REPRESENTATIONS OF RESTRICTED LIE ALGEBRAS OF CHARACTERISTIC p

It was proved by Jacobson [2] that every finite-dimensional Lie algebra of characteristic p has a finite-dimensional representation space which is not semisimple. In this note we prove a similar result for Jacobson's restricted Lie algebras [l]. The structure of a restricted Lie algebra L over a field F of characteristic p comprises, in addition to the ordinary Lie algebra structure, a map x—*-x[pl of L into itself, with formal properties corresponding to those of the map x—*xp in an associative algebra of characteristic p. Accordingly, one studies the "restricted" representations of L, in which the operator corresponding to xlp] is the pth power of the operator corresponding to x. Our result is the following.