Rational visibility of a Lie group in the monoid of self-homotopy equivalences of a homogeneous space

TL;DR

This paper investigates the injectivity of the rational homotopy map induced by a Lie group acting on a homogeneous space, explicitly determining visible degrees for simple Lie groups and rank one homogeneous spaces.

Contribution

It provides a detailed analysis of the rational homotopy injectivity of the Lie group action on homogeneous spaces, including explicit calculations for simple Lie groups and rank one cases.

Findings

Explicit determination of visible degrees for simple Lie groups.

Proof of injectivity of the rational homotopy map in specific cases.

Classification of visible degrees for rank one homogeneous spaces.

Abstract

Let M be a homogeneous space admitting a left translation by a connected Lie group G. The adjoint to the action gives rise to a map from G to the monoid of self-homotopy equivalences of M.The purpose of this paper is to investigate the injectivity of the homomorphism which is induced by the adjoint map on the rational homotopy. In particular, the visible degrees are determined explicitly for all the cases of simple Lie groups and their associated homogeneous spaces of rank one which are classified by Oniscik.