Whitehead's lemma (Lie algebra)
In homological algebra, Whitehead's lemmas (named after J. H. C. Whitehead) represent a series of statements regarding representation theory of finite-dimensional, semisimple Lie algebras in characteristic zero. Historically, they are regarded as leading to the discovery of Lie algebra cohomology. One usually makes the distinction between Whitehead's first and second lemma for the corresponding statements about first and second order cohomology, respectively, but there are similar statements pertaining to Lie algebra cohomology in arbitrary orders which are also attributed to Whitehead. The first Whitehead lemma is an important step toward the proof  of Weyl's theorem on complete reducibility. Statements. Without mentioning cohomology groups, one can state Whitehead's first lemma as follows: Let formula_1 be a finite-dimensional, semisimple Lie algebra over a field of characteristic zero, "V" a finite-dimensional module over it, and formula_2 a linear map such that Then there exists a vector formula_4 such that formula_5 for all formula_6. In terms of Lie algebra cohomology, this is, by definition, equivalent to the fact that formula_7 for every such representation. The proof uses a Casimir element (see the proof below). Similarly, Whitehead's second lemma states that under  the conditions of the first lemma, also formula_8. Another related statement, which is also attributed to Whitehead, describes Lie algebra cohomology in arbitrary order: Given the same conditions as in the previous two statements, but further let formula_9 be irreducible under the formula_1-action and let formula_1 act nontrivially, so formula_12. Then formula_13 for all formula_14. Proof. As above, let formula_1 be a finite-dimensional semisimple Lie algebra over a field of characteristic zero and formula_16 a finite-dimensional representation (which is semisimple but the proof does not use that fact). Let formula_17 where formula_18 is an ideal of formula_1 . Then, since formula_18 is semisimple, the trace form formula_21, relative to formula_22, is nondegenerate on formula_18. Let formula_24 be a basis of formula_18 and formula_26 the dual basis with respect to this trace form. Then define the Casimir element formula_27 by which is an element of the universal enveloping algebra of formula_29. Via formula_22, it acts on "V" as a linear endomorphism (namely, formula_31.) The key property is that it commutes with formula_32 in the sense formula_33 for each element formula_6. Also, formula_35 Now, by Fitting's lemma, we have the vector space decomposition formula_36 such that formula_37 is a  (well-defined) nilpotent endomorphism for formula_38 and is an automorphism for formula_39. Since formula_40 commutes with formula_32, each formula_42 is a formula_1-submodule. Hence, it is enough to prove the lemma separately for formula_44 and formula_45. First, suppose formula_40 is a nilpotent endomorphism. Then, by the early observation, formula_47; that is, formula_22 is a trivial representation. Since formula_49, the condition on formula_50 implies that formula_51 for each formula_6; i.e., the zero vector formula_53 satisfies the requirement. Second, suppose formula_40 is an automorphism. For notational simplicity, we will drop formula_22 and write formula_56. Also let formula_57 denote the trace form used  earlier. Let formula_58, which is a vector in formula_9. Then Now, and, since formula_62, the second term of the expansion of formula_63 is Thus, Since formula_27 is invertible and formula_67 commutes with formula_68, the vector formula_69 has the required property. formula_70