There are some simple facts which distinguish Lie-algebras over fields of prime characteristic from Lie-algebras over fields of characteristic zero. These are

(1) The degrees of the absolutely irreducible representations of a Lie-algebra of prime characteristic are bounded whereas, according to a theorem of H. Weyl, the degrees of the absolutely irreducible representations of a semi-simple Lie-algebra over a field of characteristic zero can be arbitrarily high.

(2) For each Lie-algebra of prime characteristic there are indecomposable representations which are not irreducible, whereas every indecomposable representation of a semi-simple Liealgebra over a field of characteristic zero is irreducible (cf. [4]).

(3) The quotient ring of the embedding algebra of a Lie-algebra over a field of prime characteristic is a division algebra of finite dimension over its center, whereas this is not the case for characteristic zero. (cf. [4]).

(4) There are faithful fully reducible representations of every Lie-algebra of prime characteristic, whereas for characteristic zero only ring sums of semi-simple Lie-algebras and abelian Lie-algebras admit faithful fully reducible representations (cf. [6], [2], [4]).

Let μ be a homomorphic mapping of some subgroup A of the group G onto a subgroup Ḃ (not necessarily distinct from A) of G; then we call μ a partial endomorphism of G. If A coincides with G, that is, if the homomorphism is defined on the whole of G, we speak of a total endomorphism; this is what is usually called an endomorphism of G. A partial (or total) endomorphism μ*extends or continues a partial endomorphism μ if the domain of μ* contains the domain of μ, that is, μ* is defined for (at least) all those elements for which μ. is defined, and moreover μ* coincides with μ where μ is defined.

In a previous paper [5] the equivalence of randomisation and normal theory distributions of linear combinations was discussed. In the present paper we discuss the asymptotic randomisation distributions of statistics used in analysis of variance and in a closely related problem which includes, in particular, the “problem of m-rankings“. Kruskal [4] has studied the first of these questions in the case where observations are replaced by ranks.

The two following formulae are to be established.

If R (m ± n) > 0, | amp z | < π,

If p ≧ q + 1, R(k ± n + 2αr) > 0, r = 1, 2, …, p, | amp z | < π,

For other values of p and q the formula holds if the integral is convergent.