Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups
by
Helge Glöckner
University of Göttingen

A real or complex analytic Lie group G in the sense of [Glöckner, H., Infinite-dimensional Lie groups without completeness restrictions, LSU Preprint, Baton Rouge, May 2001], modelled over an arbitrary, not necessarily sequentially complete locally convex space, is called a Baker-Campbell-Hausdorff (BCH-) Lie group if its exponential function induces a local diffeomorphism at 0 and the group multiplication is given locally by the BCH-series. Beside Banach-Lie groups, typical examples of BCH-Lie groups are the mapping groups C^r_K(M, G):={ f in C^r(M, G): f|_M\K = 1}, for every BCH-Lie group G, finite-dimensional smooth manifold M, compact subset K of M, and 0 <= r <= \infty; the corresponding test function groups D^r(M, G) = \cup _K C^r_K(M, G) are BCH-Lie groups as well. We show: If G is a BCH-Lie group and N a closed normal subgroup, then the topological quotient group G/N is a BCH-Lie group if and only if N is a Lie subgroup of G (with L(N) not necessarily complemented in L(G)). This result is then used to characterize those BCH-Lie groups which possess universal complexifications in the category of complex BCH-Lie groups. Although not every BCH-Lie group has a universal complexification, mapping groups and test function groups do so under natural hypotheses. In more special situations, we can prove that C^r_K(M, G)_C = C^r_K(M, G_C) and D^r(M, G)_C = D^r(M, G_C), even in the category of all complex Lie groups with complex analytic exponential functions. The results presented are taken from [Glöckner, H., Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups , LSU Preprint, May 2001]; the Banach-case had been settled earlier in [Glöckner, H., and K.-H. Neeb, Banach-Lie quotients, enlargibility, and universal complexifications , LSU Preprint 2001-4, April 2001].

http://www.math.lsu.edu/~glockner/

Date received: May 13, 2001

Copyright © 2001 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 # cagw-26.