General Structure Theory

  • Joachim Hilgert
  • Karl-Hermann Neeb
Part of the Springer Monographs in Mathematics book series (SMM)


In this chapter, we shall reach our first main goal, namely the Manifold Splitting Theorem, that a Lie group G with finitely many connected components is diffeomorphic to K×ℝ n , where K is a maximal compact subgroup. Many results we proved so far for special classes of groups, such as nilpotent, compact and semisimple ones, will be used to obtain the general case. This approach is quite typical for Lie theory. To prove a theorem for general Lie groups, one first deals with special classes such as abelian, nilpotent, and solvable groups, and then one turns to the other side of the spectrum, to semisimple Lie groups. Often the techniques required for semisimple and solvable groups are quite different. To obtain the Manifold Splitting Theorem for general Lie groups, Levi’s Theorem is a crucial tool to combine the semisimple and the solvable pieces because the Levi decomposition \({\mathfrak{g}}= {\mathfrak{r}}\rtimes {\mathfrak{s}}\) of a Lie algebra implies a corresponding decomposition G=RS of the corresponding 1-connected Lie group G. This is already the central idea for the simply connected case. To deal with a general Lie group G, we need a better understanding of the center of a connected Lie group because G is a quotient of its simply connected covering by a discrete central subgroup. We have to understand how this influences the splitting of \(\widetilde {G}\) to obtain a solution of the general case.


Solvable Group Discrete Subgroup Maximal Compact Subgroup Cartan Decomposition Solvable Subgroup 
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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of PaderbornPaderbornGermany
  2. 2.Department of MathematicsFriedrich-Alexander Universität Erlangen-NürnbergErlangenGermany

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