Nijenhuis operators on Banach homogeneous spaces

TL;DR

This paper characterizes when bounded operators on Lie algebras induce homogeneous vector bundle maps on Banach homogeneous spaces, extending previous results and linking Nijenhuis torsion to integrability of almost complex structures.

Contribution

It establishes general conditions for bounded operators to define homogeneous vector bundle maps on Banach homogeneous spaces, including cases with non-complemented subalgebras, and relates Nijenhuis torsion to integrability.

Findings

Conditions for bounded operators to define bundle maps

Equivalence of Nijenhuis torsion vanishing and values in Lie(K)

Characterization of integrability of almost complex structures

Abstract

For a Banach--Lie group $G$ and an embedded Lie subgroup $K$ we consider the homogeneous Banach manifold $\mathcal M=G/K$. In this context we establish the most general conditions for a bounded operator $N$ acting on $Lie(G)$ to define a homogeneous vector bundle map $\mathcal N:T\mathcal M\to T\mathcal M$. In particular our considerations extend all previous settings on the matter and are well-suited for the case where $Lie(K)$ is not complemented in $Lie(G)$. We show that the vanishing of the Nijenhuis torsion for a homogeneous vector bundle map $\mathcal N:T\mathcal M\to T\mathcal M$ (defined by an admissible bounded operator $N$ on $Lie(G)$) is equivalent to the Nijenhuis torsion of $N$ having values in $Lie(K)$. As an application, we consider the question of integrability of an almost complex structure $\mathcal J$ on $\mathcal M$ induced by an admissible bounded operator $J$, and we give a simple characterization of integrability in terms of certain subspaces of the complexification of $Lie(G)$ (which are not eigenspaces of the complex extension of $J$).