In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the defining module of a classical Lie group is a fundamental representation. Any finite-dimensional irreducible representation of a semisimple Lie group or Lie algebra can be constructed from the fundamental representations by a procedure due to Élie Cartan. Thus in a certain sense, the fundamental representations are the elementary building blocks for arbitrary finite-dimensional representations.
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“This is the fundamental idea of culture, insofar as it sets but one task for each of us: to further the production of the philosopher, of the artist, and of the saint within us and outside us, and thereby to work at the consummation of nature.”
—Friedrich Nietzsche (1844–1900)