Restricted Lie algebras of characteristic 𝑝

Abstract

In an earlier paper(') [3] we noted certain identities which connect addition, scalar multiplication, commutation ([ab] = ab ba), and pth powers in an arbitrary associative algebra of characteristic p ( 0). These lead naturally to the definition of a class of abstract algebras called restricted Lie algebras which in many respects bear a closer relation to Lie algebras of characteristic O than ordinary Lie algebras of characteristic p. As is shown in the present paper any restricted Lie algebra S may be obtained from an associative algebra by using the operations mentioned above. In fact 2 determines a certain associative algebra U, called its u-algebra, such that 2 is isomorphic to a subalgebra of U1, the restricted Lie algebra defined by 2; and if Q8 is any associative algebra such that Y13 contains a subalgebra homomorphic to 2 and Q8 is the enveloping algebra of this subset then U is homomorphic to Q3. The algebra U has an anti-automorphism relative to which the elements corresponding to those in 2 are skew. For ordinary Lie algebras an algebra having these properties has been defined by G. Birkhoff [2] and by Witt [5]. In their case however, the associative algebra has an infinite basis even when the Lie algebra has a finite basis whereas here U has a finite basis if and only if 2 has. Consequently every restricted Lie algebra 2 with a finite basis has a (1-1) representation by finite matrices. The theory of representations of 2 can be reduced to that of the associative algebra U. Thus, for example, there are only a finite number of inequivalent irreducible representations. The most natural way to obtain a restricted Lie algebra is as a derivation algebra of an arbitrary algebra 2, i.e., as the set of transformations D: aaD in 2I such that