Can anyone suggest a reference that defines or explains that a semisimple real Lie group can be decomposed into a product of almost simple factors? In some papers that I read recently people keep talk about "semisimple Lie groups withoug compact factors" without explanation and it appears to be some standard notion. But the only reference I can find is Margulis' book "discrete subgroups of semisimple Lie groups" where this decomposition is claimed only for algebraic groups instead of for Lie groups.

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    $\begingroup$ A semisimple Lie group is a covering of a semisimple algebraic group .... Alternatively, use that its Lie algebra is a product of simple Lie algebras. $\endgroup$ – anon Mar 7 '12 at 5:45

Concerning references, there exist many books which treat the structure and classification of semisimple Lie groups, usually with a wider agenda involving for example symmetric spaces, harmonic analysis, infinite dimensional representations. Older and newer authors include Chevalley, Helgason, Knapp, Wallach, Onishchik-Vinberg, Bump, etc. (Other books concentrate more heavily on compact groups.)

Though the coverage in such books varies a lot, the basic outline is usually similar: start with the notion of (real) Lie group and the associated Lie algebra (originally called the "infinitesimal group"), study the solvable radical, pass to semisimple groups and their Lie algebras, then complexify the situation in order to use more algebraic methods.

Ultimately the structure of a complex semisimple group lifts from the structure of a complex semisimple Lie algebra. Here relatively elementary methods, based on nondegeneracy of the Killing form, decompose the Lie algebra into a direct sum of simple ideals (which can be readily classified). Then the nice correspondence between the groups and their Lie algebras allows most of this structure to be found in the group as well, though the "simple" groups may in fact just be "almost simple". Decomposing the group directly into simple factors is not an attractive project, though it might be done indirectly using Chevalley's approach via linear algebraic groups over arbitrary algebraically closed fields.

Only after such results are in hand for the complex Lie algebras and groups can one adapt the structure and classification to the real case. I don't think it's practical to get a direct factorization of a semisimple (real) Lie group into its simple factors, but on the other hand the existence of unique compact real forms makes the less direct comparison of real and complex cases doable.

None of the standard Lie groups books can be viewed as easy reading, since by its nature Lie group theory merges ideas from analysis, topology, algebra in a sophisticated way. In any case, algebraic group ideas have their limits in the study of real Lie groups, since covering groups arise which are not algebraic.


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