Taylor spectrum for modules over Lie algebras

TL;DR

This paper extends the Taylor spectrum concept to modules over arbitrary Lie algebras, providing descriptions for nilpotent, semisimple, and Borel subalgebras, highlighting differences in solvable cases.

Contribution

It introduces a generalized Taylor spectrum for Lie algebra modules and characterizes it for various classes of Lie algebras, including new results for Borel subalgebras.

Findings

Spectrum equals simple submodules for nilpotent Lie algebras

Spectrum equals simple submodules for semisimple Lie algebras

Explicit description of spectrum for Borel subalgebras

Abstract

In this paper we generalize the notion of the Taylor spectrum to modules over an arbitrary Lie algebra and study it for finite-dimensional modules. We show that the spectrum can be described as the set of simple submodules in case of nilpotent and semisimple Lie algebras. We also show that this result does not hold for solvable Lie algebras and obtain a precise description of the spectrum in case of Borel subalgebras of semisimple Lie algebras.