Journal of Lie Theory Volume 6 (1996) 1–17 ◯^{C} 1996 Heldermann Verlag

Operators on Differential Forms for Lie Transformation Groups

Christian Gross^{∗}

Communicated by K. H. Hofmann

Abstract. For any Lie group action S: G×P → P, we introduce C^{∞}(P)- linear operators S_{•}^{i}, that transform n-forms ω_{n} ∈ A_{n}(P, V) into (n−i)-forms S_{•}^{i}ω_{n} ∈ A_{n−i}(P, Alt_{i}(𝔤, V)). We compute the exterior derivative of these generated forms and their behavior under interior products with vector fields and Lie differentiation. By combination with Lie algebra valued forms θ ∈ A_{1}(P, 𝔤) and ϕ_{p} ∈ A_{p}(P, 𝔤), we recover V -valued forms ω ∘^{S} θ ∈ A_{n}(P, V), resp., (χ_{n}^{s}∘^{S} θ)•ϕ_{p} for χ_{n}^{s} ∈ A_{n}(P, Hom(⨂^{s} 𝔤, V)) and compute their exterior derivative. The derived formulae play an important role for local evaluations of connections on fiber bundles.

Math. Subj. Class. 1991: Primary: 58A10, Secondary: 57S20, 22E15, 53C05

Fachbereich Mathematik TH Darmstadt Schlossgartenstrasse 7 D-64289 Darmstadt, Germany gross@mathematik.th-darmstadt.de

Received September 15, 1995 and in final form February 8, 1996

∗Supported by a grant of the Studienstiftung des deutschen Volkes

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag