LAUSR

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