Graded Lie algebra of Hermitian tangent valued forms

Abstract

We define the Hermitian tangent valued forms of a complex 1-dimensional line bundle equipped with a Hermitian metric. We provide a local characterization of these forms in terms of a local basis and of a local fibred chart. We show that these forms constitute a graded Lie algebra through the Frölicher–Nijenhuis bracket. Moreover, we provide a global characterization of this graded Lie algebra, via a given Hermitian connection, in terms of the tangent valued forms and forms of the base space. The bracket involves the curvature of the given Hermitian connection.