Lifting double linear vector fields to Weil like functors on double vector bundles

Abstract

The complete description is given of the product preserving gauge bundle functors F on the category 2−VB of double vector bundles in terms of the AF‐bilinear maps ⋄F:UF×VF→WF, where AF are Weil algebras and UF,VF,WF are finite dimensional (over boldR) AF‐modules. Then the so‐called “iteration” problem is solved (i.e. ⋄F′0.16em∘0.16emF by means of ⋄F and ⋄F′ is computed). That any two product preserving gauge bundle functors on 2−VB commute is derived. All gauge natural operators lifting double linear vector fields to product preserving gauge bundle functors F are completely described.

Complete lifting of double-linear semi-basic tangent valued forms to Weil like functors on double vector bundles

“…Let DVB denote the category of all double vector bundles and their almost double vector bundle maps, and let FM denote the category of fibered manifolds and fibered maps. (In [14], the notation 2-VB instead of DVB is used. )…”

“…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.…”

“…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].…”

“…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…”

“…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].…”