Semisimplicity refers to a property of Lie algebras where the algebra can be decomposed into a direct sum of simple Lie algebras, which themselves have no nontrivial ideals. This concept is crucial because semisimple Lie algebras have well-defined representation theories, making them central in the study of Lie algebras and Lie groups. Understanding semisimplicity helps in characterizing the structure and behavior of these mathematical objects.
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