Vol. 26, No. 3, 1968

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The integration of a Lie algebra representation

Jacques Tits and Lucien Waelbroeck

Vol. 26 (1968), No. 3, 595–600

Let u : G A be a differentiable representation of a Lie group into a b-algebra. The differential u0 = due of u at the neutral element e of G is a representation of the Lie algebra g of G into A. Because a Lie group is locally the union of one-parameter subgroups and since the infinitesimal generator of a differentiable (multiplicative) sub-semi-group of A determines this sub-semi-group, the representation u0 determines u if G is connected.

We shall be concerned with the converse: given a representation u0 of g, when can it be obtained by differentiating a representation u of G? We shall assume G connected and simply connected, which means that we are only interested in the local aspect of the problem.

Mathematical Subject Classification
Primary: 22.60
Received: 27 November 1967
Published: 1 September 1968
Jacques Tits
Lucien Waelbroeck