Joint spectra and nilpotent Lie algebras of linear transformations

TL;DR

This paper characterizes the joint spectra of complex nilpotent Lie algebras of linear transformations on finite-dimensional spaces, showing they coincide with the set of weights and providing new module operation interpretations.

Contribution

It explicitly computes the joint spectra for nilpotent Lie algebras and links them to weights, offering new insights into module operations.

Findings

Joint spectra coincide with weights of the Lie algebra.

Explicit computation of joint spectra for nilpotent Lie algebras.

New interpretation of module operations via joint spectra.

Abstract

Given a complex nilpotent finite dimensional Lie algebra of linear transformations $L$, in a complex finite dimensional vector space $E$, we study the joint spectra $Sp(L,E)$, $\sigma_{\delta,k}(L,E)$ and $\sigma_{\pi,k}(L,E)$. We compute them and we prove that they all coincide with the set of weights of $L$ for $E$. We also give a new interpretation of some basic module operations of the Lie algebra $L$ in terms of the joint spectra.