Letters in Mathematical Physics

, Volume 23, Issue 2, pp 99–109 | Cite as

Differentiable structure for direct limit groups

  • Loki Natarajan
  • Enriqueta Rodríguez-Carrington
  • Joseph A. Wolf
Article

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 expG: 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 expG is a diffeomorphism from a certain open neighborhood of 0∈g onto an open neighborhood of 1GG. In the course of the argument, one sees that the structure sheaf for this analytic group structure coincides with the direct limit\(\mathop {\lim }\limits_ \to\)Cω(Gα) of the sheaves of germs of analytic functions on theGα.

AMS subject classifications (1991)

Primary 22E65 Secondary 20E18, 20F40 

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Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • Loki Natarajan
    • 1
  • Enriqueta Rodríguez-Carrington
    • 1
  • Joseph A. Wolf
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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