Samelson complex structures for the tangent Lie group

TL;DR

This paper demonstrates that the tangent bundle of any compact Lie group admits a natural, left-invariant integrable complex structure, extending Samelson's construction from even to all compact Lie groups.

Contribution

It introduces a new integrable complex structure on the tangent bundle of any compact Lie group, generalizing Samelson's method beyond even-dimensional cases.

Findings

Tangent bundle of any compact Lie group admits a left-invariant integrable complex structure.

The construction generalizes Samelson's approach to all compact Lie groups.

Provides a new geometric structure on tangent bundles of Lie groups.

Abstract

It is shown that for any compact Lie group $G$ (odd or even dimensional), the tangent bundle $TG$ admits a left-invariant integrable almost complex structure, where the Lie group structure on $TG$ is the natural one induced from $G$. The aforementioned complex structure on $TG$ is inspired by Samelson's construction for even dimensional compact Lie groups.