A relation between a conjecture of Kac and the structure of the Hall algebra (Q5939911)

The Hall algebra \(H(Q)\) of a quiver \(Q\) is an algebra defined on isomorphism classes of \(Q\) representations. \textit{C. M. Ringel} proved in Invent. Math. 101, 583-591 (1990; Zbl 0735.16009) that the subalgebra generated by the simple representations is the positive part of the quantized enveloping algebra of a Kac-Moody algebra. In this paper, it is shown that the full Hall algebra is the positive part of a quantized enveloping algebra of a generalized Kac-Moody algebra. The authors use this result to reformulate a conjecture of Kac about the constant coefficient of a polynomial that counts the number of absolutely indecomposable representations of \(Q\) over \(\mathbb F_q\).