Functorial prolongations of Lie algebroids. (English) Zbl 1114.58010

Kolář, Ivan

Bureš, Jarolím (ed.) et al., Differential geometry and its applications. Proceedings of the 9th international conference on differential geometry and its applications, DGA 2004, Prague, Czech Republic, August 30–September 3, 2004. Prague: matfyzpress (ISBN 80-86732-63-0/pbk). 305-314 (2005).

The author studies the prolongation of Lie algebroids and their actions with respect to product and fiber product preserving bundle functors. His starting points are the facts that every product preserving bundle functor on the category of smooth manifolds coincides with a Weil functor by the author, P. W. Michor and J. Slovák [Natural Operations in Differential Geometry, Springer-Verlag, Berlin (1993; Zbl 0782.53013)], and fiber preserving bundle functors of order \(r\) on the category of fibered manifolds over \(m\)-dimensional bases with fiber preserving mappings which determines local diffeomorphisms of bases as morphisms are in one-to-one correspondence with the triples \((A, H, t)\) by the author and W. M. Mikulski [Differ. Geom. Appl. 11, No. 2, 105–115 (1999; Zbl 0935.58001)] where \(A\) is a Weil algebra of height \(r\), \(H\) a homomorphism of \(r\)-jet group \(G^r_m\) to the group \(\operatorname{Aut}\;A\) and \(t\) a \(G^r_m\)-equivariant algebra homomorphism of \(J^r_0(\mathbb R^m, \mathbb R)\) to \(A\). (\(A\) denotes a different object in the Chern-Weil construction of characteristic classes.)
Let \(\varphi\) be an action of a Lie algebroid \(E\to M\) on a vector bundle \(D\to M\). The author defines a formal version \(\Phi\) of \(\varphi\) and then constructs a formal prolongation \(\Phi\), that looks like the functorial version of an action of the Lie algebroid \(\mathcal W^FE:=J^rTM\times_{FTM}FE\) on the vector bundle \(FD\) if \(E\) is transitive. In the case of the Lie algebroid associated to a principal bundle \(P\to M\), it follows that \(W^F(LP)=L(W^FP)\).
For the entire collection see [Zbl 1097.53001].

Reviewer: Haruo S. Suzuki (Sapporo)

MSC:

58H05	Pseudogroups and differentiable groupoids
58A20	Jets in global analysis

Keywords:

bundle functor; prolongation; Lie algebroid

Citations:

Zbl 0782.53013; Zbl 0935.58001