New simple Lie algebras of prime characteristic

Introduction.For a number of years, the only known simple Lie algepras of characteristic p>0 not analogues of the classical algebras of characteristic zero were those of dimension pn of H. Zassenhaus [9], and of dimension np" of N. Jacobson [5], both generalizations of the ^-dimensional algebras of E. Witt.During the last few years, a number of new classes of simple Lie algebras have been determined.These are due to Kaplansky, who in [6] noted the existence of a class of algebras of dimension rpn, 1 gr g w, generalizing those of Zassenhaus and Jacobson; to M. S. Frank, who in [2]obtained algebras ©" of dimension (n -l)(pn -l); and to A. A. Albert and M. S. Frank, who in [l ] determined new classes of simple algebras Xn, %$m, So, and Sj, of dimensions (n -l)pn, p2m -2, pn -l and pn -2 respectively, and a generalization of the Zassenhaus algebras, of dimension pn.In this paper we obtain a new class of simple Lie algebras, which we shall designate by the symbol £(®, 5, /).Here ® is an additive group, of order pn> 1, which is a direct sum of a finite number of finite elementary ^-groups ®oi • • • , ®m-These groups are then finite dimensional vector spaces over the prime field %p.We let m(@) denote the index m, allow ®0 to be zero, and assume that either p>2 or ® ?^®i.For i = l, • • • , m, we take 5< to be a nonzero element of ®<, and write 5=5i+• • • +Sm.We index a basis of 8(®, 5,/) over a field $ of characteristic p by the elements of ® other than 0 and -S, denoting by v(a) the basis element corresponding to a. Furthermore we assume given, for each i, a nondegenerate skew-symmetric biadditive function /,-on (®,-, ®j) to §, such that, for i = l, • ■ • , m, there are additive functions gu hi, on ®i to %, with f,-(5.)=0 and/<(a, 8) = gi(°t)hi(0) -gi(0)h{(a) for every a and 8 in ®<-The definition of the algebra 8(®, S,f) may then be completed by defining multiplication in the algebra by v(a)v(8) = JZ?-ofi(ai, 0d ■v(a+8 -8i), where a< and 8i are the components of a and 0 in ®<, and where 50 and v(0) denote zero.Then we shall prove that 8(®, 5,/) is a central simple Lie algebra.Its dimension is pn -2 except when ® = ®0, in which case the Presented to the Society on September 1, 1955 and April 14, 1956 under the titles Some properties of new simple Lie algebras and New simple Lie algebras of order p"-2; received by the editors March 8, 1957.