Tangent Lie groups are Riemannian naturally reductive spaces

Abstract

Given a compact Lie group $G$ with Lie algebra $\mathfrak{g}$, we consider its tangent Lie group $TG\cong G{⋉}_{\mathrm{Ad}}\mathfrak{g}$. In this short note, we prove that $TG$ admits a left-invariant naturally reductive Riemannian metric $g$ and a metric connection with skew torsion $\nabla$ such that $(TG,g,\nabla )$ is naturally reductive. An alternative spinorial description of the same connection on the direct product $G\times \mathfrak{g}$ generalizes in a rather subtle way to $T{S}^7$, which is in many senses almost a tangent Lie group.

12 pages, Advances in Applied Clifford Algebras, March 2016