Negative Ricci curvature on some non-solvable Lie groups II

Abstract

We construct many examples of Lie groups admitting a left-invariant metric of negative Ricci curvature. We study Lie algebras which are semidirect products ${\mathfrak {l}}= ({\mathfrak {a}} \oplus {\mathfrak {u}} ) < imes {\mathfrak {n}}$ and we obtain examples where ${\mathfrak {u}}$ is any semisimple compact real Lie algebra, ${\mathfrak {a}}$ is one-dimensional and ${\mathfrak {n}}$ is a representation of ${\mathfrak {u}}$ which satisfies some conditions. In particular, when ${\mathfrak {u}} = {{\mathfrak {s}}}{{\mathfrak {u}}}(m)$, ${{\mathfrak {s}}}{{\mathfrak {o}}} (m)$ or ${{\mathfrak {s}}}{{\mathfrak {p}}} (m)$ and ${\mathfrak {n}}$ is a representation of ${\mathfrak {u}}$ in some space of homogeneous polynomials, we show that these conditions are indeed satisfied. In the case ${\mathfrak {u}} = {{\mathfrak {s}}}{{\mathfrak {u}}}(2)$ we get a more general construction where ${\mathfrak {n}}$ can be any nilpotent Lie algebra where ${{\mathfrak {s}}}{{\mathfrak {u}}}(2)$ acts by derivations. We also prove a general result in the case when ${\mathfrak {u}}$ is a semisimple Lie algebra of non-compact type.