The Lie theory of pro-Lie groups
by
Karl H. Hofmann
Fachbereich Mathematik, Technische Universität Darmstadt
Coauthors: Sidney A. Morris

A topological group G is a projective system of finite dimensional Lie groups iff it is complete and every identity neighorhood contains a normal subgroup N such that G/N is a Lie group iff it is (isomorphic to) a closed subgroup of a product of finite dimensional Lie group. Such a group is called pro-Lie group. The category of pro-Lie groups is closed under all limits in the category of topological groups and under passing to closed subgroups. It is not closed under passing to quotient groups. Every pro-Lie group has a Lie algebra and an exponential function whose image generates (algebraically) a dense subgroup of the identity component. The Lie algebra of a pro-Lie group is a complete topological Lie algebra in which every zero neighborhood contains a closed ideal such that the quotient is a finite dimensional Lie algebra. Such Lie algebras are called pro-Lie algebras. The Lie algebra functor preserves limits and quotients, and its left adjoint provides a functorial version of Lie's Third Theorem. A pro-Lie algebra has a radical and a Levi summand such that a Levi-Mal'cev Theorem holds. Each simply connected pro-Lie algebra is semidirect product of a simply connected countably solvable pro- Lie group and a product of simply connected Lie groups. A good deal of information is available on countably solvable pro-Lie groups. The abelian pro-Lie groups form a relevant subclass. (See Lecture Sidney A. Morris.)

1991 Mathematics Subject Classification: 22B, 22E.

Keywords: Projective limit, Lie group, pro-Lie group, pro-Lie algebra, Lie algebra functor, exponential function.

Date received: May 16, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cakb-23.