Lie symmetries of the canonical connection: codimension one abelian nilradical case

TL;DR

This paper investigates the Lie symmetries of the canonical symmetric connection on Lie groups, especially focusing on cases where the Lie algebra has a codimension one abelian nilradical, providing complete integration of symmetry conditions.

Contribution

It characterizes Lie symmetries of the geodesic system for a specific class of Lie groups with abelian nilradicals, extending understanding of their geometric properties.

Findings

Complete integration of symmetry conditions for codimension one abelian nilradical cases.

Comparison with four-dimensional Lie groups with three-dimensional abelian nilradicals.

Application of MAPLE for explicit calculations.

Abstract

This paper studies the canonical symmetric connection $\nabla$ associated to any Lie group $G$. The salient properties of $\nabla$ are stated and proved. The Lie symmetries of the geodesic system of a general linear connection are formulated. The results are then applied to $\nabla$ in the special case where the Lie algebra $\g$ of $G$, has a codimension one abelian nilradical. The conditions that determine a Lie symmetry in such a case are completely integrated. Finally the results obtained are compared with some four-dimensional Lie groups whose Lie algebras have three-dimensional abelian nilradicals, for which the calculations were performed by MAPLE.