General Structure Theory

Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

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.

Keywords

Solvable Group Discrete Subgroup Maximal Compact Subgroup Cartan Decomposition Solvable Subgroup
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. [Dix57]
Dixmier, J., L’application exponentielle dans les groupes de Lie résolubles, Bull. Soc. Math. Fr. 85 (1957), 113–121
2. [DH97]
Djokovic, D., and K. H. Hofmann, The surjectivity question for the exponential function in real Lie groups: A status report, J. Lie Theory 7 (1997), 171–199
3. [dlH00]
de la Harpe, P., “Topics in Geometric Group Theory,” Chicago Lectures in Math., The Univ. of Chicago Press, Chicago, 2000
4. [Mi03]
Milnor, J., Towards the Poincaré conjecture and the classification of 3-manifolds, Notices Amer. Math. Soc. 50:10 (2003), 1226–1233
5. [MS08]
Moskowitz, M., and R. Sacksteder, On complex exponential groups, Math. Res. Lett. 15 (2008), 1197–1210.
6. [Ra72]
Raghunathan, M. S., “Discrete Subgroups of Lie Groups,” Ergebnisse der Math. 68, Springer, Berlin, 1972
7. [RV81]
Raymond, F., and A. T. Vasquez, 3-manifolds whose universal coverings are Lie groups, Topology and its Appl. 12 (1981), 161–179
8. [Sai57]
Saito, M., Sur certains groupes de Lie résolubles, Sci. Pap. Coll. Gen. Educ. Univ. Tokyo 7 (1957), 1–11.
9. [Seg83]
Segal, D., “Polycyclic Groups, ” Cambridge Univ. Press, Cambridge, 1983
10. [Wu98]
Wüstner, M., On the surjectivity of the exponential function of solvable Lie groups, Math. Nachrichten 192 (1998), 255–266
11. [Wu03]
Wüstner, M., Supplements on the theory of exponential Lie groups, J. Algebra 265 (2003), 148–170.
12. [Wu05]
Wüstner, M., The classification of all simple Lie groups with surjective exponential map. J. Lie Theory 15 (2005), 269–278.