# On Poincare Polynomials of Hyperbolic Lie Algebras

Meltem Gungormez and Hasan R. Karadayi

Arxiv ID: 0706.2563•Last updated: 5/21/2021

We have general frameworks to obtain Poincare polynomials for Finite and also
Affine types of Kac-Moody Lie algebras. Very little is known however beyond
Affine ones, though we have a constructive theorem which can be applied both
for finite and infinite cases. One can conclusively said that theorem gives the
Poincare polynomial P(G) of a Kac-Moody Lie algebra G in the product form
P(G)=P(g) R where g is a precisely chosen sub-algebra of G and R is a rational
function. Not in the way which theorem says but, at least for 48 hyperbolic Lie
algebras considered in this work, we have shown that there is another way of
choosing a sub-algebra in such a way that R appears to be the inverse of a
finite polynomial. It is clear that a rational function or its inverse can not
be expressed in the form of a finite polynomial. Our method is based on
numerical calculations and results are given for each and every one of 48
Hyperbolic Lie algebras. In an illustrative example however, we will give how
above-mentioned theorem gives us rational functions in which case we find a
finite polynomial for which theorem fails to obtain.

#### PaperStudio AI Chat

I'm your research assistant! Ask me anything about this paper.