Negative Ricci curvature on some non-solvable Lie groups II

TL;DR

This paper constructs numerous examples of non-solvable Lie groups with negative Ricci curvature, expanding understanding of geometric properties of such groups and their Lie algebras.

Contribution

It introduces new methods to produce Lie groups with negative Ricci curvature, including cases with Levi factors su(n) or so(n) acting on nilradicals, and generalizes to semisimple Levi factors.

Findings

Examples of Lie groups with negative Ricci curvature constructed.

General results for semisimple Levi factors of non-compact type.

Extension of constructions to nilpotent Lie algebras in su(2) case.

Abstract

We construct many examples of Lie groups with compact Levi factor admitting a left-invariant metric with negative Ricci curvature. We start with a Lie algebra with Levi factor su(n) or so(n) acting on an abelian nilradical via the representation on the space of homogeneous polynomials. In the case of su(2) we obtain a more general construction where the nilradical can be any nilpotent Lie algebra. We also prove a general result in the case when the Levi factor is a semisimple Lie algebra of non-compact type.