On the radicals of exponential Lie groups

TL;DR

This paper investigates conditions under which the solvable radical of a connected exponential Lie group is itself exponential, providing new criteria and extending known results in the context of complex Lie groups.

Contribution

It introduces a condition on the quotient G/R that ensures R is exponential, and strengthens existing results on the connectedness of the center in complex exponential Lie groups.

Findings

A criterion for elements in certain subsets of solvable Lie groups to be exponential.

A condition on G/R that guarantees R is exponential.

Extension of Moskowitz and Sacksteder's result on the center of complex exponential Lie groups.

Abstract

Let $G$ be a connected exponential Lie group and $R$ be the solvable radical of $G$. We describe a condition on $G/R$ under which one can then conclude that $R$ is an exponential Lie group. The condition holds in particular when $G$ is a complex Lie group and this yields a stronger version of a result of Moskowitz and Sacksteder \cite{MS} on the center of a complex exponential Lie group being connected. Along the way we prove a criterion for elements from certain subsets of a solvable Lie group to be exponential, which would be of independent interest.