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. The usual algebraic proof makes use of the Casimir element of the universal enveloping algebra.^[1]

External links

• A blog post by Akhil Mathew

References

1. Hall 2015 Section 10.3

• Hall, Brian C. (2015). Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics 222 (2nd ed.). Springer.
• 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.