A simple Lie algebra is a non-abelian Lie algebra that does not have any non-trivial ideals, meaning it cannot be decomposed into smaller, simpler algebras. This concept is crucial because it lays the foundation for understanding the structure of more complex algebras and the classification of semisimple Lie algebras. Simple Lie algebras play a pivotal role in representation theory, as their representations can often be built from the representations of their subalgebras.
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