A semi-simple Lie group is a type of Lie group that is connected and has a finite-dimensional representation where every non-trivial representation decomposes into irreducible representations. This means that the group can be expressed as a direct product of simple Lie groups, which are those that cannot be further decomposed. These groups are significant because they play a central role in understanding the structure and classification of Lie algebras, especially in relation to maximal tori and the Weyl group.
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