Spectral invariants for finite dimensional Lie algebras

For a Lie algebra ${mathcal L}$ with basis ${x_1,x_2,cdots,x_n}$, its associated characteristic polynomial $Q_{{mathcal L}}(z)$ is the determinant of the linear pencil $z_0I+z_1text{ad} x_1+cdots +z_ntext{ad} x_n.$ This paper shows that $Q_{mathcal L}$ is invariant under the automorphism group $text{Aut}({mathcal L}).$ The zero variety and factorization of $Q_{mathcal L}$ reflect the structure of ${mathcal L}$. In the case ${mathcal L}$ is solvable $Q_{mathcal L}$ is known to be a product of linear factors. This fact gives rise to the definition of spectral matrix and the Poincar{e} polynomial for solvable Lie algebras. Application is given to $1$-dimensional extensions of nilpotent Lie algebras.