On the Positivstellensatz for Enveloping Algebras

TL;DR

This paper leverages Reznick's Theorem to establish an elliptic regularity result for representations of enveloping algebras, enabling relaxation of technical conditions for sum of squares decompositions in Lie algebra representations.

Contribution

It introduces a novel elliptic regularity result for enveloping algebra representations, relaxing previous technical constraints for sum of squares decompositions.

Findings

Elliptic regularity for enveloping algebra representations established

Relaxation of technical conditions for sum of squares decompositions

Application of Reznick's Theorem to Lie algebra representations

Abstract

We use Reznick's Theorem for positive homogeneous polynomials to prove an elliptic regularity result for representations of enveloping algebras of Lie algebras. This allows us to relax a technical condition for a sum of squares decomposition for representations of these algebras.