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Uniform Finite Generation of Threedimensional Linear Lie Groups

Published online by Cambridge University Press:  20 November 2018

R. M. Koch  and
Franklin Lowenthal
R. M. Koch
Affiliation:
University of Oregon, Eugene, Oregon
Franklin Lowenthal
Affiliation:
University of Wisconsin at Park side, Kenosha, Wisconsin
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A connected Lie group G is generated by one-parameter subgroups exp(tX1), … , exp(tXk) if every element of G can be written as a finite product of elements chosen from these subgroups. This happens just in case the Lie algebra of G is generated by the corresponding infinitesimal transformations X1, … , Xk ; indeed the set of all such finite products is an arcwise connected subgroup of G, and hence a Lie subgroup by Yamabe's theorem [9]. If there is a positive integer n such that every element of G possesses such a representation of length at most n, G is said to be uniformly finitely generated by the one-parameter subgroups.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 2 , 01 April 1975 , pp. 396 - 417
Copyright
Copyright © Canadian Mathematical Society 1975

References

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