Semisimple Lie algebras are those that can be expressed as a direct sum of simple Lie algebras, which do not have non-trivial ideals. These algebras play a crucial role in understanding representations because they have well-behaved representation theories, allowing us to study finite-dimensional representations and their characters more effectively. The structure of semisimple Lie algebras leads to significant results like the Weyl character formula and connections to root systems.
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