Spectral invariants for finite dimensional Lie algebras

TL;DR

This paper introduces spectral invariants derived from the characteristic polynomial of a Lie algebra's adjoint representation, demonstrating their invariance under automorphisms and their reflection of algebraic structure, with applications to extensions of nilpotent Lie algebras.

Contribution

It defines spectral invariants for finite-dimensional Lie algebras based on characteristic polynomials, revealing structural insights and invariance properties.

Findings

The characteristic polynomial is invariant under automorphisms.

Zero variety and factorization reflect algebra structure.

For solvable Lie algebras, the polynomial factors into linear components.

Abstract

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.