In algebra, Whitehead's lemma on a Lie algebra representation is an important step toward the proof of Weyl's theorem on complete reducibility. Let ${\displaystyle {\mathfrak {g}}}$ be a semisimple Lie algebra over a field of characteristic zero, V a finite-dimensional module over it and ${\displaystyle f:{\mathfrak {g}}\to V}$ a linear map such that ${\displaystyle f([x,y])=xf(y)-yf(x)}$. The lemma states that there exists a vector v in V such that ${\displaystyle f(x)=xv}$ for all x.

The lemma may be interpreted in terms of Lie algebra cohomology. The proof of the lemma uses a Casimir element.

## References

• Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4