Journal of Lie Theory

\def\g{{\frak g}} \def\s{{\frak s}} Given a Lie algebra $\s$, we call Lie $\s$-algebra a Lie algebra endowed with a reductive action of $\s$. We characterize the minimal $\s$-Lie algebras with a nontrivial action of $\s$, in terms of irreducible representations of $\s$ and invariant alternating forms. \endgraf As a first application, we show that if $\g$ is a Lie algebra over a field of characteristic zero whose amenable radical is not a direct factor, then $\g$ contains a subalgebra which is isomorphic to the semidirect product of $\frak{sl}_2$ by either a nontrivial irreducible representation or a Heisenberg group (this was essentially due to Cowling, Dorofaeff, Seeger, and Wright). As a corollary, if $G$ is an algebraic group over a local field $\bf K$ of characteristic zero, and if its amenable radical is not, up to isogeny, a direct factor, then $G(\bf K)$ has Property (T) relative to a noncompact subgroup. In particular, $G(\bf{K})$ does not have Haagerup's property. This extends a similar result of Cherix, Cowling and Valette for connected Lie groups, to which our method also applies. \endgraf We give some other applications. We provide a characterization of connected Lie groups all of whose countable subgroups have Haagerup's property. We give an example of an arithmetic lattice in a connected Lie group which does not have Haagerup's property, but has no infinite subgroup with relative Property (T). We also give a continuous family of pairwise non-isomorphic connected Lie groups with Property (T), with pairwise non-isomorphic (resp. isomorphic) Lie algebras.