Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups. (English) Zbl 1022.22021

The author studies the question whether a given infinite-dimensional Lie group has a complexification. He answers the question completely by giving necessary and sufficient conditions for the so-called Baker-Campbell-Hausdorff (BCH) Lie groups. This class of groups is characterized by the existence of neighborhoods in the Lie algebra on which the exponential function is a diffeomorphism and the multiplication (transported back via the exponential function) is given by the Baker-Campbell-Hausdorff formula. This class of groups includes Banach Lie groups, mapping groups, and direct limit groups but not diffeomorphism groups. In the process the author develops the basic Lie theory for BCH Lie groups as well as the basic properties for various types of mapping groups.


22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
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