Lie Groups and Subsemigroups With Subjective Exponential Function

by Karl H. Hofmann

In the structure theory of real Lie groups, there is still information lacking about the exponential function. Most notably, there are no general necessary and sufficient conditions for the exponential function to be surjective. It is surprising that for subsemigroups of Lie groups, the question of the surjectivity of the exponential function can be answered. Under natural reductions setting aside the "group part" of the problem, subsemigroups of Lie groups with surjective exponential function are completely classified and explicitly constructed in this memoir. There are fewer than one would think and the proofs are harder than one would expect, requiring some innovative twists. The main protagonists on the scene are $SL(2,R)$ and its universal covering group, almost abelian solvable Lie groups (i.e., vector groups extended by homotheties), and compact Lie groups.

Paperback

Published January 1, 1997

About the author

Prof. Dr. Karl Heinrich Hofmann is a mathematician who earned his Ph.D. in 1958 from the University of Tübingen (Germany).
Since then has been lecturing in the U.S. (Universities of Tulane and Princeton), France (Université de Paris), Australia (La Trobe University, Melbourne), Belgium (Université Catholique de Louvain-la-Neuve) and Germany (Technische Hochschule Darmstadt and Universität Tübingen).
He is also Deputy Managing Editor of the Journal of Lie Theory, Honorary Editor of the Semigroup Forum, Editor of Research and Exposition in Mathematics and Editor of Heldermann Verlag.