Abstract
Let G be a simple Lie group with finite center. We show that G can be algebraically generated as a semigroup by a one-parameter subsemigroup X + and one additional element g. In fact, given one of the two, a non-constant X + or a non-central g, there is a g, respectively X +, such that the two together generate G as a semigroup. It follows that given a non-constant one-parameter subsemigroup X + of G there is another one-parameter subsemigroup Y + such that the two generate G as a semigroup. Similarly, given a non-central element g of G there is an element h of G such that the two generate a dense subsemigroup of G. We have analogous results for semismple Lie groups. For SL(2, ℝ) and its universal cover we spell out when two one-parameter subsemigroups generate the group as a semigroup. On the way we prove results on open subsets of subsemigroups and on exponentiality, which may be of independent interest.
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ABELS, H. SUBSEMIGROUPS OF SEMISIMPLE LIE GROUPS. Transformation Groups 20, 307–318 (2015). https://doi.org/10.1007/s00031-015-9303-3
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DOI: https://doi.org/10.1007/s00031-015-9303-3