"Lie algebroid of a principal fibre bundle – three equivalent definitions"

Prace naukowe Politechniki Szczeci�skiej, Nr 11, Instytut Matematyki, 1988, 123-145.
publications

Starting with some Atiyah's construction (1957) we define the so-called Lie functor for principal fibre bundles. This functor assigns some Lie algebroid (Pradines 1967) to each principal fibre bundle and plays an analogous role as the Lie functor for Lie groups. The Lie algebroid TP_/G of a principal fibre bundle P(M,G) is constructed by using the Lie algebra X^R(P) of all right-invariant vector fields on P and is isomorphic to the Lie algebroid A(PP^-1) of the Ehresmann Lie groupoid PP^-1 assigned to P. We also give the third natural construction of Lie algebroid for a principal fibre bundle P(M,G), as the associated vector bundle W¹(P) � _G¹n (Rⁿ � g) with some suitable structures, g is the right Lie algebra of G, W¹(P) and G¹_n are the 1-st order prolongations of P and G, respectively.