# Weyl's theorem on complete reducibility

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In algebra, **Weyl's theorem on complete reducibility** is a fundamental result in the theory of Lie algebra representations. Let be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over is semisimple as a module (i.e., a direct sum of simple modules.)

The theorem is a consequence of Whitehead's lemma (see Weibel's homological algebra book). Weyl's original proof was analytic in nature: it famously used the unitarian trick.

A Lie algebra is called reductive if its adjoint representation is semisimple. Thus, the theorem says that a semisimple Lie algebra is reductive. (But this can be seen more directly.)

## References

- Jacobson, Nathan,
*Lie algebras*, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4 - {{#invoke:citation/CS1|citation

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## External links

- A blog post by Akhil Mathew