Weyl's theorem on complete reducibility

In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations. Let $\mathfrak{g}$ be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over $\mathfrak{g}$ 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
Weibel, Charles A. (1995). An Introduction to Homological Algebra. Cambridge University Press.

External links

A blog post by Akhil Mathew