On the Poincar\'{e} series of Kac-Moody Lie algebras

Abstract

In this paper, we discuss the Poincar\'{e} series of Kac-Moody Lie algebras, especially for indefinite type. Firstly, we compute the Poincar\'{e} series of certain indefinite Kac-Moody Lie algebras whose Cartan matrices have the same type of $2\times 2$ principal sub-matrices. Secondly, we show that the Poincar\'{e} series of Kac-Moody Lie algebras satisfy certain interesting properties. Lastly we give some applications of the Poincar\'{e} series to other fields. Particularly we construct some counter examples to a conjecture of Victor Kac\cite{Kac_85} and a conjecture of Chapavalov, Leites and Stekolshchik\cite{CDR_10}.