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

1. Motivation

Let P (M, G) denote a principal bundle with base manifold M = ⋃_{α∈A} U_{α}, projection π: P → M, Lie group G, right action R: P × G → P and local trivializations ψ_{α}: π^{−1}(U_{α}) → U_{α} × G with local projections π_{α} = pr_{G} ∘ψ_{α}. Recall that any connection Γ on P defines horizontal and vertical projections of vector fields, not only on P, but also on every associated fiber bundle B(M, F, G) = P ×_{G} F with fiber F and left action L: G × F → F, such that the vertical fields are tangential to the fiber. We thus obtain projections h, v of differential forms via

ωh(…, X^{i}, …) := ω(…, hX^{i}, …),  ωv(…, X^{i}, …) := ω(…, vX^{i}, …) (1)

for all V -valued forms ω ∈ A(B, V). If we compute the vertical projections locally on the bundle charts U_{α} × F, we obtain with L^{f}: G → F defined by L^{f}(g) := L(g, f) and its differential (dL^{f})_{e}: 𝔤 → T_{f}(F) at the neutral element e ∈ G:

(ϕ^{α}v^{α})_{(x,f)}(…, (X^{i}, F^{i}), …) = ϕ_{(x,f)}^{α}(…, (0, (dL^{f})_{e}A_{x}^{α}(X^{i}) + F^{i}), …) (2)

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag for all ϕ^{α} ∈ A(U_{α} × F, V) and (X^{i}, F^{i}) ∈ T_{x}(U_{α}) ⊕ T_{f}(F). Here A^{α} ∈ A_{1}(U_{α}, 𝔤) mean the Lie algebra valued gauge potentials, which one obtains from the connection 1-form ω^{Γ} ∈ A_{1}(P, 𝔤) as pullbacks A^{α} := σ_{α,e}^{⋆}ω^{Γ} under the local sections σ_{α,e}: U_{α} → π^{−1}(U_{α}) defined by σ_{α,e}(x) := ψ_{α}^{−1}(x, e).

∗Supported by a grant of the Studienstiftung des deutschen Volkes

Instead of (2), we would like to obtain a handier formula for these local projections, which is merely expressed in terms of the involved forms ϕ and A^{α}. Especially if ϕ is a (pullback of a) form on F, this is indeed possible. Moreover, we will also be able to give a formula for the exterior derivative d(ϕv). Such formulas are quite essential if one tries to combine the de-Rham cohomology of a fiber bundle with connections on that bundle.

For any X ∈ 𝔤, let L_{X} ∈ D^{1}(F) denote the induced vector field on F that is given by (L_{X})_{f} := (dL^{f})_{e}(X) and let L: 𝔤 → D^{1}(F) denote the Lie algebra antihomomorphism defined hereby. Then L ∘ A^{α} maps vector fields on U_{α} to vector fields on F. For a n-form ϕ ∈ A_{n}(F, V), we may decompose (pr_{F}^{⋆} ϕ)v^{α} according to (2) into a sum of differential forms χ_{i} ∈ A_{n}(U_{α} × F, V), i = 0, …, n, such that χ_{i} acts on n − i vertical fields and i horizontal vector fields via L ∘ A^{α}. The forms χ are obtained from ϕ in two steps: using a product of i maps L, we first transform ϕ into a (n − i)-form L_{•}^{i}ϕ ∈ A_{n−i}(F, Alt_{i}(𝔤, V)). Then we combine that form with i factors of A^{α} in such a way that for any X ∈ D^{1}(U_{α}) the maps pr_{U_{α}}^{⋆}[A^{α}(X)] ∈ C^{∞}(U_{α} × F, 𝔤) serve as input for the maps in Alt_{i}(𝔤, V). The resulting form will be denoted by [pr_{F}^{⋆} (L_{•}^{i}ϕ)] • (pr_{U_{α}}^{⋆} A^{α}).

Such a construction is possible for all Lie groups that act on a differentiable manifold from the right or the left. Thus we will choose the general framework of Lie transformation groups in the sequel. For notational convenience, we will recall the basic definitions from differential geometry for Lie group actions. Then we introduce the operators L_{•}^{i} and R_{•}^{i} for a left, resp., right action and compute, in how far they commute with exterior differentiation d of forms, interior products ı_{X} with respect to a vector field X and Lie differentiation L_{X}, which is given by L_{X} = ı_{X} ∘ d + d ∘ ı_{X}. The operator • has already been discussed in detail in [1], thus we only recall its definition and main properties for our purposes. Finally we introduce operators ∘^{L} and ∘^{R} , [such that (pr_{F}^{⋆} ϕ)v^{α} is indeed given by (pr_{F}^{⋆} ϕ)∘^{L} (pr_{U_{α}}^{⋆} A^{α})] and compute the exterior derivative of ω_{n}∘^{L} θ, resp., ω_{n}∘^{R} θ.

2. Basic definitions

Let us first recall some of the definitions we have already used above, according to Helgason [3] and Kobayashi, Numizu [4]. For any vector spaces V and W, Alt_{p}(W, V) and Sym_{p}(W, V) denote the vector spaces of all alternating, resp., symmetric p-linear maps from W^{p} to V . For convenience we define Sym^{±}(W, V) := ⨁_{p=0}^{∞} Sym_{p}^{±}(W, V) by Sym_{p}^{+}(W, V) := Sym_{p}(W, V) and Sym_{p}^{−}(W, V) := Alt_{p}(W, V).

If f: M → N is differentiable, we denote the differential of f at x ∈ M by df_{x}. We have [df_{x}(X_{x})]g = X_{x}(g ∘ f) for all X_{x} ∈ T_{x}(M), g ∈ C^{∞}(N). If in addition, f is a diffeomorphism then for X ∈ D^{1}(M) the push-out f_{⋆}X ∈ D^{1}(N) is defined by (f_{⋆}X)_{f(x)} = df_{x}(X_{x}) for all x ∈ M.

For α ∈ A_{r}(N, V), r ∈ ℕ and X_{i} ∈ T_{x}(M), the pullback f^{⋆}α ∈ A_{r}(M, V) is defined by (f^{⋆}α)_{x}(X_{1}, …, X_{r}) = α_{f(x)}(df_{x}(X_{1}), …, df_{x}(X_{r})). For α ∈ C^{∞}(N, V) we have f^{⋆}α := α ∘ f, linear extension defines the pullback on A(N, V). If we insert A(M) ⊗ V into A(M, V) in a natural way, then obviously f^{⋆}(A(N) ⊗ V) ⊆ A(M) ⊗ V . (If V is finite dimensional, we will identify A(M) ⊗ V and A(M, V).)

Let T (V) denote the tensor algebra of V . Then every linear map Λ: V →W defines a pullback Λ^{⋆}: Hom(T (W), Z) → Hom(T (V), Z): for K ∈ Hom(⨂^{p} W, Z), X_{i} ∈ V we have Λ^{⋆}K(X_{1}, …, X_{p}) = K(Λ(X_{1}), …, Λ(X_{p})), so Λ^{⋆}(Sym^{±}(W, Z)) ⊆ Sym^{±}(V, Z). Λ also defines a push-out Λ_{⋆}: A(M, V) → A(M, W) by Λ_{⋆}ω = Λ∘ω. Again Λ_{⋆}(A(M) ⊗ V) ⊆ A(M) ⊗ W, where we have Λ_{⋆}(α ⊗ v) = α ⊗ Λ(v) for all α ∈ A(M), v ∈ V . For an example, let E_{j} ∈ W, j = 1, …, s and let E_{1} ⊗ ⋯ ⊗ E_{s}: Hom(⨂^{s} W, V) → V denote the canonical evaluation morphism. For any differential form χ_{r}^{s} ∈ A_{r}(M, Hom(⨂^{s} W, V)) define χ_{r}^{E_{1},...,E_{s}} ∈ A_{r}(M, V) to be the push-out of χ_{r}^{s} under this morphism: χ_{r}^{E_{1},...,E_{s}} := (E_{1} ⊗ ⋯ ⊗ E_{s})_{⋆}χ_{r}^{s}, i. e., for all x ∈ M and X^{i} ∈ D^{1}(M), i = 1, …, r,

(χ_{r}^{E_{1},...,E_{s}})_{x}(X_{x}^{1}, …, X_{x}^{r}) := (E_{1} ⊗ ⋯ ⊗ E_{s}) ∘ (χ_{r}^{s})_{x}(X_{x}^{1}, …, X_{x}^{r}).  (3)

In the sequel, G will always mean a Lie group with Lie algebra 𝔤, left and right multiplication λ, ρ: G → G and inversion η: G → G. For S = L, R let S: G × P → P denote a left, resp., right Lie group action. We identify S with S: G → Diff(P). Also for notational convenience, we always write G on the left, even if S denotes a right action. In that case, we put sgn(S) := 1, whereas sgn(S) := −1, if S denotes a left action. Since S is differentiable, all maps S^{p}: G → P, p ∈ P, resp., S_{g}: P → P, g ∈ G, defined by S^{p}(g) := S_{g}(p) := S(g, p), are differentiable, resp., diffeomorphisms.

G is called a Lie transformation group of the manifold P. If P is a vector space and the action is linear, we speak of a representation of G, e. g., the adjoint action Ad: G → Gl(𝔤) is a left representation. The trivial action means the natural projection pr_{P}: G × V → V .

An action is effective if S_{g} = id_{P} only for g = e. In that case G may be thought of as a subgroup of Diff(P). An action is free if (in addition) S_{g}(p) = p only for g = e for all p ∈ P. Via λ and ρ every Lie group acts freely on itself.

ω ∈ A(P, V) is called G-invariant or simply invariant if S_{g}^{⋆}ω = ω for all g ∈ G. Denote their set by A(P, V)_{inv} . Analoguosly for any subgroup H < G, we define A(P, V)_{H−inv} to be the set of H -invariant forms, i. e., those forms that are invariant under the restriction of S onto H × P. Especially we will use this notation for G_{e}-invariant forms, where G_{e} is the connected component of e ∈ G.

A(P)_{inv} and A(P)_{inv} ⊗V are graded subalgebras of A(P), resp., A(P)⊗V (whenever a wedge product ∧_{V} of V -valued forms is given by a bilinear mapping m: V × V → V ), with d(A(P)_{inv}) ⊆ A(P)_{inv} . Analogous statements hold for A(P)_{H−inv} and A(P)_{H−inv} ⊗ V , which are modules of A(P)_{inv} . Obviously A(P)_{inv} ⊆ A(P)_{H−inv} and A(P, V)_{inv} ⊆ A(P, V)_{H−inv} for any subgroup H < G.

Lemma 2.1. If S: G×P →P is a Lie group action then S_{⋆}: G×D^{1}(P)→D^{1}(P), S^{⋆}∘η: G×A(P, V) → A(P, V), S′: G×A(P, Hom(T (𝔤), V)) → A(P, Hom(T (𝔤), V)) and S′′: G × A(P, 𝔤) → A(P, 𝔤) defined by

(S_{⋆})_{g}(X) := (S_{g})_{⋆}X  for all X ∈ D^{1}(P),

(S^{⋆} ∘ η)_{g}(ω) := (S_{g^{−1}})^{⋆}ω  for all ω ∈ A(P, V),
S′_{g}(χ) := (S_{g^{−1}})^{⋆}(Ad(g^{sgn(S)})^{⋆})_{⋆}χ for all χ ∈ A(P, Hom(T (𝔤), V)) and
S′′_{g}(φ) := (S_{g^{−1}})^{⋆} Ad(g^{− sgn(S)})_{⋆}φ for all φ ∈ A(P, 𝔤),

are all representations of G on the same side.

Let S, S′ be two actions of G on spaces X, resp., X′ on the same side. A mapping f: X → X′ is called G-equivariant, if

[figure: G × X′; X′; G × X; X; S′; S; id ×f; f]

commutes, i. e., if S′(g, f(x)) = f(S(g, x)) for all x ∈ X and g ∈ G.

If S is a Lie group action on P and R is a right representation on W, then a differential form ω ∈ A(P, W) is called G-equivariant, if S_{g}^{⋆}χ = R(g^{sgn(S)})_{⋆}χ for all g ∈ G (resp., if S_{g}^{⋆}χ = L(g^{− sgn(S)})_{⋆}χ for a left representation L). Thus — referring to the right representation Ad^{⋆} on W = Hom(T (𝔤), V) — we call χ ∈ A(P, Hom(T (𝔤), V)) G-equivariant, if χ is invariant under S′. Analogously, φ ∈ A(P, 𝔤) will be called G-equivariant if φ is invariant under S′′. We denote the set of equivariant forms by A(P, W)_{equiv} . It is a module over A(P)_{inv} .

If G is compact with Haar measure μ we have projections onto invariant and G-equivariant forms defined in the following way:

ω_{inv} := ∫_{G} S_{g}^{⋆}ω dμ(g)  for all ω ∈ A(P, V),
χ_{equiv} := ∫_{G}(Ad(g^{− sgn(S)})^{⋆})_{⋆}S_{g}^{⋆}χ dμ(g) for all χ ∈ A(P, Hom(T (𝔤), V)),
φ_{equiv} := ∫_{G} Ad(g^{sgn(S)})_{⋆}S_{g}^{⋆}φ dμ(g) for all φ ∈ A(P, 𝔤).

As already introduced in the previous section, every X ∈ 𝔤 induces a canonical complete vector field S_{X} ∈ D^{1}(P) by (S_{X})_{p} := (dS^{p})_{e}(X), so

(S_{X})_{p}(f) = (dS^{p})_{e}(X)(f) = \frac{d}{dt}f(S_{e^{tX}}(p))|_{t=0} for all f ∈ C^{∞}(P), p ∈ P,
[S_{X}, Y]_{p} = lim_{t→0} \frac{1}{t}\{Y_{p}−((S_{e^{tX}})_{⋆}Y)_{p}\} = lim_{t→0} \frac{1}{t}\{((S_{e^{−tX}})_{⋆}Y)_{p}−Y_{p}\} for all Y ∈ D^{1}(P).

R: 𝔤 → D^{1}(P) and −L: 𝔤 → D^{1}(P) are Lie algebra homomorphisms and

[R_{X}, R_{Y}] = R_{[X,Y]}, [L_{X}, L_{Y}] = L_{[Y,X]} = −L_{[X,Y]} for all X, Y ∈ 𝔤,
(R_{g^{−1}})_{⋆}R_{X} = R_{Ad(g)X}, (L_{g})_{⋆}L_{X} = L_{Ad(g)X}  for all g ∈ G, X ∈ 𝔤.

Obviously S = L, R only depends on the restriction of S onto G_{e} × P. For all forms ω ∈ A(P) ⊗ V we have L_{S_{X}}ω = [\frac{d}{dt}((S_{e^{tX}})^{⋆}ω)|_{t=0}] and for all X, Y ∈ 𝔤:

[L_{S_{X}}, L_{S_{Y}}] = L_{[S_{X},S_{Y} ]} = sgn(S)L_{S_{[X,Y ]}}, [L_{S_{X}}, d] = 0,  (4)
[L_{S_{X}}, ı_{S_{Y}}] = ı_{[S_{X},S_{Y} ]} = sgn(S)ı_{S_{[Y,X]}}, L_{S_{X}} = ı_{S_{X}} ∘ d + d ∘ ı_{S_{X}}.  (5)

We call a differential form ω ∈ A(P) 𝔤-invariant if L_{S_{X}}ω = 0 for all X ∈ 𝔤. Analogously, ω will be called horizontal if ı_{S_{X}}ω = 0 for all X ∈ 𝔤. Denote their sets by A(P)_{𝔤−inv} , resp., A(P)h and let A(P)h_{𝔤−inv} := A(P)_{𝔤−inv} ∩ A(P)h.

The notion of “horizontal” forms is due to the fact that for a principal bundle P (M, G), the horizontal forms in the sense of (1) are exactly those forms ω with ı_{R_{X}}ω = 0 for all X ∈ 𝔤 with respect to the free right action R on P.

Since ı_{X} and L_{X} are (skew-)derivations of A(P) and G_{e} = ⟨exp 𝔤⟩, we get:

Lemma 2.2. A(P)_{𝔤−inv} , A(P)h and A(P)h_{𝔤−inv} are graded subalgebras of A(P) with d(A(P)_{𝔤−inv}) ⊆ A(P)_{𝔤−inv} and d(A(P)h_{𝔤−inv}) ⊆ A(P)h_{𝔤−inv} . Analogous statements hold for A(P)_{𝔤−inv} ⊗ V and ∧_{V} , etc.

A(P)_{inv} ⊗ V ⊆ A(P)_{𝔤−inv} ⊗ V = A(P)_{G_{e}−inv} ⊗ V for every vector space V . If G is connected then A(P)_{inv} ⊗ V = A(P)_{𝔤−inv} ⊗ V .

Lemma 2.3. S: 𝔤 → D^{1}(P) induces a G-equivariant C^{∞}(P)-module homomorphism S′: C^{∞}(P, 𝔤) → C^{∞}(P)S(𝔤) ⊆ D^{1}(P) (with respect to S′′ and S_{⋆}). If G acts effectively on P then S is injective. If G acts freely on P then even (dS^{p})_{e} is injective for all p ∈ P, thus X ≠ 0 yields (S_{X})_{p} ≠ 0 for all p ∈ P; for every basis {E_{i}}_{i,...,dim 𝔤} for 𝔤, {S_{E_{i}}}_{i,...,dim 𝔤} is then a basis for the free C^{∞}(P)- module C^{∞}(P)S(𝔤) and the induced S′ is an isomorphism of free C^{∞}(P)-modules. Proof. Assume that G acts effectively. Let X ∈ 𝔤 and suppose (S_{X})_{p}(f) = 0 for all f ∈ C^{∞}(P) and all p ∈ P. For p = S(e^{sX}, p′) this yields \frac{d}{dt}f(S_{e^{(t+s)X}}(p′))|_{t=0} = \frac{d}{dt}f(S_{e^{tX}}(p′))|_{t=s} = 0 for all f ∈ C^{∞}(P), p′ ∈ P and s ∈ ℝ. Thus S(e^{tX}, p′) = p′ for all p′ ∈ P and t ∈ ℝ, and thus X = 0 since S is effective. Analogously for a free action, one proves injectivity of (dS^{p})_{e} for all p ∈ P using (S_{X})_{S(e^{sX},p)} = dS_{e^{sX}}(S_{X})_{p}. But then all S_{E_{i}} are independent over C^{∞}(P), since they are independent for all p ∈ P.

Finally we need the notion of 𝔤-equivariant forms. Just as Ad: G → Gl(𝔤) induces the representation ad: 𝔤 → gl(𝔤) with ad(X)(Y) = [X, Y], every representation S: G → Gl(W) of a Lie group G induces a representation s = dS_{e}: 𝔤 → gl(W) of 𝔤 such that S ∘ exp X = e^{sX} for all X ∈ 𝔤. We will identify s with the corresponding bilinear mapping s: 𝔤 × W → W given by s(X, w) := s_{X}(w) := (dS^{w})_{e}(X). From this point of view, R and L: 𝔤 → D^{1}(P) = der C^{∞}(P) are the (infinite dimensional) representations induced by the Lie group representations R^{⋆} and L^{⋆}: G → Aut(C^{∞}(P)).

Let ad^{⋆}: 𝔤 × Hom(T (𝔤), V) → Hom(T (𝔤), V) denote the bilinear mapping induced by Ad^{⋆}. Then for X, X_{i} ∈ 𝔤, p ∈ ℕ_{0} and K ∈ Hom(⨂^{p} 𝔤, V), we have

(ad_{X}^{⋆} K)(X_{1}, …, X_{p}) = ∑_{i=1}^{p} K(X_{1}, …, [X, X_{i}], …, X_{p}).  (6)

If S is a Lie group action of G on P and S′: G → Gl(W) is a representation, then a differential form χ ∈ A(P) ⊗ W will be called 𝔤-equivariant if

L_{S_{X}}χ = sgn(S) sgn(S′)s′(X)_{⋆}χ for all X ∈ 𝔤.  (7)

Thus χ ∈ A(P) ⊗ Hom(T (𝔤), V) is 𝔤-equivariant if L_{S_{X}}χ = sgn(S)(ad_{X}^{⋆})_{⋆}χ for all X ∈ 𝔤. We will denote the vector space of 𝔤-equivariant forms by A(P)_{𝔤−equiv}⊗W. Analogously to Lemma 2.2 we obtain:

Lemma 2.4. A(P)_{𝔤−equiv} ⊗ W is a A(P)_{𝔤−inv} -module with d(A(P)_{𝔤−equiv} ⊗ W) ⊆ A(P)_{𝔤−equiv}⊗W and A(P)_{equiv}⊗W ⊆ A(P)_{𝔤−equiv}⊗W = A(P)_{G_{e}−equiv}⊗W for all vector spaces W. If G is connected then A(P)_{equiv}⊗W = A(P)_{𝔤−equiv}⊗W.

It is an elementary result in differential geometry (e. g., cf. [4, p. 34]) that

(L_{X} ω)(P^{1}, …, P^{n}) = X (ω(P^{1}, …, P^{n})) − ∑_{i=1}^{n} ω(P^{1}, …, [X , P^{i}], …, P^{n})

for all X , P^{i} ∈ D^{1}(P) and ω ∈ A(P) ⊗ V . This yields the following corollaries:

Corollary 2.5. If ω ∈ A_{n}(P)_{𝔤−inv} ⊗ V , then for all X ∈ 𝔤 and P^{i} ∈ D^{1}(P)

S_{X}(ω(P^{1}, …, P^{n})) = ∑_{i=1}^{n} ω(P^{1}, …, [S_{X}, P^{i}], …, P^{n}).

Corollary 2.6. Let χ_{n}^{s} ∈ A_{n}(P) ⊗ Hom(⨂^{s} 𝔤, V) be 𝔤-equivariant. Then for all p ∈ P, P^{i} ∈ D^{1}(P) and X, E_{i} ∈ 𝔤:

(L_{S_{X}}χ_{n}^{s})(P^{1}, …, P^{n})(p)(E_{1} ⊗ ⋯ ⊗ E_{s}) =

= {S_{X}(χ_{n}^{s}(P^{1}, …, P^{n})) − ∑_{i=1}^{n} χ_{n}^{s}(P^{1}, …, [S_{X}, P^{i}], …, P^{n})}(p)(E_{1} ⊗ ⋯ ⊗ E_{s})
= sgn(S) ∑_{j=1}^{s} χ_{n}^{s}(P^{1}, …, P^{n})(p)(E_{1} ⊗ ⋯ ⊗ E_{j−1} ⊗ [X, E_{j}] ⊗ E_{j+1} ⊗ ⋯ ⊗ E_{s}).

3. Mapping invariant forms onto equivariant forms

Now everything is prepared for the definition of the operators S_{•}^{i} which map V - valued forms onto Alt_{i}(𝔤, V)-valued forms:

Definition 3.1. Let S be a Lie group action of G on P and ω_{n} ∈ A_{n}(P, V). We define S_{•}^{i}ω_{n} ∈ A_{n−i}(P, Alt_{i}(𝔤, V)), i ≤ n, for all P^{j} ∈ D^{1}(P), E_{k} ∈ 𝔤 and p ∈ P by

[(S_{•}^{i}ω_{n})(P^{1}, …, P^{n−i})(p)](E_{1}, …, E_{i}) := \frac{n!}{(n−i)!}ω_{n}(S^{1}, …, S^{i}, P^{1}, …, P^{n−i})(p) ∈ V,

where S^{i} := S_{E_{i}}. Thus S_{•}^{i}ω_{n} ∈ A_{n−i}(P) ⊗ Alt_{i}(𝔤, V) if ω_{n} ∈ A_{n}(P) ⊗ V . For i > n we put S_{•}^{i}ω_{n} = 0.

The factor \frac{n!}{(n−i)!} is inherited from the definition of the interior product with vector fields: recall (S_{•}^{i}ω_{n})_{n−i}^{E_{1},...,E_{i}} ∈ A_{n−i}(P, V) for E_{k} ∈ 𝔤 from (3), then we have

(S_{•}^{i}ω_{n})_{n−i}^{E_{1},...,E_{i}} = (ı_{S^{i}} ∘ ⋯ ∘ ı_{S^{1}})ω_{n}.  (8)
If {E_{k}} is a base for 𝔤, we obtain for ω ∈ A_{n}(P) and v ∈ V :

[S_{•}^{i}(ω⊗v)](P^{1}, …, P^{n−i}) = \frac{n!}{(n−i)!} ∑_{k_{1}<···<k_{i}} ω(S^{k_{1}}, …, S^{k_{i}}, P^{1}, …, P^{n−i})⊗[(E_{k_{1}}∧⋯∧E_{k_{i}})↦v].

Lemma 3.2. For all i ≤ n, S_{•}^{i}: A_{n}(P, V) → A_{n−i}(P, Alt_{i}(𝔤, V)) is C^{∞}(P)- linear. For ω_{n} ∈ A_{n}(P, V), χ_{n}^{s} ∈ A_{n}(P, Alt_{s}(𝔤, V)) and i + j ≤ n, we have

S_{•}^{0}ω_{n} = ω_{n},  (S_{•}^{n}ω_{n})(p) = n! [(S^{p})^{⋆}ω_{n}]_{e} for all p ∈ P,  (9)
S_{•}^{i}(Λ_{⋆}ω_{n}) = Λ_{⋆}(S_{•}^{i}ω_{n})  for all Λ ∈ Hom(V, W),  (10)
S_{g}^{⋆}(S_{•}^{i}ω_{n}) = (Ad(g^{sgn(S)})^{⋆})_{⋆}[S_{•}^{i}(S_{g}^{⋆}ω_{n})],  thus  (11)
S_{g}^{⋆}(S_{•}^{i}χ_{n}^{s}) = (Ad(g^{sgn(S)})^{⋆})_{⋆}(S_{•}^{i}χ_{n}^{s}), if S_{g}^{⋆}χ_{n}^{s} = (Ad(g^{sgn(S)})^{⋆})_{⋆}χ_{n}^{s}. (12)

Let f^{i,j}: Alt_{i+j}(𝔤, V) ↪ Alt_{i}(𝔤, Alt_{j}(𝔤, V)) denote the injection defined by

[f^{i,j}(a)](E_{1}, …, E_{i})(F_{1}, …, F_{j}) := a(E_{1}, …, E_{i}, F_{1}, …, F_{j}) for a ∈ Alt_{i+j}(𝔤, V).

Then

f_{⋆}^{i,j}(S_{•}^{i+j}ω_{n}) = (−1)^{ij}S_{•}^{i}(S_{•}^{j}ω_{n}).  (13)

Lemma 3.3. For all i ≤ n we have:

S_{•}^{i}(A_{n}(P, V)_{inv}) ⊆ A_{n−i}(P, Alt_{i}(𝔤, V))_{equiv},
S_{•}^{i}(A_{n}(P)_{inv} ⊗ V) ⊆ A_{n−i}(P)_{equiv} ⊗ Alt_{i}(𝔤, V),
S_{•}^{i}(A_{n}(P)_{𝔤−inv} ⊗ V) ⊆ A_{n−i}(P)_{𝔤−equiv} ⊗ Alt_{i}(𝔤, V).

Proof. (12) yields that S_{•}^{i}ω_{n} is G-equivariant if ω_{n} is invariant under S. Now the operators S_{•}^{i} only depend on the restriction of S to G_{e} ×P. Thus Lemmas 2.2 and 2.4 prove that S_{•}^{i}ω_{n} is 𝔤-equivariant if ω_{n} is 𝔤-invariant.

Let us compute in how far the operators S_{•}^{i} commute with the exterior differentiation, interior products with vector fields and Lie differentiation.

Lemma 3.4. Let S be a Lie group action of G on P. For all ω_{n} ∈ A_{n}(P)⊗V , i ≤ n + 1 and E_{k} ∈ 𝔤 we have {S_{•}^{i}(dω_{n}) − (−1)^{i}d(S_{•}^{i}ω_{n})}_{n+1−i}^{E_{1},...,E_{i}} =

= − ∑_{j=1}^{i}(−1)^{j}{[S_{•}^{i−1}(L_{S^{j}}ω_{n})]_{n+1−i}^{E_{1},...,_{E_{j}},...,E_{i}} + sgn(S) ∑_{k=j+1}^{i}(S_{•}^{i−1}ω_{n})_{n+1−i}^{E_{1},...,_{E_{j}},...,[E_{j},E_{k}],...,E_{i}}}

= − ∑_{j=1}^{i}(−1)^{j}{[L_{S^{j}}(S_{•}^{i−1}ω_{n})]_{n+1−i}^{E_{1},...,_{E_{j}},...,E_{i}} − sgn(S) ∑_{k=j+1}^{i}(S_{•}^{i−1}ω_{n})_{n+1−i}^{E_{1},...,_{E_{j}},...,[E_{j},E_{k}],...,E_{i}}},

where indicates that the term is omitted. Proof. From (8), the fact that d commutes with the push-outs (E_{1} ⊗ ⋯ ⊗ E_{i})_{⋆} and the identities (5) we get by induction:

{S_{•}^{i}(dω_{n}) − (−1)^{i}d(S_{•}^{i}ω_{n})}_{n+1−i}^{E_{1},...,E_{i}} = − ∑_{j=1}^{i}(−1)^{j}(ı_{S^{i}} ∘ ⋯ ∘ L_{S^{j}} ∘ ⋯ ∘ ı_{S^{1}})ω_{n}

= − ∑_{j=1}^{i}(−1)^{j}(⋯∘î_{S^{j}}∘⋯)(L_{S^{j}}ω_{n})−∑_{j=1}^{i}(−1)^{j} ∑_{k=1}^{j−1}(ı_{S^{i}}∘⋯∘î_{S^{j}}∘⋯∘ı_{[S^{j},S^{k}]}∘⋯∘ı_{S^{1}})ω_{n}.

Interchanging j and k in the last sum and [S^{j}, S^{k}] = sgn(S)S_{[E_{j},E_{k}]} yield the first equation. The second is proved analogously.

If χ_{n}^{s} ∈ A_{n}(P) ⊗ Hom(⨂^{s} 𝔤, V) is 𝔤-equivariant, Corollary 2.6 yields

[S_{•}^{i−1}(L_{S^{j}}χ_{n}^{s})]_{n+1−i}^{E_{1},...,_{E_{j}},...,E_{i+s}} = sgn(S) ∑_{k=1}^{s}(S_{•}^{i−1}χ_{n}^{s})^{E_{1},...,_{E_{j}},...,[E_{j},E_{i+k}],...,E_{i+s}},

(we again identify Hom(⨂^{i+s} 𝔤, V) and Hom(⨂^{i} 𝔤, Hom(⨂^{s} 𝔤, V))). We obtain:

Corollary 3.5. For 𝔤-equivariant χ_{n}^{s} ∈ A_{n}(P)⊗Hom(⨂^{s} 𝔤, V) and i ≤ n+1,

{[S_{•}^{i}(dχ_{n}^{s})] − (−1)^{i}d(S_{•}^{i}χ_{n}^{s})}_{n+1−i}^{E_{1},...,E_{i+s}} =
= − sgn(S) ∑_{j=1}^{i} ∑_{k=j+1}^{i+s}(−1)^{j}(S_{•}^{i−1}χ_{n}^{s})_{n+1−i}^{E_{1},...,_{E_{j}},...,[E_{j},E_{k}],...,E_{i+s}}.

Thus for 𝔤-invariant ω_{n}, dω_{n} = 0 yields d(S_{•}ω_{n}) = 0, too.

Analogously one proves using ı_{X} ∘ ı_{Y} = −ı_{X} ∘ ı_{Y} and [L_{X} , ı_{Y}] = ı_{[X ,Y]}:

Lemma 3.6. For all ω_{n} ∈ A_{n}(P) ⊗ V , X ∈ D^{1}(P), E_{k} ∈ 𝔤, and i ≤ n,

[S_{•}^{i}(ı_{X} ω_{n})]_{n−1−i}^{E_{1},...,E_{i}} = (−1)^{i}[ı_{X} (S_{•}^{i}ω_{n})]_{n−1−i}^{E_{1},...,E_{i}},
[S_{•}^{i}(L_{X} ω_{n}) − L_{X} (S_{•}^{i}ω_{n})]_{n−i}^{E_{1},...,E_{i}} = ∑_{j=1}^{i}(−1)^{j}[S_{•}^{i−1}(ı_{[X ,S^{j}]}ω_{n})]_{n−i}^{E_{1},...,_{E_{j}},...,E_{i}}.

If X = S_{X} with X ∈ 𝔤, we get [S_{•}^{i}(ı_{S_{X}}ω_{n})]_{n−1−i}^{E_{1},...,E_{i}} = (S_{•}^{i+1}ω_{n})_{n−1−i}^{X,E_{1},...,E_{i}},

[S_{•}^{i}(L_{S_{X}}ω_{n}) − L_{S_{X}}(S_{•}^{i}ω_{n})]_{n−i}^{E_{1},...,E_{i}} = − sgn(S) ∑_{j=1}^{i}(S_{•}^{i}ω_{n})_{n−i}^{E_{1},...,[X,E_{j}],...,E_{i}}.

4. Mapping equivariant forms onto invariant forms

For our purposes we also need operators in the opposite direction, that produce V -valued forms from Hom(T (𝔤), V)-valued and 𝔤-valued forms. This can be done in a very general way and does not require a Lie group action (cf. [1]). Given forms χ_{r}^{s} ∈ A_{r}(P, Hom(⨂^{s} W, V) and ϕ_{p} = ∑_{i=1}^{m} ϕ^{i} ⊗ E_{i} ∈ A_{p}(P) ⊗ W with p, r, s − 1 ∈ ℕ_{0}, we define a V -valued form χ_{r}^{s} • ϕ_{p} in the following way [recall (3)]:

χ_{r}^{s} • ϕ_{p} = ∑_{i_{1},...,i_{s}=1}^{m} χ_{r}^{E_{i_{1}},...,E_{is}} ∧ ϕ^{i_{1}} ∧ ⋯ ∧ ϕ^{i_{s}} ∈ A(P, V).  (14)

Thus if χ_{r}^{s} ∈ A_{r}(P) ⊗ Hom(⨂^{s} W, V) then also χ_{r}^{s} • ϕ_{p} ∈ A_{r+sp}(P) ⊗ V . Linear extension defines the operator • for χ ∈ A(P, Hom(T (W), V)). Note that if χ_{r}^{s} ∈ A_{r}(P, Sym_{s}^{ς}(W, V), ς = ±, with s > 1 and ς(−1)^{p} = −1 then χ_{r}^{s} • ϕ_{p} = 0.

Since • behaves well under pullbacks and push-outs, one easily proves that • maps equivariant forms onto invariant forms (cf. [1, Lemma 7.1]):

Lemma 4.1. Let S: G × P → P be a Lie group action and L: G → Gl(W) be a left representation. If φ_{r} ∈ A_{r}(P, W) and χ ∈ A(P, Hom(T (W), V)) are equivariant (i. e., S_{g}^{⋆}φ_{r} = L(g^{− sgn(S)})_{⋆}φ_{r} and S_{g}^{⋆}χ = (L(g^{sgn(S)})^{⋆})_{⋆}χ for all g ∈ G), then χ • φ_{r} is invariant. E. g., if χ ∈ A(P, Hom(T (𝔤), V))_{equiv} and φ_{r} ∈ A_{r}(P, 𝔤)_{equiv} then χ • φ_{r} is invariant.

Analogously, if χ and φ_{r} are both 𝔤-equivariant then χ • φ_{r} is 𝔤-invariant. We are interested especially in the case where χ_{n}^{s} = S_{•}^{s}ω_{n} and we combine

both operators in the following form:

Definition 4.2. Let S be a Lie group action of G on P. Then for ω_{n} ∈ A_{n}(P, V) and θ ∈ A_{1}(P, 𝔤) we define

ω_{n} ∘^{S} θ := ∑_{i=0}^{n} \frac{(−1)^{i(n−i)}}{i!} (S_{•}^{i}ω_{n}) • θ ∈ A_{n}(P, V).

Analogously, for f: M → P and θ ∈ A_{1}(M, 𝔤), resp., linear Λ: V → W we write

(f^{⋆}ω_{n})∘^{S} θ := ∑_{i=0}^{n} \frac{(−1)^{i(n−i)}}{i!} f^{⋆}(S_{•}^{i}ω_{n}) • θ ∈ A_{n}(M, V), resp.,
(Λ_{⋆}ω_{n})∘^{S} θ := ∑_{i=0}^{n} \frac{(−1)^{i(n−i)}}{i!} Λ_{⋆}[(S_{•}^{i}ω_{n}) • θ] ∈ A_{n}(P, W), etc.

Linear extension defines ω ∘^{S} θ for ω ∈ A(P, V).

Due to Lemmas 3.3 and 4.1, ω∘^{S} θ is (𝔤-)invariant if ω is (𝔤-)invariant and θ is (𝔤-)equivariant. ω∘^{S} θ is the differential form that one obtains if ω_{p}, p ∈ P, does not act on the tangent vectors X_{p}^{i} ∈ T_{p}(P) themselves but on X_{p}^{i} + (dS^{p})_{e}θ_{p}(X_{p}^{i}). Indeed, we have:

Lemma 4.3. Let p ∈ P and X^{i} ∈ D^{1}(P). Then

(ω ∘^{S} θ)_{p}(…, X_{p}^{i}, …) = ω_{p}(…, X_{p}^{i} + (dS^{p})_{e}θ_{p}(X_{p}^{i}), …).  (15)

Proof. Let ω ∈ A_{n}(P, V). Then ω_{p}(…, X_{p}^{i} + (dS^{p})_{e}θ_{p}(X_{p}^{i}), …) =

= ∑_{i=0}^{n} \binom{n}{i} ∑_{ρ∈S_{n}} \frac{(−1)^{ρ}}{n!} ω_{p}((dS^{p})_{e}θ_{p}X_{p}^{ρ(1)}, …, (dS^{p})_{e}θ_{p}X_{p}^{ρ(i)}, X_{p}^{ρ(i+1)}, …, X_{p}^{ρ(n)})
= ∑_{i=0}^{n} \frac{1}{i!} ∑_{ρ∈S_{n}} \frac{(−1)^{ρ}}{n!} (S_{•}^{i}ω)_{p}(X_{p}^{ρ(i+1)}, …, X_{p}^{ρ(n)})[θ_{p}X_{p}^{ρ(1)}, …, θ_{p}X_{p}^{ρ(i)}]
= ∑_{i=0}^{n} \frac{(−1)^{i(n−i)}}{i!} [(S_{•}^{i}ω) • θ](X_{p}^{1}, …, X_{p}^{n}) = (ω ∘^{S} θ)_{p}(X_{p}^{1}, …, X_{p}^{n}).

Lemma 4.4. For all ω ∈ A(P, V), ϕ ∈ A_{1}(P, 𝔤) and horizontal θ ∈ A_{1}(P, 𝔤),

[(ω ∘^{S} θ)∘^{S} ϕ] = ω ∘^{S} (θ + ϕ).  (16)

Proof. If θ is horizontal then θ_{p}(dS^{p})_{e} = 0 for all p ∈ P. Thus for all vector fields X^{i}, Lemma 4.3 yields [(ω ∘^{S} θ) ∘^{S} ϕ]_{p}(…, X_{p}^{i}, …) = ω_{p}(…, X_{p}^{i} + (dS^{p})_{e}ϕ_{p}(X_{p}^{i})+(dS^{p})_{e}θ_{p}(X_{p}^{i})+(dS^{p})_{e}θ_{p}(dS^{p})_{e}ϕ_{p}(X_{p}^{i}), …) = ω_{p}(…, X_{p}^{i}+(dS^{p})_{e}(ϕ_{p}+ θ_{p})(X_{p}^{i}), …) = [ω ∘^{S} (θ + ϕ)]_{p}(…, X_{p}^{i}, …).

Again for S = L, R, let Θ^{S} ∈ A_{1}(G, 𝔤) denote the left, resp., right canonical 1-form on the Lie group G that is given by Θ^{S} = ψ^{S}(id_{𝔤}), where ψ^{S}: Alt(𝔤, V) → A(G, V)_{inv} (invariance with respect to left, resp., right multiplication) means the isomorphism that is inverse to the evaluation at e; i. e., Θ_{g}^{L}(X_{g}) = dλ_{g^{−1}}(X_{g}) and Θ_{g}^{R}(X_{g}) = dρ_{g^{−1}}(X_{g}) for all g ∈ G and X_{g} ∈ T_{g}(G). We thus have Θ^{R} = Ad •Θ^{L}, i. e., Θ_{g}^{R} = Ad(g) ∘ Θ_{g}^{L} for all g ∈ G. If f: M → G is differentiable, f^{⋆}Θ^{R} = (Ad ∘f) • f^{⋆}Θ^{L} and f^{⋆}Θ^{L} = (Ad ∘f^{−1}) • f^{⋆}Θ^{R}, where f^{−1} := η ∘ f: M → G. Also with the constant map 1 ∈ C^{∞}(G):

ψ^{S}(K) = (1 ⊗ K) • Θ^{S} ∈ A(G, V)_{inv}  for all K ∈ Alt(𝔤, V).  (17)

Let us first give an application of ∘^{S} . Suppose S is a Lie group action of G on P and ω ∈ A_{n}(P, V)_{inv} . For any differentiable g: M → G and f: M → P one would like to compute [S ∘ (g, f)]^{⋆}ω. Then in order to split this form into its portions that belong to f^{⋆}, resp., g^{⋆}, one needs ∘^{S} . In fact, the following holds:

Theorem 4.5. Let S be a Lie group action of G on P, S′ a representation of G on V on the same side and ω ∈ A(P, V) be equivariant. If g: M → G and f: M → P are differentiable, then S′_{g} ∈ A(M, Gl(W)) and

[S ∘ (g, f)]^{⋆}ω = S′_{g} • (f^{⋆}ω ∘^{S} g^{⋆}Θ_{G}^{S}) = (S′_{g} • f^{⋆}ω)∘^{S} g^{⋆}Θ_{G}^{S}.  (18)

Proof. Let X^{i} ∈ D^{1}(M) and x ∈ M. Then {[S ∘ (g, f)]^{⋆}ω}_{x}(…, X_{x}^{i}, …) =

= ω_{S(g(x),f(x))}(…, [(dS_{g(x)})_{f(x)}df_{x} + (dS^{f(x)})_{g(x)}dg_{x}]X_{x}^{i}, …)
= (S_{g(x)}^{⋆}ω)_{f(x)}(…, df_{x}X_{x}^{i} + d(S_{g^{−1}(x)} ∘ S^{f(x)})_{g(x)}dg_{x}X_{x}^{i}, …)
= S′_{g(x)} ∘ [ω_{f(x)}(…, df_{x}X_{x}^{i} + (dS^{f(x)})_{e}(g^{⋆}Θ_{G}^{S})_{x}X_{x}^{i}, …)]
= S′_{g(x)} ∘ [(f^{⋆}ω ∘^{S} g^{⋆}Θ_{G}^{S})_{x}(…, X_{x}^{i}, …)] = [S′_{g} • (f^{⋆}ω ∘^{S} g^{⋆}Θ_{G}^{S})]_{x}(…, X_{x}^{i}, …).

Corollary 4.6. If S be a Lie group action of G on P and ω ∈ A(P, V) is invariant then for any differentiable g: M → G and f: M → P

[S ∘ (g, f)]^{⋆}ω = f^{⋆}ω ∘^{S} g^{⋆}Θ_{G}^{S}.  (19)

Suppose that under the conditions of Theorem 4.5, (S^{p})^{⋆}ω_{n} is independent of p ∈ P. Then (S^{p})^{⋆}ω_{n} ∈ A_{n}(G, V) is invariant: (L^{p})^{⋆}ω = ψ^{R}(K), resp., (R^{p})^{⋆}ω = ψ^{L}(K) for a K ∈ Alt_{n}(𝔤, V). Moreover, we find (S′_{g})_{⋆}K = Ad(g^{− sgn S})^{⋆}K, so for the i = n term in the definition of ∘^{S} in Theorem 4.5 we get from (9) with −S := R for S = L, and vice versa:

S′_{g} • [f^{⋆}(S_{•}^{n}ω_{n}) • g^{⋆}Θ_{G}^{S}] = g^{⋆}ψ^{−S}(K).

The i = 0 term reads S′_{g} • f^{⋆}ω, so for ω ∈ A_{1}(G, 𝔤) we obtain

Corollary 4.7. Let L, R: G × P → P be a left, resp., right action of G on P and f: M → P and g: M → G be differentiable; K ∈ Alt_{1}(𝔤, 𝔤) be invertible and ω ∈ A_{1}(P, 𝔤). Then K(Ad ∘g)K^{−1} ∈ A_{0}(M, Alt_{1}(𝔤, 𝔤)) and we have

1. If (L^{p})^{⋆}ω = ψ^{R}(K) and L_{c}^{⋆}ω = K Ad(c)K^{−1} ∘ ω for all p ∈ P, c ∈ G, then

[L ∘ (g, f)]^{⋆}ω = K(Ad ∘g)K^{−1} • f^{⋆}ω + g^{⋆}ψ^{R}(K).

2. If (R^{p})^{⋆}ω = ψ^{L}(K) and R_{c}^{⋆}ω =KAd(c^{−1})K^{−1}∘ω for all p ∈ P, c ∈ G, then

[R ∘ (g, f)]^{⋆}ω = K(Ad ∘g^{−1})K^{−1} • f^{⋆}ω + g^{⋆}ψ^{L}(K).

For P = G and L = λ, resp., R = ρ, Corollary 4.7 gives a proof for the formulas for (f · g)^{⋆}Θ^{S} and (f^{−1})^{⋆}Θ^{S}: put K = id_{𝔤} and observe that (f · f^{−1})^{⋆} = e^{⋆} = 0, where e: M → {e} ⊆ G is the constant map onto the neutral element. Then Corollary 4.7 yields:

(f · g)^{⋆}Θ^{L} = (Ad ∘g^{−1}) • f^{⋆}Θ^{L} + g^{⋆}Θ^{L},
(f · g)^{⋆}Θ^{R} = f^{⋆}Θ^{R} + (Ad ∘f) • g^{⋆}Θ^{R},
(f^{−1})^{⋆}Θ^{L} = −(Ad ∘f) • f^{⋆}Θ^{L} = −f^{⋆}Θ^{R},
(f^{−1})^{⋆}Θ^{R} = −(Ad ∘f^{−1}) • f^{⋆}Θ^{R} = −f^{⋆}Θ^{L}.

We already stated another application of ∘^{S} in Section 1.: if Γ is a connection on a principal bundle P (M, G) and B(M, F, G) is an associated fiber bundle with a left Lie group action L: G × F → F (we also use L for the natural extensions to the bundle charts U_{α} × F), then for any ϕ ∈ A(F, V), the local vertical projections of pr_{F}^{⋆} ϕ ∈ A(U_{α} × F) are given by

(pr_{F}^{⋆} ϕ)v^{α} = (pr_{F}^{⋆} ϕ)∘^{L} (pr_{U_{α}}^{⋆} A^{α}).  (20)

This follows immediately from Lemma 4.3 and v_{(x,f)}^{α}(X, Y) = (0, Y +(dL^{f})_{e}A_{x}^{α}(X)) for all (X, Y) ∈ T_{x}(U_{α}) ⊕ T_{f}(F). If ϕ is invariant, then one easily computes that

L_{g}^{⋆}[(pr_{F}^{⋆} ϕ)v^{α}] = (pr_{F}^{⋆} ϕ)∘^{L} [(pr_{U_{α}}^{⋆}(Ad(g^{−1})_{⋆}A^{α})].  (21)

Let U_{αβ} := U_{α} ∩U_{β} ≠ Ø and T_{βα} := (ψ_{β}|_{U_{αβ}})^{−1})∘(ψ_{α}|_{U_{αβ}}) denote the maps for the change of bundle charts. If g_{βα}: U_{αβ} → G are the transition functions, then the maps T_{βα} are given by

T_{βα} = (pr_{U_{αβ}}, L ∘ (g_{βα} ∘ pr_{U_{αβ}}, pr_{F})) = L ∘ (g_{βα} ∘ pr_{U_{αβ}}, id_{U_{αβ}×F}). (22)

For computations on fiber bundles one needs to know how differential forms transform under such a change of bundle charts, e. g., in order to check whether a collection of local forms defines a global form. In view of this question we obtain from (19) for ϕ ∈ A(F, V)_{inv} :

T_{βα}^{⋆}(pr_{F}^{⋆} ϕ) = (pr_{F}^{⋆} ϕ)∘^{L} (g_{βα} ∘ pr_{U_{αβ}})^{⋆}Θ^{L}.  (23)

Recall from the theory of connections that the gauge potentials A^{α} transform according to A^{α} = (Ad ∘g_{αβ}) • A^{β} + g_{βα}^{⋆}Θ^{L}, where we omitted the restriction to U_{αβ}. In fact this is a consequence of Corollary 4.7.2 for ω = ω^{Γ}, K = id_{𝔤} and f = σ_{β,e} because σ_{α,e} = R ∘ (g_{βα}, σ_{β,e}). Further observe that in view of Lemma 4.4, (g_{βα} ∘ pr_{U_{αβ}})^{⋆}Θ^{L} and (pr_{U_{α}}^{⋆} A^{α}) are both horizontal. Although due to (21) for invariant ϕ ∈ A(F, V), the vertical form (pr_{F}^{⋆} ϕ)v^{α} needs not be invariant and thus Corollary 4.6 does not apply, one quickly checks tracing the proof of Theorem 4.5, that

T_{βα}^{⋆}[(pr_{F}^{⋆} ϕ)v^{β}] = [(pr_{F}^{⋆} ϕ)∘^{L} [(pr_{U_{αβ}}^{⋆}(Ad(g_{αβ})_{⋆}A^{β})]]∘^{L} (g_{βα} ∘ pr_{U_{αβ}})^{⋆}Θ^{L}
= [(pr_{F}^{⋆} ϕ)∘^{L} [pr_{U_{αβ}}^{⋆}((Ad ∘g_{αβ}) • A^{β} + g_{βα}^{⋆}Θ^{L})] = (pr_{F}^{⋆} ϕ)v^{α}.

We have thus proved Theorem 4.8. If ϕ ∈ A(F, V) is invariant, {(pr_{F}^{⋆} ϕ)v^{α} ∈ A(U_{α} × F, V)}_{α∈A}, resp., {(π_{α}^{⋆}ϕ)v^{α} ∈ A(π^{−1}(U_{α}), V)}_{α∈A} defines a global form ϕv ∈ A(B, V). If ϕ is invariant and locally vertical, then {π_{α}^{⋆}ϕ}_{α∈A} is global.

Generalizations of this theorem to combinations of equivariant differential forms as in Lemma 4.1 are possible. E. g., if χ ∈ A(F, Hom(T (𝔤), V)) is equivariant and F^{α} ∈ A_{2}(U_{α}, 𝔤) denote the local gauge fields that are obtained from the curvature 2-form Ω^{Γ} as F^{α} := σ_{α,e}^{⋆}Ω^{Γ}, then {[(π_{α}^{⋆}χ)v^{α}] • (π^{⋆}F^{α}) ∈ A(π^{−1}(U_{α}), V)}_{α∈A} defines a global form χv • F on the bundle B, cf. [2]. Locally this form is given by

[(pr_{F}^{⋆} χ)v^{α}] • (pr_{U_{α}}^{⋆} F^{α}) = [(pr_{F}^{⋆} ϕ)∘^{L} (pr_{U_{α}}^{⋆} A^{α})] • (pr_{U_{α}}^{⋆} F^{α}).  (24)

5. Differentiation of the combined forms

From the previous applications it should be clear that it is important to control interior products, Lie derivatives and above all, the exterior derivatives of the differential forms χ_{r}^{s} • ϕ_{p} and ω_{n} ∘^{S} θ. E. g., one is interested in the exterior derivative d(ϕv) from Theorem 4.8 if ϕ is closed. Thus the computation of d(ω_{n}∘^{S} θ) and, more generally, of d[(χ_{n}^{s}∘^{S} θ) • ϕ_{p}] will be the main task of this last section. Unfortunately, the most general formulas turn out to be quite voluminous. For this reason, we will then discuss the important special cases.

We need to generalize • to give formulas for χ_{r}^{s} • (ϕ_{p} + ψ_{p}) and d(χ_{r}^{s} • ϕ_{p}), cf. [1]. First we observe that χ_{r}^{s} ∈ A_{r}(P, Sym_{s}^{ς}(𝔤, V)), ς = ±, naturally defines

χ_{r}^{s′;s′′} ∈ A_{r}(P, Sym_{s′}^{ς}(𝔤, Sym_{s′′}^{ς}(𝔤, V))) for all s′, s′′ ∈ ℕ_{0}, s′ + s′′ = s. (25)

For any such combination of s′ and s′′, χ_{r}^{s} • (ϕ_{p}^{q} + ψ_{p}^{q}) will contain terms, where s′ factors of ϕ_{p}^{q} and s′′ terms of ψ_{p}^{q} serve as input for χ_{r}^{s}. In order to cover this situation, we need the following two definitions.

Generally, for χ_{r}^{s′;s′′} ∈ A_{r}(P, Hom(⨂^{s′} 𝔤, Hom(⨂^{s′′} 𝔤, V))), s′, s′′ ∈ ℕ, r ∈ ℕ_{0}, and any E_{i} ∈ 𝔤, i = 1, …, s′, we define

χ_{r}^{E_{1},...,E_{s′};s′′} := [(E_{1} ⊗ ⋯ ⊗ E_{s′})^{⋆}]_{⋆}χ_{r}^{s′;s′′} ∈ A_{r}(P, Hom(⨂^{s′′} 𝔤, V))).

[Thus if χ_{r}^{s′,s′′} ∈ A_{r}(P) ⊗ Hom(⨂^{s′} 𝔤, Hom(⨂^{s′′} 𝔤, V)) then χ_{r}^{E_{1},...,E_{s′};s′′} ∈ A_{r}(P) ⊗ Hom(⨂^{s′′} 𝔤, Z).] For any such differential form χ_{r}^{s′;s′′} and any ϕ_{p} ∈ A_{p}(P) ⊗ 𝔤, let V′ := Hom(⨂^{s′′} 𝔤, Z) and χ̃_{r}^{s′} := χ_{r}^{s′;s′′} ∈ A_{r}(P, Hom(⨂^{s′} 𝔤, V′)), and define

χ_{r}^{s′;s′′} ◀ ϕ_{p} := χ̃_{r}^{s′} • ϕ_{p} ∈ A_{r+s′p}(P, Hom(⨂^{s′′} 𝔤, V))).

If ϕ_{p} = ∑_{j=1}^{m} ϕ^{j} ⊗ E_{j} ∈ A_{p}(P) ⊗ W then we obtain

χ_{r}^{s′;s′′} ◀ ϕ_{p} = ∑_{j_{1},...,j_{s′}=1}^{m} χ_{r}^{E_{j_{1}},...,E_{j_{s′}};s′′} ∧ ϕ^{j_{1}} ∧ ⋯ ∧ ϕ^{j_{s′}},  (26)

which shows that χ_{r}^{s′;s′′} ◀ ϕ_{p} ∈ A_{r+s′p}(P) ⊗ Hom(⨂^{s′′} 𝔤, V)) if χ_{r}^{s′;s′′} ∈ A_{r}(P) ⊗ Hom(⨂^{s′} 𝔤, Hom(⨂^{s′′} 𝔤, V)).

Gross 13

We also introduce generalizations \binom{s}{k}_{±} of the ordinary binomial coefficients:

\binom{s}{k}_{+} := \binom{s}{k}, \binom{s}{k}_{−}:= \{ \begin{matrix}0, & if s even and k odd, \\ \binom{[s/2]}{[k/2]}, & else (for r ∈ ℝ, [r] := max_{z∈ℤ}\{z ≤ r\}). \end{matrix} (27)

Thus \binom{s}{k}_{±} = \binom{s}{s−k}_{±} as before. Now if χ_{r}^{s} ∈ A_{r}(P, Sym_{s}^{ς}(𝔤, V)) and ℓ := ς(−1)^{p} then χ_{r}^{s} • (ϕ_{p} + ψ_{p}) can be written as

χ_{r}^{s} • (ϕ_{p} + ψ_{p}) = ∑_{k=0}^{s} \binom{s}{k}_{ℓ}(χ_{r}^{k;s−k} ◀ ϕ_{p}) • ψ_{p} = ∑_{k=0}^{s} \binom{s}{k}_{ℓ}(χ_{r}^{k;s−k} ◀ ψ_{p}) • ϕ_{p}.

If χ_{r}^{s} ∈ A_{r}(P) ⊗ Sym_{s}^{ς}(𝔤, V) we obtain for d(χ_{r}^{s} • ϕ_{p}), ı_{X} (χ_{r}^{s} • ϕ_{p}) and L_{X} (χ_{r}^{s} • ϕ_{p}):

d(χ_{r}^{s} • ϕ_{p}) = (dχ)_{r+1}^{s} • ϕ_{p} + (−1)^{r} \begin{matrix}( & {s \atop 1} & ) \\ ℓ \end{matrix} [χ_{r}^{1;s−1} ◀ (dϕ)_{p+1}] • ϕ_{p},  (28)
ı_{X} (χ_{r}^{s} • ϕ_{p}) = (ı_{X} χ)_{r−1}^{s} • ϕ_{p} + (−1)^{r} \begin{matrix}( & {s \atop 1} & ) \\ ℓ \end{matrix} [χ_{r}^{1;s−1} ◀ (ı_{X} ϕ)_{p−1}] • ϕ_{p}, (29)
L_{X} (χ_{r}^{s} • ϕ_{p}) = (L_{X} χ)_{r}^{s} • ϕ_{p} + \begin{matrix}( & {s \atop 1} & ) \\ ℓ \end{matrix} [χ_{r}^{1;s−1} ◀ (L_{X} ϕ)_{p}] • ϕ_{p}.  (30)

Note that whenever χ_{r}^{s} • ϕ_{p} ≠ 0, \binom{s}{k}_{ℓ} = \binom{s}{k}. If χ_{r}^{s} ∈A_{r}(P)⊗Hom(⨂^{s} 𝔤, V), we get

d(χ_{r}^{s} • ϕ_{p}) = (dχ_{r}^{s})• ϕ_{p} + (−1)^{r} ∑_{j=0}^{s−1}(−1)^{jp}[(χ_{r}^{j;s−j}◀ ϕ_{p})^{1;s−j−1}◀ dϕ_{p}] • ϕ_{p}, (31)
ı_{X} (χ_{r}^{s} • ϕ_{p}) = (ı_{X} χ_{r}^{s})• ϕ_{p} + (−1)^{r}∑_{j=0}^{s−1}(−1)^{jp}[(χ_{r}^{j;s−j}◀ ϕ_{p})^{1;s−j−1}◀ ı_{X} ϕ_{p}] • ϕ_{p},(32)
L_{X} (χ_{r}^{s} • ϕ_{p}) = (L_{X} χ_{r}^{s}) • ϕ_{p} + ∑_{j=0}^{s−1}(−1)^{jp}[(χ_{r}^{j;s−j} ◀ ϕ_{p})^{1;s−j−1} ◀ L_{X} ϕ_{p}] • ϕ_{p}. (33)

Now for the operator ∘^{S} : if θ is horizontal one quickly verifies analogously to the proof of Lemma 4.4 that ı_{S_{X}}(ω ∘^{S} θ) = (ı_{S_{X}}ω)∘^{S} θ. Thus we have: Lemma 5.1. If χ_{r}^{s} ∈ A_{r}(P) ⊗ Sym_{s}^{ς}(𝔤, V) and θ is horizontal then

ı_{S_{X}}[(χ_{r}^{s}∘^{S} θ)•ϕ_{p}] = [(ı_{S_{X}}χ)_{r−1}^{s}∘^{S} θ]•ϕ_{p} +(−1)^{r} \begin{matrix}( & {s \atop 1} & ) \\ ℓ \end{matrix} [(χ_{r}^{s}∘^{S} θ)_{r}^{1;s−1} ◀(ı_{S_{X}}ϕ)_{p−1}]•ϕ_{p};

If χ_{r}^{s} ∈ A_{r}(P) ⊗ Hom(⨂^{s} 𝔤, V) and both θ and ϕ_{p} are horizontal then

ı_{S_{X}}[(χ_{r}^{s} ∘^{S} θ) • ϕ_{p}] = [(ı_{S_{X}}χ)_{r−1}^{s} ∘^{S} θ] • ϕ_{p}.

Recall ∧_{V} from Section 2. and let ∧_{𝔤} denote the exterior product for 𝔤valued differential forms which is induced by ad: 𝔤 × 𝔤 → 𝔤. Lemma 5.2. Let χ_{n}^{1} ∈ A_{n}(P, Hom(𝔤, V)) and {E_{k}}_{k=1,...,dim 𝔤} be a basis for 𝔤. Then for θ_{q} = ∑_{k=1}^{dim 𝔤} θ_{q}^{k} ⊗ E_{k} ∈ A_{q}(P, 𝔤) and ϕ_{p} = ∑_{l=1}^{dim 𝔤} ϕ_{p}^{l} ⊗ E_{l} ∈ A_{p}(P, 𝔤),

χ_{n}^{1} • (θ_{q} ∧_{𝔤} ϕ_{p}) = ∑_{j=1}^{dim 𝔤} χ_{n}^{E_{j}} ∧ (θ_{q} ∧_{𝔤} ϕ_{p})^{j} = ∑_{k,l=1}^{dim 𝔤} χ_{n}^{[E_{k},E_{l}]} ∧ θ_{q}^{k} ∧ ϕ_{p}^{l}.  (34)

Proof. Let [E_{k}, E_{l}] = ∑_{j=1}^{dim 𝔤} c_{kl}^{j} E_{j} with structure constants c_{kl}^{j}. Then by definition, θ_{q} ∧_{𝔤} ϕ_{p} = ∑_{k,l=1}^{dim 𝔤} θ_{q}^{k} ∧ ϕ_{p}^{l} ⊗ [E_{k}, E_{l}] = ∑_{j,k,l=1}^{dim 𝔤} c_{kl}^{j} θ_{q}^{k} ∧ ϕ_{p}^{l} ⊗ E_{j} =: ∑_{j}^{dim 𝔤}(θ_{q} ∧_{𝔤} ϕ_{p})^{j} ⊗ E_{j}, thus

χ_{n}^{1}•(θ_{q}∧_{𝔤}ϕ_{p}) = ∑_{j=1}^{dim 𝔤}χ_{n}^{E_{j}}∧(θ_{q}∧_{𝔤}ϕ_{p})^{j} = ∑_{j,k,l=1}^{dim 𝔤}c_{kl}^{j} χ_{n}^{E_{j}}∧θ_{q}^{k}∧ϕ_{p}^{l} = ∑_{k,l=1}^{dim 𝔤}χ_{n}^{∑_{j=1}^{dim 𝔤}c_{kl}^{j}E_{j}}∧θ_{q}^{k}∧ϕ_{p}^{l}.

We will divide the computation of the exterior derivatives into two steps:

Proposition 5.3. Let S be a Lie group action of G on P, θ_{q} ∈ A_{q}(P, 𝔤), ϕ_{p} ∈ A_{p}(P, 𝔤) and χ_{n}^{s} ∈ A_{n}(P) ⊗ Hom(⨂^{s} 𝔤, V) 𝔤-equivariant. Then for all i ≤ n + 1 with ℓ = (−1)^{q−1}

{[d(S_{•}^{i}χ_{n}^{s}) − (−1)^{i}S_{•}^{i}(dχ_{n}^{s})]_{n+1}^{i;s} ◀ θ_{q}}^{s} • ϕ_{p} =

= sgn(S)\{ − \binom{i}{2}_{ℓ}\{[(S_{•}^{i−1}χ_{n}^{s})_{n+1−i}^{i−2;s+1}◀θ_{q}]^{1;s}◀(θ_{q}∧_{𝔤}θ_{q})\}^{s} •ϕ_{p} +

+\binom{i}{1}_{ℓ} ∑_{k=1}^{s}(−1)^{qp(k−1)}\{[[(S_{•}^{i−1}χ_{n}^{s})_{n+1−i}^{i−1;s}◀θ_{q}]^{k−1;s−k+1}◀ϕ_{p}]^{1;s−k}◀(θ_{q}∧_{𝔤}ϕ_{p})\}^{s−k}•ϕ_{p}\}.

Proof. With the notation of the previous lemma, we evaluate the left side using (14). Then by Corollary 3.5,

∑_{l_{1},...,l_{i+s}}^{dim 𝔤} ; {d(S_{•}^{i}χ_{n}^{s}) − (−1)^{i}S_{•}^{i}(dχ_{n}^{s})}_{n+1−i}^{E_{l_{1}},...,E_{l_{i+s}}}∧⋯∧θ_{q}^{l_{i}} ∧ϕ_{p}^{l_{i+1}} ∧⋯∧ϕ_{p}^{l_{i+s}} =

= sgn(S) ∑_{j=1}^{i} ∑_{k=j+1}^{i+s} (−1)^{i+j} ∑_{l_{1},...,l_{i+s}}^{dim 𝔤} (S_{•}^{i−1}χ_{n}^{s})_{n+1−i}^{E_{l_{1}},...,_{E_{l_{j}}},...,[E_{l_{j}},E_{l_{k}}],...,E_{l_{i+s}}}∧⋯∧θ_{q}^{l_{i}} ∧ϕ_{p}^{l_{i+1}} ∧⋯

= − sgn(S) ∑_{j=1}^{i} ∑_{k=j+1}^{i}ℓ^{k−j+1} ∑_{l_{1},...,l_{i+s}}^{dim 𝔤} (S_{•}^{i−1}χ_{n}^{s})_{n+1−i}^{E_{l_{1}},...,_{E_{l_{j}}} ,...,_{E_{l_{k}}} ,...,E_{l_{i}},[E_{l_{j}} ,E_{l_{k}}];...,E_{l_{i+s}}} ∧

∧ θ_{q}^{l_{1}} ∧ ⋯ θ̂_{q}^{l_{j}} ⋯ θ̂_{q}^{l_{k}} ⋯ ∧ θ_{q}^{l_{i}} ∧ θ_{q}^{l_{j}} ∧ θ_{q}^{l_{k}} ∧ ϕ_{p}^{l_{i+1}} ∧ ⋯ ∧ ϕ_{p}^{l_{i+s}}

+ sgn(S) ∑_{j=1}^{i} ℓ^{i−j} ∑_{k=1}^{s}(−1)^{qp(k−1)} ∑_{l_{1},...,l_{i+s}}^{dim 𝔤} (S_{•}^{i−1}χ_{n}^{s})_{n+1−i}^{E_{l_{1}},...,_{E_{l_{j}}},...,E_{l_{i}};...,[E_{l_{j}},E_{l_{i+k}}],...,E_{l_{i+s}}} ∧

∧ θ_{q}^{l_{1}} ∧ ⋯ θ̂_{q}^{l_{j}} ⋯ ∧ θ_{q}^{l_{i}} ∧ ϕ_{p}^{l_{i+1}} ∧ ⋯ ∧ θ_{q}^{l_{j}} ∧ ϕ_{p}^{l_{i+k}} ∧ ⋯ ∧ ϕ_{p}^{l_{i+s}}

= − sgn(S) ∑_{j=1}^{i} ∑_{k=j+1}^{i}ℓ^{k−j+1} ∑_{l_{1},...,l_{i+s−1}}^{dim 𝔤} (S_{•}^{i−1}χ_{n}^{s})_{n+1−i}^{E_{l_{1}},...,E_{l_{i−1}};E_{l_{i}},...,E_{l_{i+s}}} ∧
∧ θ_{q}^{l_{1}} ∧ ⋯ ∧ θ_{q}^{l_{i−2}} ∧ (θ_{q} ∧_{𝔤} θ_{q})^{l_{i−1}} ∧ ϕ_{p}^{l_{i}} ∧ ⋯ ∧ ϕ_{p}^{l_{i+s−1}}

+ sgn(S) ∑_{j=1}^{i} ℓ^{i−j} ∑_{k=1}^{s}(−1)^{qp(k−1)} ∑_{l_{1},...,l_{i+s−1}}^{dim 𝔤} (S_{•}^{i−1}χ_{n}^{s})_{n+1−i}^{E_{l_{1}},...,E_{l_{i−1}};E_{l_{i}},...,E_{l_{i+s}}} ∧

∧ θ_{q}^{l_{1}} ∧ ⋯ ∧ θ_{q}^{l_{i−1}} ∧ ϕ_{p}^{l_{i}} ∧ ⋯ ∧ (θ_{q} ∧_{𝔤} ϕ_{p})^{l_{i+k−1}} ∧ ⋯ ∧ ϕ_{p}^{l_{i+s−1}},

by (34). Since ∑_{j=1}^{i} ∑_{k=j+1}^{i} ℓ^{k−j+1} =\binom{i}{2}_{ℓ} and ∑_{j=1}^{i}ℓ^{i−j} =\binom{i}{1}_{ℓ}, all follows from (26).

Corollary 5.4. Suppose θ ∈ A_{1}(P, 𝔤) and χ_{n}^{s} ∈ A_{n}(P)⊗Sym_{s}^{ς}(𝔤, V) in Proposition 5.3, then with ℓ = ς(−1)^{p} for all i ≤ n + 1

{[d(S_{•}^{i}χ_{n}^{s})]^{i;s} ◀ θ}^{s} • ϕ_{p} − (−1)^{i}{[S_{•}^{i}(dχ_{n}^{s})]^{i;s} ◀ θ}^{s} • ϕ_{p} =
= − sgn(S) \binom{i}{2}\{[(S_{•}^{i−1}χ_{n}^{s})_{n+1−i}^{i−2;s+1} ◀ θ]^{1;s} ◀ (θ ∧_{𝔤} θ)\}^{s} • ϕ_{p}
+ sgn(S) i \binom{s}{1}_{ℓ}\{[(S_{•}^{i−1}χ_{n}^{s})^{i−1;s} ◀ θ]^{1;s−1} ◀ (θ ∧_{𝔤} ϕ_{p})\}^{s−1} • ϕ_{p}.

Proof. This follows from (−1)^{p(k−1)}[[(S_{•}^{i−1}χ_{n}^{s})_{n+1−i}^{i−1;s}◀θ]^{k−1;s−k+1}◀ϕ_{p}]^{1;s−k}◀(θ∧_{𝔤}ϕ_{p}) = ℓ^{k−1}[[(S_{•}^{i−1}χ_{n}^{s})_{n+1−i}^{i−1;s}◀θ]^{1;s−1}◀(θ∧_{𝔤}ϕ_{p})]^{k−1;s−k}◀ϕ_{p} and ∑_{k=1}^{s} ℓ^{k−1} = \binom{s}{1}_{ℓ}.

Theorem 5.5. Let S be a Lie group action of G on P, θ ∈ A_{1}(P, 𝔤), ϕ_{p} ∈ A_{p}(P, 𝔤) and χ_{n}^{s} ∈ A_{n}(P) ⊗ Sym_{s}^{ς}(𝔤, V) 𝔤-equivariant. If ℓ := ς(−1)^{p}, then

d[(χ_{n}^{s} ∘^{S} θ) • ϕ_{p}] − [(dχ_{n}^{s})∘^{S} θ] • ϕ_{p} =
= \{[(S_{•}χ_{n}^{s})∘^{S} θ]^{1;s} ◀ (dθ − sgn(S) \frac{1}{2} θ ∧_{𝔤} θ)\}^{s} • ϕ_{p}
+(−1)^{n}\binom{s}{1}_{ℓ} [(χ_{n}^{s} ∘^{S} θ)^{1;s−1} ◀ (dϕ_{p} − sgn(S) θ ∧_{𝔤} ϕ_{p})]^{s−1} • ϕ_{p}.

Proof. By linearity of d and • in its left argument we obtain for the left side

∑_{i=0}^{n} \frac{(−1)^{in−i}}{i!} d\{[(S_{•}^{i}χ_{n}^{s})^{i;s} ◀ θ] • ϕ_{p}\} − ∑_{i=0}^{n+1} \frac{(−1)^{in}}{i!} \{[S_{•}^{i}(dχ_{n}^{s})]^{i;s} ◀ θ\} • ϕ_{p} =

= ∑_{i=0}^{n} \frac{(−1)^{in−i}}{i!} [d(S_{•}^{i}χ_{n}^{s})^{i;s}◀ θ] • ϕ_{p} + \binom{s}{1}_{ℓ} ∑_{i=0}^{n} \frac{(−1)^{in−n−i}}{i!} \{[(S_{•}^{i}χ_{n}^{s})^{i;s}◀ θ]^{1;s−1}◀ dϕ_{p}\} • ϕ_{p}

− ∑_{i=1}^{n} \frac{(−1)^{in−n−i}}{(i−1)!} \{[(S_{•}^{i}χ_{n}^{s})^{i−1;s+1}◀θ]^{1;s}◀ dθ\} • ϕ_{p} − ∑_{i=0}^{n+1} \frac{(−1)^{in}}{i!} \{[S_{•}^{i}(dχ_{n}^{s})]^{i;s}◀ θ\} • ϕ_{p}

by (28). With Corollary 5.4 we get

∑_{i=0}^{n} \frac{(−1)^{in−i}}{i!} [d(S_{•}^{i}χ_{n}^{s})^{i;s} ◀ θ] • ϕ_{p} − ∑_{i=0}^{n+1} \frac{(−1)^{in}}{i!} \{[S_{•}^{i}(dχ_{n}^{s})]^{i;s} ◀ θ\} • ϕ_{p} =

= − ∑_{i=2}^{n+1} \frac{(−1)^{in−i}}{(i−2)!} \{[(S_{•}^{i−1}χ_{n}^{s})^{i−2;s+1} ◀ θ]^{1;s} ◀ (sgn(S) \frac{1}{2} θ ∧_{𝔤} θ)\} • ϕ_{p}

+\binom{s}{1}_{ℓ} ∑_{i=1}^{n+1} \frac{(−1)^{in−i}}{(i−1)!} \{[(S_{•}^{i−1}χ_{n}^{s})^{i−1;s} ◀ θ]^{1;s−1} ◀ (sgn(S) θ ∧_{𝔤} ϕ_{p})\} • ϕ_{p}.

Finally we put all together and use S_{•}^{i+1}χ_{n}^{s} = (−1)^{i}S_{•}^{i}(S_{•}χ_{n}^{s}) from (13).

For χ_{n}^{s} ∈ A_{n}(P) ⊗ Hom(⨂^{s} 𝔤, V), the last term in Theorem 5.5 reads

∑_{k=1}^{s}(−1)^{n+p(k−1)}{[[(χ_{n}^{s} ∘^{S} θ)^{k−1;s−k+1} ◀ ϕ_{p}]^{1;s−k} ◀ (dϕ_{p} − sgn(S) θ ∧_{𝔤} ϕ_{p})]^{s−k} • ϕ_{p}}

as a consequence of Proposition 5.3, cf. (31). In any case we get the following Corollary 5.6. If S is a Lie group action of G on P, χ_{n}^{s} ∈ A_{n}(P)_{𝔤−equiv} ⊗ Hom(⨂^{s} 𝔤, V), and θ ∈ A_{1}(P, 𝔤), ϕ_{p} ∈ A_{p}(P, 𝔤) with dϕ_{p} = sgn(S) θ ∧_{𝔤} ϕ_{p}, then

d[(χ_{n}^{s} ∘^{S} θ) • ϕ_{p}] = [(dχ_{n}^{s})∘^{S} θ] • ϕ_{p} + \{[(S_{•}χ_{n}^{s})∘^{S} θ]^{1;s}◀ (dθ − sgn(S)\frac{1}{2}θ ∧_{𝔤} θ)\}^{s}• ϕ_{p}.

Now suppose, θ is a pullback of an invariant 1-form on G. Then the Maurer-Cartan identities dΘ^{S} = sgn(S)\frac{1}{2}Θ^{S} ∧_{𝔤} Θ^{S} and (17) give

Corollary 5.7. Let S be a Lie group action of G on P, f: P → G differentiable, K ∈ End(𝔤) and χ_{n}^{s} ∈ A_{n}(P) ⊗ Hom(⨂^{s} 𝔤, V) 𝔤-equivariant.

1. If χ_{n}^{s} ∈ A_{n}(P) ⊗ Sym_{s}^{ς}(𝔤, V) and ϕ_{p} ∈ A_{p}(P, 𝔤), then

d[(χ_{n}^{s} ∘^{S} f^{⋆}ψ^{S}(K)) • ϕ_{p}] = [(dχ_{n}^{s})∘^{S} f^{⋆}ψ^{S}(K)] • ϕ_{p}

+(−1)^{n}\binom{s}{1}_{ℓ} [(χ_{n}^{s} ∘^{S} f^{⋆}ψ^{S}(K))^{1;s−1} ◀ (dϕ_{p} − sgn(S) f^{⋆}ψ^{S}(K) ∧_{𝔤} ϕ_{p})]^{s−1}• ϕ_{p},

d[(χ_{n}^{s} ∘^{S} f^{⋆}Θ^{S}) • ϕ_{p}] = [(dχ_{n}^{s})∘^{S} f^{⋆}Θ^{S}] • ϕ_{p}

+(−1)^{n}\binom{s}{1}_{ℓ} [(χ_{n}^{s} ∘^{S} f^{⋆}Θ^{S})^{1;s−1} ◀ (dϕ_{p} − sgn(S)f^{⋆}Θ^{S} ∧_{𝔤} ϕ_{p})]^{s−1}• ϕ_{p}.

2. For ϕ_{p} ∈ A_{p}(P, 𝔤) with dϕ_{p} = sgn(S)f^{⋆}ψ^{S}(K) ∧_{𝔤} ϕ_{p}, e. g. for ϕ_{2} = d(f^{⋆}ψ^{S}(K)),

d[(χ_{n}^{s} ∘^{S} f^{⋆}ψ^{S}(K)) • ϕ_{p}] = [(dχ_{n}^{s})∘^{S} f^{⋆}ψ^{S}(K)] • ϕ_{p},
d[(χ_{n}^{s} ∘^{S} f^{⋆}Θ^{S}) • ϕ_{p}] = [(dχ_{n}^{s})∘^{S} f^{⋆}Θ^{S}] • ϕ_{p}.

Finally, in the case s = 0, Theorem 5.5 yields

Corollary 5.8. If S is a Lie group action of G on P and ω_{n} ∈ A_{n}(P) ⊗ V is 𝔤-invariant, then for all θ ∈ A_{1}(P, 𝔤)

d(ω_{n} ∘^{S} θ) = (dω_{n})∘^{S} θ + [(S_{•}ω_{n})∘^{S} θ]^{1} ◀ (dθ − \frac{1}{2} sgn(S) θ ∧_{𝔤} θ).

For any f: P → G, K ∈ End(𝔤), especially K = id_{𝔤}, we thus obtain

d(ω_{n} ∘^{S} f^{⋆}ψ^{S}(K)) = (dω_{n})∘^{S} f^{⋆}ψ^{S}(K), d(ω_{n} ∘^{S} f^{⋆}Θ^{S}) = (dω_{n})∘^{S} f^{⋆}Θ^{S}.

Recall that the gauge fields F^{α} ∈ A_{2}(U_{α}, 𝔤) are given by F^{α} = dA^{α} + \frac{1}{2}A^{α} ∧_{𝔤} A^{α}. For that reason we are interested especially in the case where ϕ_{2} = dθ − \frac{1}{2} sgn(S) θ ∧_{𝔤} θ. Using θ ∧_{𝔤} (θ ∧_{𝔤} θ) = 0 one easily checks that this yields dϕ_{2} = sgn(S) θ ∧_{𝔤} ϕ_{2}. Thus Corollary 5.6 reads

d[(χ_{n}^{s} ∘^{S} θ) • ϕ_{2}] = [(dχ_{n}^{s})∘^{S} θ] • ϕ_{2} + [(S_{•}χ_{n}^{s})∘^{S} θ] • ϕ_{2}.

Now S_{•}χ_{n}^{s} ∈ A_{n−1}(P, Hom(𝔤, Hom(⨂^{s} 𝔤, V))) ≅ A_{n−1}(P, Hom(⨂^{s+1} 𝔤, V)). Since ϕ_{2} has even degree, only the symmetric part of Hom(⨂^{s+1} 𝔤, V) counts [e.g., confer (14)]. So [(S_{•}χ_{n}^{s})∘^{S} θ]•ϕ_{2} = Sym_{⋆}[(S_{•}χ_{n}^{s})∘^{S} θ]•ϕ_{2} = [Sym_{⋆}(S_{•}χ_{n}^{s})∘^{S} θ]•ϕ_{2}, because ∘^{S} only acts on A(P) and commutes with any operation on Hom(⨂^{s+1} 𝔤, V). This leads to the following definition:

Definition 5.9. For χ_{n}^{s} ∈ A_{n}(P, Hom(⨂^{s} 𝔤, V)) and S: G × P → P, we define

S_{•}^{∨}χ_{n}^{s} := Sym_{⋆}(S_{•}χ_{n}^{s}) ∈ A_{n−1}(P, Sym_{s+1}(𝔤, V)).

Corollary 5.10. If S is a Lie group action of G on P, χ_{n}^{s} ∈ A_{n}(P)_{𝔤−equiv} ⊗ Hom(⨂^{s} 𝔤, V), θ ∈ A_{1}(P, 𝔤) and ϕ_{2} = dθ − \frac{1}{2} sgn(S) θ ∧_{𝔤} θ ∈ A_{2}(P, 𝔤), then

d[(χ_{n}^{s} ∘^{S} θ) • ϕ_{2}] = [(dχ_{n}^{s})∘^{S} θ] • ϕ_{2} + [(S_{•}^{∨}χ_{n}^{s})∘^{S} θ] • ϕ_{2}.

Extend the symmetric product ∨ in Sym(𝔤, ℝ) ≅ S(𝔤^{∗}) to Sym(𝔤, V), whenever a bilinear map ϕ: V ×V → V is given. Equip A(P)⊗Sym(𝔤, V) with the gradation induced by A(P), then we obtain from (8) since ı_{X} is a skew-derivation of degree −1:

Lemma 5.11. S_{•}^{∨} is a skew-derivation of degree −1 of A(P)_{equiv} ⊗ Sym(𝔤, V) and A(P)⊗Sym(𝔤, V), e. g. for α_{n} ∈A_{n}(P)⊗Sym(𝔤, V) and ω ∈A(P)⊗Sym(𝔤, V),

S_{•}^{∨}(α_{n} ∧_{∨} ω) = (S_{•}^{∨}α_{n}) ∧_{∨} ω + (−1)^{n}α_{n} ∧_{∨} (S_{•}^{∨}ω).

In view of our applications to connections on bundles we have thus proved:

Theorem 5.12. Let Γ be a connection on a principal fiber bundle P (M, G) and let B(M, F, G) be an associated bundle, V any vector space, χ_{n}^{s} ∈ A_{n}(F) ⊗ Hom(⨂^{s}𝔤, V) be G-equivariant and ϕ_{n} ∈ A_{n}(F)⊗V be invariant under G. Then

d(χ_{n}^{s}v • F) = [(dχ_{n}^{s})v]_{n+1}^{s} • F + [(L_{•}χ_{n}^{s})v]_{n−1}^{s+1} • F,
= [(dχ_{n}^{s})v]_{n+1}^{s} • F + [(L_{•}^{∨}χ_{n}^{s})v]_{n−1}^{s+1} • F,
d(ϕ_{n}v) = (dϕ_{n})v + [(L_{•}ϕ_{n})v]_{n−1}^{1} • F.

References

[1] Gross, C., A generalization of the exterior product of differential forms combining Hom-valued forms, submitted.

[2] Gross, C., Connections on fiber bundles and canonical lifts of differential forms, submitted.

[3] Helgason, S., “Differential Geometry and Symmetric Spaces,” Academic Press, 1962.

[4] Kobayashi, S. and Nomizu, K., “Foundations of Differential Geometry,” Vol. I, John Wiley & Sons, 1963.

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