On Poincare Polynomials of Hyperbolic Lie AlgebrasGungormez M* and Karadayi HR
Department of Physics, Faculty of Sciences and Letters, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey
- *Corresponding Author:
- Gungormez M
Department of Physics
Faculty of Sciences and Letters
Istanbul Technical University
Maslak, 34469 Istanbul, Turkey
Tel: +90 212 2853220
Fax: +90 212 2856386
E-mail: [email protected]
Received Date: January 11, 2017; Accepted Date: February 17, 2017; Published Date: February 27, 2017
Citation: Gungormez M, Karadayi HR (2017) On Poincare Polynomials of Hyperbolic Lie Algebras. J Generalized Lie Theory Appl 11: 255. doi:10.4172/1736-4337.1000255
Copyright: © 2017 Gungormez M, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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.