On generalized Witt algebras
Rimhak Ree · Transactions of the American Mathematical Society · 1956
RIMHAK REE Introduction.Let 4> be a field of characteristic p>0.The Witt algebra over is a Lie algebra with basis eo, ei, ■ ■ ■ , ep-i and relations e, o cy = (j-i)ei+j, where i+j is to be calculated modulo p. H. Zassenhaus [5, p. 47] generalized the Witt algebra to algebras with basis {ea}, where a runs over a subgroup of the additive group of the ground field "£, and with the relations ea o ep= (fj -a)ea+p.Another generalization was obtained by N. Jacobson [3].In his investigations Witt [l ] used implicitly the fact that the Witt algebra is the derivation algebra of the group algebra of a cyclic group of order p.In the paper cited above, Jacobson proved that the derivation algebra of the group algebra of an elementary p-group, by which we shall mean throughout this paper an abelian group of the type (p, p, ■ ■ -, p), is simple if the order of the group is greater than 2.Recently, I. Kaplansky [4, p. 471 ] gave an ingenious generalization of the Witt algebra, which includes the generalizations obtained by Zassenhaus and Jacobson.Let 7= \i,j, • • ■ } be a set of indices, and ® a total(2) additive group of functionals on 7 with values in the ground field .Kaplansky considers the Lie algebra 2 over $ with basis {(i, a)}, where iEI, &E®, and the multiplication (0.0.1) (i, a) O (j, t) = r(i)(j, a + r) -a(j)(i, a + r).It appears that 2 is simple except when 7 consists of a single element and 4> is of characteristic 2. Zassenhaus' algebra is the case when 7 consists of a single element, while Jacobson's is the case where ® consists of all functionals with values in the prime field of .We shall call the above algebra 2 a generalized Witt algebra.In order that 2 he finite dimensional it is necessary and