On the Poincare series of Kac-Moody Lie algebras
In this paper, we discuss the Poincare series of Kac-Moody Lie algebras, especially for indefinite type. Firstly, we compute the Poincare series of certain indefinite Kac-Moody Lie algebras whose Cartan matrices have the same type of 2 × 2 principal sub-matrices. Secondly, we show that the Poincare series of Kac-Moody Lie algebras satisfy certain interesting properties. Lastly we give some applications of the Poincare series to other fields. Particularly we construct some counter examples to a conjecture of Victor Kac(5) and a conjecture of Chapavalov, Leites and Stekolshchik(16).