A relation between a conjecture of Kac and the structure of the Hall algebra

Abstract

In this paper we show that the Hall algebra of a quiver, as defined by Ringel, is the positive part of the quantived enveloping algebra of a generalized Kac-Moody Lie algebra. We give a potential application of this result to a conjecture of Kac which states that the constant coefficient of the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra. (C) 2001 Elsevier Science B.V. All rights reserved.