, Volume 23, Issue 2, pp 99-109

Differentiable structure for direct limit groups

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Abstract

A direct limit \(G = \mathop {\lim }\limits_ \to G_\alpha\) of (finite-dimensional) Lie groups has Lie algebra \(\mathfrak{g} = \mathop {\lim }\limits_ \to \mathfrak{g}_\alpha\) and exponential map exp G : g→G. BothG and g carry natural topologies.G is a topological group, and g is a topological Lie algebra with a natural structure of real analytic manifold. In this Letter, we show how a special growth condition, natural in certain physical settings and satisfied by the usual direct limits of classical groups, ensures thatG carries an analytic group structure such that exp G is a diffeomorphism from a certain open neighborhood of 0∈g onto an open neighborhood of 1 G G. In the course of the argument, one sees that the structure sheaf for this analytic group structure coincides with the direct limit \(\mathop {\lim }\limits_ \to\) C ω(G α) of the sheaves of germs of analytic functions on theG α.

L.N. partially supported by a University of California Dissertation Year Fellowship.
E.R.C. partially supported by N.S.F. Grant DMS 89 09432.
J.A.W. partially supported by N.S.F. Grant DMS 88 05816.