Tangent lie groups are Riemannian naturally reductive spaces

Abstract

Given a compact Lie group G with Lie algebra gg, we consider its tangent Lie group TG≅G⋉AdgTG. In this short note, we prove that TG admits a left-invariant naturally reductive Riemannian metric g and a metric connection with skew torsion ∇∇ such that (TG,g,∇) is naturally reductive. An alternative spinorial description of the same connection on the direct product G×g generalizes in a rather subtle way to TS7, which is in many senses almost a tangent Lie group.