Vol. 20, No. 2, 1967

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Locally trivial Cr groupoids and their representations

Joel John Westman

Vol. 20 (1967), No. 2, 339–349

We develop a theory of representations of a locally trivial Cr groupoid, Z, on a Cr fiber bundle, E (with fiber Y , Lie group G, and base space M = the set of units of Z).

A covariant functor, A, is defined, sendin gE into a locally trivial Cr groupoid A(E) = the groupoid of admissible maps between fibers of E, with a natural Cr structure. A Cγ bundle map h : E Eis sent into a Cr isorqiorphism A(h) : A(E) A(E). Properties of tke functor A are studied.

A Cr representation of Z on E is defined as a Cr homomorphism ρ : Z A(E). Let Zee be the group of elements in Z with e as the left and right unit. We obtain the important result that a Cr homomorphism ρe : Zee A(Ee) has an (essentially) unique extension to a Cr representation of Z on a Cr fiber bundle E, where Eis determined by Z and ρe. This leads to interesting applications in differential geometry. The representations of Lk, the (locally trivial C) groupoid of invertible k-jets of C maps of a C manifold, M, into itself, provide (but are not the same as) natural fiber bundles of order k in the sense of Nijenhuis.

Mathematical Subject Classification
Primary: 57.70
Secondary: 53.45
Received: 19 March 1965
Revised: 2 February 1966
Published: 1 February 1967
Joel John Westman