Cohomology of Restricted Lie Algebras

Abstract

Introduction. We shall be concerned with restricted Lie algebras of characteristic p (7& 0), in the sense of N. Jacobsoll, [6]. The structure of a restricted Lie algebra L comprises, in addition to the usual Lie algebra structure, a map of L into itself, denoted x -> xP], with properties that correspond, ill a fashion we shall make precise later, to the properties of the map x> xP in an associative algebra of characteristic p. The appropriate representation theory for such algebras is accordingly obtained by confining one's attention to the restricted representations in which the transformation corresponding to x[P] is the p-th power of the transformation corresponding to x. Just as the ordinary representations of a Lie algebra correspond to the representations of its universal enveloping (or Birkhoff-Witt) algebra, the restricted representations of a restricted Lie algebra correspond to the representations of a certain homomorphic image of its universal enveloping algebra; its u-algebra, in the sense of Jacobson, [6]. It has been shown by HI. Cartan and S. Eilenberg that the cohoinology groups of a Lie algebra cain be computed, as described iM sectioin 1 below, from any augmented free acyclic complex over its universal envelopinig algebra. If, in this context, one replaces the universal enveloping algebra by the ualgebra one obtains new groups, which we shall call the ilestricted cohomology groups. One is thus led to the problem of determiniing what becomes of the usual elementary interpretations of the 1and 2dimnensional cohomolog,y groups in terms of extensions of modules and Lie algebras when ouie passes from the ordinary situations to the corresponding r restricted onies. This is what we propose to do here.' The results are what one would naturally expect them to be, except perhaps for those which deal with the relations between the ordinary objects and tlhe corresponding irestricted ones, such as are expressed in Theorems 3. 1 and 3. 2.