Differentiable structure for direct limit groups

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.