Wild quivers: on a conjecture of kac and ringel-hall algebra

Abstract

The results in this thesis center around Kac's conjectures. Although we do not succeed in proving them, we find some equivalent statements, and also some related results which we think are interesting in their own right. After a first introductory chapter containing basic definitions we present in Chapter 2 some more evidence on the positivity conjecture: we prove that it holds for an m-loop quiver with dimension vector up to 5 and m arbitrary. In Chapter 3, which is based on [SV dB99b] we use the theory of symmetric functions to find a combinatorial reformulation of the constant term conjecture. The final statement amounts to a signed counting of certain words. While it seems quite plausible that such a counting may be carried out, we have not succeeded in doing it. An interesting side result of this chapter is an explicit expression for the Hall-Littlewood polynomial corresponding to a hook in terms of Schur functions. This generalizes a conjecture by Carbonara (proved by MacDonald) [Car98] . It is a tantalizing question if a similar result exists for more general partitions. The Ringel-Hall algebra of a quiver is an algebra whose basis consists of the isomorphism classes of indecomposable representations. After adding a diagonal part and performing a Drinfeld double construction one obtains a Hopf algebra which looks very much like the quantum enveloping algebra of a Lie algebra [Gre95, Kap97, Xia97]. In Chapter 4 we carry out this construction in detail. As a result we obtain the precise relations which hold in the resulting algebra. These relations are used in the next chapter. In Chapter 5, based on [SV dB99a], we introduce three isomorphisms between the (double) Ringel-Hall algebras of quivers with the same underlying graph. By composing these isomorphisms we obtain a new construction of Lusztig's braid group action on a quantum enveloping algebra. In Chapter 6, based on [SVdBOl] we show that the Ringel-Hall algebra is a quantized enveloping algebra of a generalized Kac-Moody Lie algebra. Such objects had been introduced by Kang [Kan95J in the generic case. Our methods are different and more general that those of Kang since they also apply in the non-generic case. One of our main results is a Weyl-Borcherds character formula for the Hall algebra. This yields another reformulation of Kac's constant term conjecture, this time in terms of the multiplicities of the imaginary simple roots of the Ringel-Hall algebra. Our work in Chapters 4, 5, 6 has been the basis for [DXOla, DXOlb, DXOlc, XYOl] where our results are generalized to arbitrary finite dimensional hereditary algebras (using similar methods). The connection with our work is given by the path algebra which is a certain hereditary algebra naturally associated to any quiver.