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.[1]

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

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Without mentioning cohomology groups, one can state Whitehead's first lemma as follows: Let   be a finite-dimensional, semisimple Lie algebra over a field of characteristic zero, V a finite-dimensional module over it, and   a linear map such that

 .

Then there exists a vector   such that   for all  . In terms of Lie algebra cohomology, this is, by definition, equivalent to the fact that   for every such representation. The proof uses a Casimir element (see the proof below).[2]

Similarly, Whitehead's second lemma states that under the conditions of the first lemma, also  .

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   be irreducible under the  -action and let   act nontrivially, so  . Then   for all  .[3]

As above, let   be a finite-dimensional semisimple Lie algebra over a field of characteristic zero and   a finite-dimensional representation (which is semisimple but the proof does not use that fact).

Let   where   is an ideal of  . Then, since   is semisimple, the trace form  , relative to  , is nondegenerate on  . Let   be a basis of   and   the dual basis with respect to this trace form. Then define the Casimir element   by

 

which is an element of the universal enveloping algebra of  . Via  , it acts on V as a linear endomorphism (namely,  .) The key property is that it commutes with   in the sense   for each element  . Also,  

Now, by Fitting's lemma, we have the vector space decomposition   such that   is a (well-defined) nilpotent endomorphism for   and is an automorphism for  . Since   commutes with  , each   is a  -submodule. Hence, it is enough to prove the lemma separately for   and  .

First, suppose   is a nilpotent endomorphism. Then, by the early observation,  ; that is,   is a trivial representation. Since  , the condition on   implies that   for each  ; i.e., the zero vector   satisfies the requirement.

Second, suppose   is an automorphism. For notational simplicity, we will drop   and write  . Also let   denote the trace form used earlier. Let  , which is a vector in  . Then

 

Now,

 

and, since  , the second term of the expansion of   is

 

Thus,

 

Since   is invertible and   commutes with  , the vector   has the required property.  

Notes

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  1. ^ Jacobson 1979, p. 93
  2. ^ Jacobson 1979, p. 77, p. 95
  3. ^ Jacobson 1979, p. 96
  4. ^ Jacobson 1979, Ch. III, § 7, Lemma 3.

References

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  • Jacobson, Nathan (1979). Lie algebras (Republication of the 1962 original ed.). Dover Publications. ISBN 978-0-486-13679-0. OCLC 867771145.