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 l= (a⊕ u) ⋉ n and we obtain examples where u is any semisimple compact real Lie algebra, a is one-dimensional and n is a representation of u which satisfies some conditions. In particular, when u= su(m) , so(m) or sp(m) and n is a representation of u in some space of homogeneous polynomials, we show that these conditions are indeed satisfied. In the case u= su(2) we get a more general construction where n can be any nilpotent Lie algebra where su(2) acts by derivations. We also prove a general result in the case when u is a semisimple Lie algebra of non-compact type.