COMPLETE LIFTING OF DOUBLE-LINEAR SEMI-BASIC TANGENT VALUED FORMS TO WEIL LIKE FUNCTORS ON DOUBLE VECTOR BUNDLES

. Let F be a product preserving gauge bundle functor on double vector bundles. We introduce the complete lifting F ϕ : FK → ∧ p T ∗ FM ⊗ TFK of a double-linear semi-basic tangent valued p -form ϕ : K → ∧ p T ∗ M ⊗ TK on a double vector bundle K with base M . We prove that this com- plete lifting preserves the Frolicher–Nijenhuis bracket. We apply the results obtained to double-linear connections.


Introduction
We assume that any manifold considered in the paper is Hausdorff, second countable, finite dimensional, without boundary and smooth (i.e. of class C ∞ ). All maps between manifolds are assumed to be smooth (of class C ∞ ). Definition 1.1. An almost double vector bundle is a system K = (K r , K l , E r , E l ) of vector bundles K r = (K, τ r , E r ), K l = (K, τ l , E l ), E r = (E r , τ l , M ) and E l = (E l , τ r , M ) such that τ l • τ r = τ r • τ l (this means that the respective diagram is commutative). We call M the basis of K.
If K = (K r , K l , E r , E l ) is another almost double vector bundle, an almost double vector bundle map K → K is a map f : K → K such that there are maps f r : E r → E r , f l : E l → E l and f : M → M such that (f, f r ) : K r → K r , (f, f l ) : K l → K l , (f r , f ) : E r → E r and (f l , f ) : E l → E l are vector bundle maps. We call f : M → M the base map of f .
For example, we have the trivial almost double vector bundle K = (K r , K l , E r , E l ), ) and E l = (R m1 ×R n1 , τ r , R m1 ), and where τ r , τ l , τ r , τ l are the obvious projections. We will denote this trivial almost double vector bundle by R m1,m2,n1,n2 .

Definition 1.2.
A double vector bundle is a locally trivial almost double vector bundle K. This means that there are nonnegative integers m 1 , m 2 , n 1 , n 2 such By [14], the ppgb-functors F on DVB are in bijection with the A F -bilinear maps F : U F ×V F → W F , where A F are Weil algebras and U F , V F and W F are finitely dimensional (over R) A F -modules. Moreover, the ppgb-functors F on DVB have values in DVB. For any such F , if K is a DVB-object with basis M , then F K is a DVB-object with basis F M = T A F M ; see [14].
Let F be a ppgb-functor on DVB and let F : U F × V F → W F be the corresponding A F -bilinear map. Let K be a DVB-object. Then any doublelinear vector field Z on K can be lifted to the double-linear vector field FZ on F K via F -prolongation of flow. By [14], for any a ∈ A F we have the affinor af(a) : T F K → T F K on F K. We have af(a 1 a 2 ) = af(a 1 ) • af(a 2 ) and af(1) is the identity affinor. If f : The main result of the paper is the following one (see Theorem 4.5): Let F be a ppgb-functor on DVB. Let ϕ : K → ∧ p T * M ⊗ T K be a doublelinear semi-basic tangent valued p-form on a double vector bundle K with basis M . Then there exists one and only one double-linear semi-basic tangent valued p-form for any vector fields X 1 , . . . , X p on M and any a 1 , . . . , a p ∈ A F . Definition 1.7. We call Fϕ (as above) the complete lift of ϕ to F . Next we study the complete lifting F. We prove that F commutes with the Frolicher-Nijenhuis bracket (see Theorem 5.1) and apply this fact to double-linear connections Γ : K → T * M ⊗ T K in K (see Theorem 6.3).
By the local description of double vector bundles, presented in [8], the notion of double vector bundles in the sense of the present paper is equivalent to the one in the book [11]. Product preserving (gauge) bundle functors are studied in [1,6,7,9,10,12,13,14,16,17,18]. Liftings of vector fields to product preserving (gauge) bundle functors are studied in [5,10,14]. Complete lifting of general connections on fibered manifolds to Weil functors is studied in [7]. Complete lifting of semi-basic tangent valued p-forms on fibered manifolds to Weil functors is studied in [2,3]. Complete lifting of linear semi-basic tangent valued forms to product preserving gauge bundle functors on vector bundles is studied in [15]. The Frolicher-Nijenhuis bracket on projectable tangent valued forms is studied in [4].

Preliminaries
Let K be a double vector bundle. Let M be the basis of K and π : K → M be the projection. Proof. Using DVB-charts, we may assume K = R m1,m2,n1,n2 . Let The lemma is now clear. Now, we treat K as a fibered manifold over M or (generally) let π : K → M be an arbitrary fibered manifold.

Definition 2.2. A projectable semi-basic tangent valued
Given a projectable semi-basic tangent valued p-form ϕ : is the underlying vector field of the projectable vector field ϕ(X 1 , . . . , X p ) for any vector fields X 1 , . . . , X p on M .

Lemma 2.3. Given a projectable semi-basic tangent-valued
for any vector fields X 1 , . . . , X p+q on M , where sums are over all permutations σ : {1, . . . , p + q} → {1, . . . , p + q} and sgn σ is the signum of σ. We end this section with the DVB-version of the well-known fact of the simplicity of vector fields.

Lemma 2.6. Let Z be a double linear vector field on a double vector bundle K such that the underlying vector field Z on basis
The proof is quite similar to that of the manifold case. We may assume The last flow is the one of (F ϕ) * FZ.
Proof. If {ϕ t } is the flow of Z, then {ϕ αt } is the flow of αZ. So, {F ϕ αt } is the flow of F(αZ) and of αFZ. Hence, F is R-linear because of the homogeneous function theorem and the nonlinear Peetre theorem [7]. Proof. We may assume that the underlying vector field Z is nowhere vanishing. Then using DVB-charts and Lemma 2.6 we may assume that Z = ∂ ∂x 1 and K = R m1,m2,n1,n2 . Then Proof. We may assume that K = R m1,m2,n1,n2 , Z = ∂ ∂x 1 and then the formula is the well-know one for usual Weil functors on manifolds. For other values of Z 2 , using formula (3.2) (below) and the known formula aFZ(a 1 F f ) = aa 1 F (Z(f )) for usual Weil functors on manifolds, we get Then

2)
where π : K → M is the projection (we treat M as a DVB-object and π as a DVB-map in the obvious way) and F f : F M → F R = A F . Here (in the right of the formula) a · y := af(a)(y) for a ∈ A F and y ∈ T F K.
Proof. By Lemma 2.1, f • π · Z is double linear. So, both sides of (3.2) make sense. By the linearity of F, we may assume that Z is not π-vertical. Then by Lemma 2.6 we may assume that K = R m1,m2,n1,n2 and Z = ∂ ∂x 1 . Then we may additional assume that K = M is a manifold, Z is a vector field on M and F is a Weil functor on manifolds. Then our lemma is the (well known for Weil functors on manifolds) formula F(f Z) = F f · FZ.   Let x 1 , . . . , x m1 , u 1 , . . . , u m2 , v 1 , . . . , v n1 , w 1 , . . . , w n2 be the usual coordinates on R m1,m2,n1,n2 .

Lemma 4.1. Fω is the unique A F -valued p-form on F M such that
Because of the local expression (2.1) of double-linear vector fields and of the Definition 1.4 of double-linear semi-basic tangent valued p-forms, any double-linear semi-basic tangent valued p-form ϕ on R m1,m2,n1,n2 is of the form For any such ϕ we define its complete lift Fϕ by

Proposition 4.3. The complete lift Fϕ as in (4.2) is the unique double-linear semi-basic tangent valued
for any a 1 , . . . , a p ∈ A F and any X 1 , . . . , X p ∈ X (R m1 ).

Proof.
We have Now, applying the uniqueness case of Proposition 4.3 (or, better, the sentence of the proof of the uniqueness case of Proposition 4.3) we end the proof.
We are now in a position to prove the following result.
for any vector fields X 1 , . . . , X p on M and any a 1 , . . . , a p ∈ A F . Proof. Using DVB-charts on K, we spread the complete lifting of double-linear semi-basic tangent valued p-forms on R m1,m2,n1,n2 to the one on K. This is possible because of Lemma 4.4.

The complete lifting of double-linear semi-basic tangent valued p-forms preserves the Frolicher-Nijenhuis bracket
Let F be a ppgb-functor on DVB. Then F : DVB → DVB.
Let ϕ : K → ∧ p T * M ⊗ T K be a double-linear semi-basic tangent valued p-form on K and let ψ : K → ∧ q T * M ⊗ T K be a double-linear semi-basic tangent valued q-form on K. We can lift ϕ and ψ to F K and obtain a double-linear semi-basic tangent valued p-form Fϕ on F K and a double-linear semi-basic tangent valued q-form Fψ on F K.  for any vector fields X 1 , . . . , X p+q on M and any a 1 , . . . , a p+q ∈ A F , where a := a 1 · . . . · a p+q . Then, since the vector fields af(a) • FX generate (over C ∞ (F M )) the space X (F M ), formula (5.1) holds.

An application to double-linear general connections
Let F be a ppgb-functor on DVB. In Definition 1.5, we introduced the concept of double-linear connections Γ in a double vector bundle K. Lemma 6.1. Given a double linear connection Γ in K, its complete lift FΓ is a double-linear connection in F K.
Proof. Since Γ(X) is a double-linear vector field on K with the underlying vector field equal to X, we have that FΓ(af(a) • FX) = af(a) · F(Γ(X)) is a double-linear vector field with the underlying vector field equal to af(a) • FX. Consequently, for any vector field Y ∈ X (F M ), FΓ(Y ) is a double linear vector field with the underlying vector field equal to Y .