construction of this Hopf algebra, and denote the double by \(U(Q)\). The main interest of the paper is the following: if two quivers \(Q\) and \(Q'\) have the same underlying graph, how are the two doubles related to each other? The authors show that one can use finitely many times the usual reflection functor associated to a source vertex to define an isomorphism between \(U(Q)\) and \(U(Q')\). Furthermore, the authors define an automorphism \(\widehat t_i\) of \(U(Q)\) for each vertex \(i\) in \(Q\) and show that the \(\widehat t_i\) restrict to the automorphism of the quantum enveloping algebra, which is Lusztig's braided group action. In fact, the authors prove that the \(\widehat t_i\) satisfy the braid relations for Dynkin cases on the whole of \(U(Q)\) instead of just for the quantum enveloping algebra. They conjecture also that this is true for all cases. Perhaps with the knowledge of representation theory of algebras and a careful analysis of the paper under review one might solve this conjecture.
scientific article; zbMATH DE number 1384442