On Poincare Polinomials of Hyperbolic Lie Algebras

TL;DR

This paper explicitly calculates Poincare polynomials for hyperbolic Lie algebras, revealing a notable pattern where these polynomials resemble ratios of finite Poincare polynomials and polynomials of finite degree.

Contribution

It provides explicit formulas for Poincare polynomials of hyperbolic Lie algebras $HA_2$ and $HA_3$, highlighting their structured form and relation to finite Lie algebra polynomials.

Findings

Poincare polynomials for $HA_2$ and $HA_3$ are explicitly computed.

These polynomials are ratios involving finite Poincare polynomials.

The orders of these polynomials are 11 and 19 for $HA_2$ and $HA_3$, respectively.

Abstract

Poincare polinomials of hyperbolic Lie algebras, which are given by $HA_2$ and $HA_3$ in the Kac's notation, are calculated explicitly. The results show that there is a significant form for hyperbolic Poincare polinomials. Their explicit forms tend to be seen as the ratio of a properly chosen finite Poincare polinomial and a polinomial of finite degree. To this end, by choosing the Poincare polinomials of $D_4$ and $D_5$ Lie algebras, we show that these polinomials come out to be of order 11 and 19 respectively for $HA_2$ and $HA_3$.