A note on a theorem of Bourbaki

TL;DR

This paper discusses a theorem of Bourbaki in the context of Poincare series of Hyperbolic Lie algebras, highlighting their ratio form involving finite Lie algebra series and polynomials.

Contribution

It provides remarks on Bourbaki's theorem, connecting it to the structure of Poincare series of Hyperbolic Lie algebras and finite Lie algebras.

Findings

Poincare series of Hyperbolic Lie algebras are ratios involving finite Lie algebra series and polynomials

Examples illustrate the relationship between Hyperbolic and finite Lie algebra Poincare series

Remarks clarify the application of Bourbaki's theorem to these series

Abstract

We have recently show that Poincare series of Hyperbolic Lie algebras have the form of a ratio between Poincare series of a chosen finite Lie algebra and a polynomial of finite degree. By the aid of some properly chosen examples, we now give some remarks on a related theorem of Bourbaki.