The Harish-Chandra homomorphism is a crucial map in the representation theory of Lie algebras, particularly connecting a semisimple Lie algebra with its associated universal enveloping algebra. This homomorphism plays a pivotal role in understanding the representations of semisimple Lie algebras by providing a bridge between the algebraic structures and the analytic aspects of representation theory. It allows for the extraction of useful information about the representations of a Lie group from those of its Lie algebra.
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