Uniqueness of ad-invariant metrics (Q6619627)

The authors study Lie algebras over the fields \(\mathbf{K}=\mathbf{R},\, \mathbf{C}\) carrying an ad-invariant metric. In addition, they require that the group of automorphisms acts on the space of ad-invariant metrics either transitively or with cohomogeneity one.\N\NIn the former case, they show that if an ad-invariant metric on a Lie algebra \(\mathfrak{g}\) is solitary (i.e., every self-adjoint ad-invariant map can be written as the self-adjoint part of a derivation of \(\mathfrak{g}\)) then so are all ad-invariant metrics on \(\mathfrak{g}\). In that case the Lie algebra itself is called solitary. For \(\mathfrak{g}\) irreducible, this is equivalent to admitting a unique ad-invariant metric up to automorphisms (and sign, if \(\mathbf{K}=\mathbf{R}\)), with signature uniquely determined up to sign.\N\NIn the latter case, it is proved that such Lie algebras admit exactly one or two ad-invariant metrics up to automorphisms and a scalar factor, according to whether all derivations are traceless or not.\N\NNext, motivated by the fact that the cotangent of a Lie algebra can always be endowed with a canonical ad-invariant metric, the authors study the solitary condition for cotangent Lie algebras.\N\NThey prove that a Lie algebra \(\mathfrak{g}\) with an ad-invariant metric is solitary if and only if its cotangent Lie algebra is solitary. In general, a Lie algebra is called \(T^*\)-solitary if its cotangent Lie algebra is solitary. It turns out that a semisimple Lie algebra is never \(T^*\)-solitary and every 2-step nilpotent Lie algebra is \(T^*\)-solitary. One of the main results of the paper is in fact that a \(T^*\)-solitary Lie algebra is necessarily solvable (the converse is false).\N\NThe final part of the paper is devoted to the study of the \(T^*\)-solitary condition on low dimensional Lie algebras: every solvable Lie algebra of dimension \(\le 6\) is \(T^*\)-solitary. In particular, every ad-invariant metric on a solvable Lie algebra of dimension \(\le 6\) is solitary. Every real nilpotent Lie algebra of dimension \(\le 10\) is \(T^*\)-solitary. In particular, any ad-invariant metric on a real nilpotent Lie algebra of dimension \(\le 10\) is solitary.