On Poincare Polynomials of Hyperbolic Lie Algebras

TL;DR

This paper develops a framework for calculating Poincare polynomials of hyperbolic Lie algebras, revealing new algebraic structures and providing explicit results for 48 cases using numerical methods.

Contribution

It extends existing methods to hyperbolic Lie algebras, showing how to choose sub-algebras to express Poincare polynomials as rational functions or their inverses.

Findings

Poincare polynomials for 48 hyperbolic Lie algebras are computed.

A new approach to express these polynomials as rational functions or inverses.

Numerical methods are used to analyze algebraic structures beyond affine types.

Abstract

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.