Last updates: 14 April 2011
1.1 A Lie algebra is a vector space over with a bracket which satisfies The first relation is the skew-symmetric relation and is equivalent to for all provided . The second relation is called the Jacobi identity.
1.2 A derivation of a Lie algebra is a map such that
1.3 A -module or representation of is a pair consisting of a vector space over and a linear map such that for all . It is common to suppress the in writing the action on . With the suppressed notation the condition is for all and all .
1.4 If and are -modules then the tensor product is a -module with action given by for all . In the suppressed notation the action of on is given by for all and .
1.5 The trivial module for is a -dimensional vector space with -action given by .
1.6 The dual of a -module is the vector space and the -action given by for all , and . Here denotes the evaluation of the linear functional at the element .
1.7 The adjoint representation of is the -module where the action on is given by for all .
2.1 Let be a Lie algebra and let be a -module. Elements of are called -cochains. The maps given by determine a cochain complex It is common to suppress the subscript on the map when the is irrelevant or the context is clear. In the cases the map is given explicitly by for all , and .
2.2 One can prove directly (see [Kn] Lemma 4.5 p. 172) or by using some tricks (see [HGW] Lemma 4.1.4) tha for all positive integers . Thus, if we define for all , then is well defined. The elements of are -cocycles, the elements of are -coboundaries, and
2.3 Consider the complex where is the adjoint representation of the Lie algebra . In this case the -cocycles are maps such that , for all . Thus -cocycles are the maps such that i.e., the derivations. For any two elements Thus . Thus the -coboundaries are the inner derivations.
2.4 The case where is the second tensor power of the adjoint representation of is useful for understanding Lie bialgebras. Let be the tensor product of the adjoint representation with itself; specifically acts on by for all . Then using the explicit formula for , An element is a -cocycle if , i.e., if This is the origin of the terminology -cocycle condition in the definition of a Lie bialgebra.
[Bou] N. Bourbaki, Lie groups and Lie algebras, Chapters I-III, Springer-Verlag, 1989.
There is a very readable and informative chapter on Lie algebra cohomolgy in the forthcoming book
[HGW] R. Howe, R. Goodman, and N. Wallach, Representations and Invariants of the Classical Groups, manuscript, 1993.
The following book is a standard introductory book on Lie algebras; a book which remains a standard reference. This book has been released for a very reasonable price by Dover publishers. There is a short description of Lie algebra cohomology in Chapt. III §11, pp.93-96.
[J] N. Jacobson, Lie algebras, Interscience Publishers, New York, 1962.
A very readable and complete text on Lie algebra cohomology is