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 $z_0I+z_1\text{ ad} x_1+\cdots +z_n\text{ ad} x_n.$ 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.
Categories: math.RT