Whitehead's lemma (Lie algebras)

In algebra, Whitehead's lemma on a Lie algebra representation (named after J. H. C. Whitehead) is an important step toward the proof of Weyl's theorem on complete reducibility.

Let ${\mathfrak {g}}$ be a semisimple Lie algebra over a field of characteristic zero, V a finite-dimensional module over it, and $f\colon {\mathfrak {g}}\to V$ a linear map such that

f([x,y])=xf(y)-yf(x)

.

The lemma states that there exists a vector v in V such that $f(x)=xv$ 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

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